From 864d732fc6029a1d264a411ec78d894e3f821288 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:31:56 +0000 Subject: [PATCH 001/112] Initial commit --- README.md | 1 + 1 file changed, 1 insertion(+) create mode 100644 README.md diff --git a/README.md b/README.md new file mode 100644 index 00000000..142fb739 --- /dev/null +++ b/README.md @@ -0,0 +1 @@ +# MathsProblems \ No newline at end of file From db4fac3abe39a983e7a02a53e7688a79fddf544a Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:32:21 +0000 Subject: [PATCH 002/112] Update README.md --- README.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index 142fb739..8946e95a 100644 --- a/README.md +++ b/README.md @@ -1 +1,3 @@ -# MathsProblems \ No newline at end of file +# MathsProblems + +Maths problems for SAMI Maths Club App From ec221a4c23b494f93cd2663fb77f4be50a12b6c1 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:33:01 +0000 Subject: [PATCH 003/112] Create sols --- content/facilitator/sols | 1 + 1 file changed, 1 insertion(+) create mode 100644 content/facilitator/sols diff --git a/content/facilitator/sols b/content/facilitator/sols new file mode 100644 index 00000000..8b137891 --- /dev/null +++ b/content/facilitator/sols @@ -0,0 +1 @@ + From 6f0aa3000ed434488d792ec3f7fe4c842b653f4e Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:33:25 +0000 Subject: [PATCH 004/112] Create student --- content/student | 1 + 1 file changed, 1 insertion(+) create mode 100644 content/student diff --git a/content/student b/content/student new file mode 100644 index 00000000..a85b331e --- /dev/null +++ b/content/student @@ -0,0 +1 @@ +probs From 0dda7a4080e30f0fbdebc56decaf22d02f82455c Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:34:12 +0000 Subject: [PATCH 005/112] Delete student --- content/student | 1 - 1 file changed, 1 deletion(-) delete mode 100644 content/student diff --git a/content/student b/content/student deleted file mode 100644 index a85b331e..00000000 --- a/content/student +++ /dev/null @@ -1 +0,0 @@ -probs From 0f6745c27d0683b3e956867c8053e3f2537047cd Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:34:33 +0000 Subject: [PATCH 006/112] Create prob --- content/student/prob | 1 + 1 file changed, 1 insertion(+) create mode 100644 content/student/prob diff --git a/content/student/prob b/content/student/prob new file mode 100644 index 00000000..8b137891 --- /dev/null +++ b/content/student/prob @@ -0,0 +1 @@ + From 870535ea439cbd5d2ca0879f37c252a031eba6a8 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:35:01 +0000 Subject: [PATCH 007/112] Create img --- cover_images/img | 1 + 1 file changed, 1 insertion(+) create mode 100644 cover_images/img diff --git a/cover_images/img b/cover_images/img new file mode 100644 index 00000000..8b137891 --- /dev/null +++ b/cover_images/img @@ -0,0 +1 @@ + From 40f6f311f7073f808e8f95ec26befff1553b9b96 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:35:34 +0000 Subject: [PATCH 008/112] Create img --- images/img | 1 + 1 file changed, 1 insertion(+) create mode 100644 images/img diff --git a/images/img b/images/img new file mode 100644 index 00000000..8b137891 --- /dev/null +++ b/images/img @@ -0,0 +1 @@ + From 9a99d6da8677c1e735dc164ccb6d7cbdc5af145e Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:37:03 +0000 Subject: [PATCH 009/112] Delete prob --- content/student/prob | 1 - 1 file changed, 1 deletion(-) delete mode 100644 content/student/prob diff --git a/content/student/prob b/content/student/prob deleted file mode 100644 index 8b137891..00000000 --- a/content/student/prob +++ /dev/null @@ -1 +0,0 @@ - From 6d47234d0600974ba78dce6429857c791d71de8d Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:38:30 +0000 Subject: [PATCH 010/112] Create diffy-squares --- content/student/diffy-squares | 37 +++++++++++++++++++++++++++++++++++ 1 file changed, 37 insertions(+) create mode 100644 content/student/diffy-squares diff --git a/content/student/diffy-squares b/content/student/diffy-squares new file mode 100644 index 00000000..58744776 --- /dev/null +++ b/content/student/diffy-squares @@ -0,0 +1,37 @@ +--- +title: Diffy Squares +type: problem +printOrder: n/a +added: 2021-01 +--- + +# Diffy Squares + +Create a Diffy Square by following this process: + +1. Draw a square and choose four numbers to place on each vertex. + +2. Along each edge, write the positive difference of the numbers at the vertices that the edge connects. Join these four numbers up to form a new square. + +3. Now repeat this process with the new square. Find the positive difference between each pair of numbers which share an edge, write these differences on the edge and connect up the new numbers. + +4. At some point, you will reach a square with four zeros on the edges. At this point, you should stop. + +5. That's it. You should now count the number of squares which you drew, not including the largest square on the outside. This is the *length* of the sequence of squares. + +Here is an example, starting with the numbers 7, 15, 9 and 13: + + + +We drew two smaller squares, so this sequence has length 2. + +Try following this process with different starting numbers. + +What is the longest sequence of squares that you can find? + + +Tip: For longer sequences, it may be easier to write your sequences as a column of numbers, where each row represents a square. + +For our example of length 2, we could have written: + + From 533023164b59c2cb3ffedd961948a4731938ba04 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:39:04 +0000 Subject: [PATCH 011/112] Create diffy-squares --- content/facilitator/diffy-squares | 103 ++++++++++++++++++++++++++++++ 1 file changed, 103 insertions(+) create mode 100644 content/facilitator/diffy-squares diff --git a/content/facilitator/diffy-squares b/content/facilitator/diffy-squares new file mode 100644 index 00000000..0446835e --- /dev/null +++ b/content/facilitator/diffy-squares @@ -0,0 +1,103 @@ +# Diffy Squares + +## Introduction + +This problem, at the simplest level, is a great way to practice subtraction and, for higher level students, can reveal some interesting mathematics. + +With regards to the question asked, it is quite difficult to find particularly long seqences and not at all obvious when they will occur. In the solutions section, you fill find an example of the longest sequence possible under certain assumptions for the starting numbers. However, allowing any starting numbers to be used, this challenge can be as difficult as you want it to be. + +It is technically possible to choose four numbers that make an infinite sequence where you never end up at $0, 0, 0, 0$. However, demonstrating this is tricky. [Here](https://mathforlove.com/2011/10/squares-of-difference-iii-a-surprising-solution/) is a link to a blog discussing the problem. + +## Solution + +Using just single-digit integers, the longest sequence length possible is 7. +Here is an example of such a sequence (using the column notation): + +$0, 1, 4, 9$ +$1, 3, 5, 9$ +$2, 2, 4, 8$ +$0, 2, 4, 6$ +$2, 2, 2, 6$ +$0, 0, 4, 4$ +$0, 4, 0, 4$ +$4, 4, 4, 4$ +$0, 0, 0, 0$ + +It is interesting to point out that the same four starting numbers will not always give the same length of sequence. Taking $0, 4, 1, 9$ as the starting line for example: + +$0, 4, 1, 9$ +$4, 3, 8, 9$ +$1, 5, 1, 5$ +$4, 4, 4, 4$ +$0, 0, 0, 0$ + +We managed to make this sequence quite a bit shorter just by swapping the position of the $1$ and the $4$. You might like to think about how many different lengths of sequence it is possible to get from the same four starting numbers (see Extention section for solution). + +If we allow any integers less that 100 to be used, the longest sequence length possible is 12. +Here is an example of such a sequence: + +$0, 7, 20, 44$ +$7, 13, 24, 44$ +$6, 11, 20, 37$ +$5, 9, 17, 31$ +$4, 8, 14, 26$ +$4, 6, 12, 22$ +$2, 6, 10, 18$ +$4, 4, 8, 16$ +$0, 4, 8, 12$ +$4, 4, 4, 12$ +$0, 0, 8, 8$ +$0, 8, 0, 0$ +$8, 8, 8, 8$ +$0, 0, 0, 0$ + +Introducing fractions is an effective way to increase the length of sequences which we can obtain. Using fractions we can find sequences with numbers less that 10 which are longer than the longest sequence possible using integers up to 100. +Here is an example: + +$0, \frac{17}{12}, 4, \frac{35}{4}$ +$\frac{17}{12}, \frac{31}{12}, \frac{19}{4}, \frac{35}{4}$ +$\frac{7}{6}, \frac{13}{6}, 4, \frac{22}{3}$ +$1, \frac{11}{6}, \frac{10}{3}, \frac{37}{6}$ +$\frac{5}{6}, \frac{3}{2}, \frac{17}{6}, \frac{31}{6}$ +$\frac{2}{3}, \frac{4}{3}, \frac{7}{3}, \frac{13}{3}$ +$\frac{1}{3}, 1, 2, \frac{11}{3}$ +$\frac{1}{3}, 1, \frac{5}{3}, 3$ +$\frac{2}{3}, \frac{2}{3}, \frac{4}{3}, \frac{8}{3}$ +$0, \frac{2}{3}, \frac{4}{3}, 2$ +$\frac{2}{3}, \frac{2}{3}, \frac{2}{3}, 2$ +$0, 0, \frac{4}{3}, \frac{4}{3}$ +$0, \frac{4}{3}, 0, \frac{4}{3}$ +$\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}$ +$0, 0, 0, 0$ + +(In particular, 13 is the longest a sequence can be using numbers which are integer multiples of $\frac{1}{12}$ and less than 10, above is such an example) + +## Extension + +How many sequences of different length are possible from the same four starting numbers? + +If we take a general starting square, using the numbers $a, b, c, d$. + +Say that if we start with line $(a, b, c, d)$, this means that $a$ is in the top-left corner, $b$ is in the top-right, $c$ the bottom-right and $d$ the bottom-left. + +By rotating the square, we will not change the length of the sequence generated, and we can see therefore that the starting lines $(b, c, d, a)$, $(c, d, a, b)$ and $(d, a, b, c)$ will give sequences of the same length as that given by $(a, b, c, d)$. + + + +We could also reflect the square in any of its four lines of symmetry without changing the length of the resulting square. So the starting lines $(a, d, c, b)$, $(d, c, b, a)$, $(c, b, a, d)$ and $(b, a, d, c)$ will all give sequences of the same length as $(a, b, c, d)$. + + + +So for each starting line $(a, b, c, d)$ there are 8 lines in total using the numbers $a$, $b$, $c$ and $d$ which give sequences of the same length. + +But how many starting lines are possible in total from the numbers $a$, $b$, $c$ and $d$? + +If we choose the number in the top-left corner first, we have four choices $a$, $b$, $c$ or $d$. Next, if we choose the number in the top-right we have a choice of the three remaining number. Likewise, for the bottom-right, we have a choice from two and the bottom-left must be the one remaining number. + +So, the total number of possible lines is $4 \times 3 \times 2 \times 1 = 24$ + +But since we know that these lines come in groups of 8, all of which have the same length. The largest number of different lengths possible must be $\frac{24}{3}$ which is 3. + +Another way to see this would be to fix the position of $a$ to the top-left, which we can do since rotation does not change the length of the sequence as we have seen. We can then ask, of the remaining numbers $b$, $c$ and $d$, how many different ways can we choose the two numbers which $a$ will share an edge with? + +The answer is 3. Either $b$ and $c$, $b$ and $d$ or $c$ and $d$. From 7b686d5c7ff0c9168faaacbe74f252a77c497358 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:39:11 +0000 Subject: [PATCH 012/112] Delete sols --- content/facilitator/sols | 1 - 1 file changed, 1 deletion(-) delete mode 100644 content/facilitator/sols diff --git a/content/facilitator/sols b/content/facilitator/sols deleted file mode 100644 index 8b137891..00000000 --- a/content/facilitator/sols +++ /dev/null @@ -1 +0,0 @@ - From 4c53def4c6dad38081d5e3e4e0945572332a002f Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:42:54 +0000 Subject: [PATCH 013/112] Add files via upload --- cover_images/DiffySquareColumnForm.png | Bin 0 -> 9220 bytes cover_images/DiffySquareExample.png | Bin 0 -> 14578 bytes cover_images/DiffySquareReflection.png | Bin 0 -> 43113 bytes cover_images/DiffySquareRotation.png | Bin 0 -> 41510 bytes 4 files changed, 0 insertions(+), 0 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file changed, 1 deletion(-) delete mode 100644 images/img diff --git a/images/img b/images/img deleted file mode 100644 index 8b137891..00000000 --- a/images/img +++ /dev/null @@ -1 +0,0 @@ - From ea4b69c559117a87462031e79e88a701b105d82d Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:45:26 +0000 Subject: [PATCH 016/112] Add files via upload --- images/diffy-squares-1.png | Bin 0 -> 14578 bytes images/diffy-squares-2.png | Bin 0 -> 9220 bytes images/diffy-squares-3.png | Bin 0 -> 41510 bytes images/diffy-squares-4.png | Bin 0 -> 43113 bytes 4 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 images/diffy-squares-1.png create mode 100644 images/diffy-squares-2.png create mode 100644 images/diffy-squares-3.png create mode 100644 images/diffy-squares-4.png diff --git a/images/diffy-squares-1.png b/images/diffy-squares-1.png new file mode 100644 index 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za8B~m$m(!%IdM7BjFr3j(=}${fB0cJK|;~DV*l?>(k`(o{Rtu z#-SLII%(@uPI(Veo(JcTkUH$j_2}a_-X!&R0`VrDVDD9FtSH5`nTg56z?9JHGro$IgB3+Jy%;}G1v`@4DRMJPs29IGx z>c?wI>T*-eTN~ySwjgR;_c4ul0#?dy{G>_U?uya9_(h5{O=aEm>z5v1?_rtFsVbi{ z^gEq%iL?C2sE2)~lS`0)FF_=pr{nj9GOPCxx?XUcP2 zkC)ut;wMIb=9JEGtP}Z=Y#MS)PZxiWQ5`$K^sJ49#yuIltFT`qUCu***)z@f_)gX> zMbhYFJ-&6ZKD(vEHE`@kz_85WuDOaqjXtXjlJg}yIKITqVVCS`4r{n>1A)&yd|taQ$83!5-s%+IGT@NL*$A_?H1Z6q5V zxbYgE2nU%ib4Z_Q_8aVDPs6SMea+T;vmTW|K{&AyP- Date: Mon, 25 Jan 2021 14:46:49 +0000 Subject: [PATCH 021/112] Update diffy-squares --- content/facilitator/diffy-squares | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/facilitator/diffy-squares b/content/facilitator/diffy-squares index 0446835e..7d882502 100644 --- a/content/facilitator/diffy-squares +++ b/content/facilitator/diffy-squares @@ -82,11 +82,11 @@ Say that if we start with line $(a, b, c, d)$, this means that $a$ is in the top By rotating the square, we will not change the length of the sequence generated, and we can see therefore that the starting lines $(b, c, d, a)$, $(c, d, a, b)$ and $(d, a, b, c)$ will give sequences of the same length as that given by $(a, b, c, d)$. - + We could also reflect the square in any of its four lines of symmetry without changing the length of the resulting square. So the starting lines $(a, d, c, b)$, $(d, c, b, a)$, $(c, b, a, d)$ and $(b, a, d, c)$ will all give sequences of the same length as $(a, b, c, d)$. - + So for each starting line $(a, b, c, d)$ there are 8 lines in total using the numbers $a$, $b$, $c$ and $d$ which give sequences of the same length. From a882297d9a2066d605feac6e9dddb79e63110648 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:51:38 +0000 Subject: [PATCH 022/112] Rename diffy-squares to diffy-squares.md --- content/facilitator/{diffy-squares => diffy-squares.md} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename content/facilitator/{diffy-squares => diffy-squares.md} (100%) diff --git a/content/facilitator/diffy-squares b/content/facilitator/diffy-squares.md similarity index 100% rename from content/facilitator/diffy-squares rename to content/facilitator/diffy-squares.md From 811fd5661e0073127f539ee291adb7dd52ce93d9 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:53:10 +0000 Subject: [PATCH 023/112] Update and rename diffy-squares to diffy-squares.md --- content/student/{diffy-squares => diffy-squares.md} | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) rename content/student/{diffy-squares => diffy-squares.md} (91%) diff --git a/content/student/diffy-squares b/content/student/diffy-squares.md similarity index 91% rename from content/student/diffy-squares rename to content/student/diffy-squares.md index 58744776..9b5fdd8e 100644 --- a/content/student/diffy-squares +++ b/content/student/diffy-squares.md @@ -21,7 +21,7 @@ Create a Diffy Square by following this process: Here is an example, starting with the numbers 7, 15, 9 and 13: - + We drew two smaller squares, so this sequence has length 2. @@ -34,4 +34,4 @@ Tip: For longer sequences, it may be easier to write your sequences as a column For our example of length 2, we could have written: - + From 2f7ec68ae1438b44412827765bf3ee5463616e47 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:53:37 +0000 Subject: [PATCH 024/112] Update diffy-squares.md --- content/facilitator/diffy-squares.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/facilitator/diffy-squares.md b/content/facilitator/diffy-squares.md index 7d882502..29e3f12e 100644 --- a/content/facilitator/diffy-squares.md +++ b/content/facilitator/diffy-squares.md @@ -82,11 +82,11 @@ Say that if we start with line $(a, b, c, d)$, this means that $a$ is in the top By rotating the square, we will not change the length of the sequence generated, and we can see therefore that the starting lines $(b, c, d, a)$, $(c, d, a, b)$ and $(d, a, b, c)$ will give sequences of the same length as that given by $(a, b, c, d)$. - + We could also reflect the square in any of its four lines of symmetry without changing the length of the resulting square. So the starting lines $(a, d, c, b)$, $(d, c, b, a)$, $(c, b, a, d)$ and $(b, a, d, c)$ will all give sequences of the same length as $(a, b, c, d)$. - + So for each starting line $(a, b, c, d)$ there are 8 lines in total using the numbers $a$, $b$, $c$ and $d$ which give sequences of the same length. From 60e5abebda7c7a44e22b9018e8139207ffbfdefd Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:55:27 +0000 Subject: [PATCH 025/112] Create kaprekars-number --- content/facilitator/kaprekars-number | 54 ++++++++++++++++++++++++++++ 1 file changed, 54 insertions(+) create mode 100644 content/facilitator/kaprekars-number diff --git a/content/facilitator/kaprekars-number b/content/facilitator/kaprekars-number new file mode 100644 index 00000000..29820dd7 --- /dev/null +++ b/content/facilitator/kaprekars-number @@ -0,0 +1,54 @@ +# Kaprekar's Number + +## Introduction + +This problem is a good way to practice subtraction, it also demonstrates an interesting result. + +After trying a few examples, it becomes quite apparent that all four-digit numbers seem to lead to the same number under repeated application of this opperation, namely 6174 or *Kaprekar's Number*. + +Students will notice a similar pattern for three-digit numbers, with the resulting number being 495. They may also be able to demonstrate that this will happen for any three-digit number (see Extension section), although this is a difficult task for the four-digit problem. + +## Solution + +Starting with 4252, as in the question, and applying the operation defined in the question (known as the *Kaprekar Mapping*) the process would be as follows: + +$5422 - 2245 = 3177$ +$7731 - 1377 = 6354$ +$6543 - 3456 = 3087$ +$8730 - 0378 = 8352$ +$8532 - 2358 = 6174$ +$7641 - 1467 = 6174$ +... + +So we can see that the sequence eventually reaches 6174 at which point, if we continued, it would repeat 6174 indefinately. + +## Extension + +For three-digit numbers, we can demonstrate that repeated application of the Kaprekar mapping will always lead to 495. + +Consider a general three-digit number $*abc* = 100a + 10b + c$ +where $9 \geq a \geq b \geq c \geq 0$ + +Applying the Kaprekar mapping gives the second number in our sequence: + +$(100a + 10b + c) - (100c + 10b + a) = 100(a - c) + (c - a)$ + +We can simplify this to: $99(a - c)$ + +So, after one application of the Kaprekar mapping, the only possible results are the multiples of 99: + +099, 198, 297, 396, 495, 594, 693, 792, 891, 990 + +In fact, each of these numbers uses the same digits as exactly one other, and will give the same result under the Kaprekar mapping. So, we only need to consider how half of them will be changed by a second application. + +$990 - 099 = 891$ +$981 - 189 = 792$ +$972 - 279 = 693$ +$963 - 369 = 594$ +$954 - 459 = 495$ + +We could express this information as a flow diagram: + + + +After one application of the Kaprekar mapping, the sequence will be one of the numbers on the flow diagram and then will follow it to reach 495. In total, any number will reach 495 in a maximum of 5 applications of the Kaprekar mapping. For example, 545 will require 5 applications. From 4ef680a07ce6089fb98d5e07e22d1cc361bfcf58 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:55:40 +0000 Subject: [PATCH 026/112] Rename kaprekars-number to kaprekars-number.md --- content/facilitator/{kaprekars-number => kaprekars-number.md} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename content/facilitator/{kaprekars-number => kaprekars-number.md} (100%) diff --git a/content/facilitator/kaprekars-number b/content/facilitator/kaprekars-number.md similarity index 100% rename from content/facilitator/kaprekars-number rename to content/facilitator/kaprekars-number.md From de62b5069df6fad9bd1acfd0d6da56c74f49a27e Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 14:56:17 +0000 Subject: [PATCH 027/112] Create kaprekars-number.md --- content/student/kaprekars-number.md | 26 ++++++++++++++++++++++++++ 1 file changed, 26 insertions(+) create mode 100644 content/student/kaprekars-number.md diff --git a/content/student/kaprekars-number.md b/content/student/kaprekars-number.md new file mode 100644 index 00000000..6cd779ab --- /dev/null +++ b/content/student/kaprekars-number.md @@ -0,0 +1,26 @@ +--- +title: Kaprekar's Number +type: problem +printOrder: n/a +added: 2021-01 +--- + +# Kaprekar's Number + +Choose a four digit number where the digits are not all the same (so not 1111, 2222, ...). + +Rearrange the digits to get the largest and smallest numbers that these digits can make. + +Subtract the smallest number from the largest to get a new number, and carry on repeating this operation for each new number. + +For example, starting with 4252: + +5422 - 2245 = 3177 +7731 - 1377 = 6354 +6543 - 3456 = 3087 +... + +What do you notice as you continue this process? +What happens when you use different starting numbers? + +Does something similar happen for three digit numbers? 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For most of the +shapes, there is a clear starting point where a certain space can only be filled by one of the +blocks, and a solution can then be logically worked through. + +It should be noted that the choice of blocks is arbitrary and only done in reference to the +game ‘Tetris’. The essence of this puzzle can be maintained using blocks of smaller or larger +size, depending on the age and ability of students. + +## Solution + +The shape which cannot be made by fitting together the blocks given is the Star. It may be +interesting for students to try and reason this logically (see Extension). + +The other four solutions, for the Tree, Boat, Spaceship, and Flower, are as follows: + + + +Note that there may be other solutions not given here, any which uses each block once is equally valid. + +## Extension + +In the case of the star, we can reason as to why is cannot be created from the blocks given: + + + +Firstly, looking at the bottom two points, there is no way to fill this space other than to use +the **S** and **Z** blocks. + +If we fill these bottom points, we can consider the remaining three, which have a similar +three-square line structure. We cannot fit the **O** or **T** blocks in these spaces, so we need to +use the **I**, **J** and **L** blocks to fill these three points. + +We could attempt to do this in a few ways: + +Firstly, we could fill the top point with the **I** block and use the **J** and **L** blocks in the two side points (left on the diagram). At this point, we will not be able to fit the **O** block in the remaining space, so this way will not work. + +If we instead fill the top point with either the **J** or **L** block and use the remaining one to fill the side points, along with the **I** block, we will get a situation like on the right of the +diagram. Notice that, had we used the **L** block at the top, rather that the **J**, this would have +given a situation which is a reflection of the case shown in the diagram. In either case, there +is only one space where we can fit the **O** block and after this the remaining space will not fit +the **T** block, so this way will also not work. + +Therefore, we have exhausted all possibilities and can conclude that the star shape is +impossible to make from the blocks given. + + From 36c71113cae0c6111fee983436651d3f57ae862c Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 15:01:01 +0000 Subject: [PATCH 031/112] Create tetris-shapes.md --- content/student/tetris-shapes.md | 23 +++++++++++++++++++++++ 1 file changed, 23 insertions(+) create mode 100644 content/student/tetris-shapes.md diff --git a/content/student/tetris-shapes.md b/content/student/tetris-shapes.md new file mode 100644 index 00000000..1ddce730 --- /dev/null +++ b/content/student/tetris-shapes.md @@ -0,0 +1,23 @@ +--- +title: Tetris Shapes +type: problem/game +printOrder: n/a +added: 2021-01 +--- + +# Tetris Shapes + +Using these seven blocks (from the arcade game *Tetris*), can you create the shapes at the bottom of this page? + + + +You are allowed to rotate the blocks but not to reflect them, so you cannot change the 'L-block' into a 'J-block' by flipping it over. + +Four of these five shapes can be created using exactly one of each block. +(left to right: Tree, Boat, Spaceship, Star, Flower) + + + +Can you spot which one is not possible? + +What other shapes can you make using these blocks? From 7a4c68363345a1cde5b99dc1f35df2dbdeb489d1 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 15:01:32 +0000 Subject: [PATCH 032/112] Update tetris-shapes.md --- content/facilitator/tetris-shapes.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/facilitator/tetris-shapes.md b/content/facilitator/tetris-shapes.md index f2b0ccfa..cf471038 100644 --- a/content/facilitator/tetris-shapes.md +++ b/content/facilitator/tetris-shapes.md @@ -17,7 +17,7 @@ interesting for students to try and reason this logically (see Extension). The other four solutions, for the Tree, Boat, Spaceship, and Flower, are as follows: - + Note that there may be other solutions not given here, any which uses each block once is equally valid. 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diff --git a/content/facilitator/konigsberg-number.md b/content/facilitator/konigsberg-number.md new file mode 100644 index 00000000..64ee2df2 --- /dev/null +++ b/content/facilitator/konigsberg-number.md @@ -0,0 +1,41 @@ +# Bridges of Konigsberg + +## Introduction + +This is a classic mathematics problem. Its resolution in 1736 by Leonhard Euler is considered a foundation of the mathematical field of graph theory. + +Without knowing the problem, it is initially quite enjoyable to try any find a solution. However, there is in fact no way to cross every bridge once, as Euler showed (see Solution section). + +You could say that the ‘solution’ to this problem is to wait for the second world war, where two of the bridges were destroyed, making it possible to find a route crossing each remaining bridge once. + +## Solution + +As mentioned, there is no solution to the problem as it is outlined in the question. To demonstrate this, it is helpful to reduce the situation to a mathematical structure called a *graph* (a different type of graph to what you are probably used to). + + + +We have replaced the sections of land with circles and the bridges with connecting lines. We usually refer to these circles as *vertices*, and the lines as *edges*. + +Say we start our walk on the middle vertex and walk to the top one, and then to the vertex on the right. Our path might look like this: + + + +Notice how, since we have walked both to and from the top vertex, we have used two of the edges connected to it. Similarly, once we leave the rightmost vertex, we will use another edge and so will have removed two of its connected edges: + + + +We can see that, as we walk, we always use pairs of edges connected to a particular vertex. The only time that this is not true is for the vertices that we begin and end on. + +Because of this, after walking we will always have used an even number of edges connected to a given vertex (unless they are one that we start or end on). So, if we want to remove all the edges, the vertices in the graph must have an even number of edges connected to them. The only two vertices that can have an odd number of edges are the starting and ending ones. + +This was Euler's discovery, that to be able to pass through all the edges in a graph exactly once, there can be no more than two vertices which have an odd number of edges connected to them. + +In the case of Konigsberg, three of the vertices have 3 edges and one (the central island) has 5. Since all four of the vertices have an odd number of edges, it is impossible to find a path which passes over each edge exactly once. + +## Extension + +Test this result by creating other graphs and trying to find a route passing over each edge once, so-called *Eulerian paths*. See if you can predict whether a given graph will have a Eulerian path. + +Where a Eulerian path is possible, do you notice anything else about the number of vertices which can have an odd number of edges? Try finding a path for a graph which only has one vertex with an odd number of edges, if you cannot, can you see why? + +*Hint: we said that only the starting and ending vertices can have an odd number of connected edges, what happens in the two possible cases where these are different vertices and where these are the same vertex?* From 59adbefa28f259a026a6bf15c7ecdca1cac70f3e Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 15:05:10 +0000 Subject: [PATCH 035/112] Update konigsberg-number.md --- content/facilitator/konigsberg-number.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/content/facilitator/konigsberg-number.md b/content/facilitator/konigsberg-number.md index 64ee2df2..3cc8a384 100644 --- a/content/facilitator/konigsberg-number.md +++ b/content/facilitator/konigsberg-number.md @@ -12,17 +12,17 @@ You could say that the ‘solution’ to this problem is to wait for the second As mentioned, there is no solution to the problem as it is outlined in the question. To demonstrate this, it is helpful to reduce the situation to a mathematical structure called a *graph* (a different type of graph to what you are probably used to). - + We have replaced the sections of land with circles and the bridges with connecting lines. We usually refer to these circles as *vertices*, and the lines as *edges*. Say we start our walk on the middle vertex and walk to the top one, and then to the vertex on the right. Our path might look like this: - + Notice how, since we have walked both to and from the top vertex, we have used two of the edges connected to it. Similarly, once we leave the rightmost vertex, we will use another edge and so will have removed two of its connected edges: - + We can see that, as we walk, we always use pairs of edges connected to a particular vertex. The only time that this is not true is for the vertices that we begin and end on. From bd8e15bd0064384529ba862c4509771a0028335d Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 15:06:04 +0000 Subject: [PATCH 036/112] Create konigsberg-bridges.md --- content/student/konigsberg-bridges.md | 20 ++++++++++++++++++++ 1 file changed, 20 insertions(+) create mode 100644 content/student/konigsberg-bridges.md diff --git a/content/student/konigsberg-bridges.md b/content/student/konigsberg-bridges.md new file mode 100644 index 00000000..ceef4218 --- /dev/null +++ b/content/student/konigsberg-bridges.md @@ -0,0 +1,20 @@ +--- +title: Bridges of Konigsberg +type: problem +printOrder: n/a +added: 2021-01 +--- + +# Bridges of Konigsberg + +The Prussian city of Konigsberg (now Kaliningrad, Russia) is divided into four landmasses by the river Pregel. + + + +You can see that there are seven bridges connecting these sections of the city. + +Can you find a way to walk through the city crossing each bridge exactly once? + +Do you think that it is possible? + +Assume that there are no other ways to cross the river than by walking over one of these bridges. You may begin and end anywhere in the city. From 7ec24f8053d242b80e09f60b99fd5ec7c9c357c6 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 15:06:20 +0000 Subject: [PATCH 037/112] Update konigsberg-bridges.md --- content/student/konigsberg-bridges.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/student/konigsberg-bridges.md b/content/student/konigsberg-bridges.md index ceef4218..bc9f7b35 100644 --- a/content/student/konigsberg-bridges.md +++ b/content/student/konigsberg-bridges.md @@ -9,7 +9,7 @@ added: 2021-01 The Prussian city of Konigsberg (now Kaliningrad, Russia) is divided into four landmasses by the river Pregel. - + You can see that there are seven bridges connecting these sections of the city. From fb4558ff08ba53ff102df8b730fba33eb3278edc Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 15:06:36 +0000 Subject: [PATCH 038/112] Update konigsberg-bridges.md --- content/student/konigsberg-bridges.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/student/konigsberg-bridges.md b/content/student/konigsberg-bridges.md index bc9f7b35..bcff2f84 100644 --- a/content/student/konigsberg-bridges.md +++ b/content/student/konigsberg-bridges.md @@ -9,7 +9,7 @@ added: 2021-01 The Prussian city of Konigsberg (now Kaliningrad, Russia) is divided into four landmasses by the river Pregel. - + You can see that there are seven bridges connecting these sections of the city. From 65ce42c207ed5c81991e69cbada765c3fec3055c Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 15:10:00 +0000 Subject: [PATCH 039/112] Add files via upload --- cover_images/konigsberg-bridges.svg | 81 +++++++++++++++++++++++++++++ 1 file changed, 81 insertions(+) create mode 100644 cover_images/konigsberg-bridges.svg diff --git a/cover_images/konigsberg-bridges.svg b/cover_images/konigsberg-bridges.svg new file mode 100644 index 00000000..90760f8f --- /dev/null +++ b/cover_images/konigsberg-bridges.svg @@ -0,0 +1,81 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From c6fca6f3d2b1e8a9632c189379c1ceb8d20c6f59 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 15:10:14 +0000 Subject: [PATCH 040/112] Delete img --- cover_images/img | 1 - 1 file changed, 1 deletion(-) delete mode 100644 cover_images/img diff --git 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cover_images/kaprekars-number.svg create mode 100644 cover_images/tetris-shapes.svg diff --git a/cover_images/diffy-squares.svg b/cover_images/diffy-squares.svg new file mode 100644 index 00000000..dfc97f51 --- /dev/null +++ b/cover_images/diffy-squares.svg @@ -0,0 +1,95 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/cover_images/kaprekars-number.svg b/cover_images/kaprekars-number.svg new file mode 100644 index 00000000..e54dc601 --- /dev/null +++ b/cover_images/kaprekars-number.svg @@ -0,0 +1 @@ + \ No newline at end of file diff --git a/cover_images/tetris-shapes.svg b/cover_images/tetris-shapes.svg new file mode 100644 index 00000000..54d080a3 --- /dev/null +++ b/cover_images/tetris-shapes.svg @@ -0,0 +1,46 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From 96a952b7382362e82ee5355863d59e46ca6b96b6 Mon Sep 17 00:00:00 2001 From: dankelly01 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In fact, we will se that it cannot be done in any fewer. + +The majority of the map can be coloured using just two colours, take blue and green: + + + +At this point, we wish to colour the region in the top left corner. Since it shares edges with both blue and green regions, we must introduce a third colour, say red. This is also true for the region in the bottom right corner. + + + +Now we want to colour the star-shaped region. However, the star shares edges will a blue, a green and a red region so we must introduce a fourth colour. Yellow seems appropriate. + + + +And so, we have coloured this map using exactly four colours. Clearly, we could have used more colours (we could, for example, use a different colour for each of the 29 section). But, having seen that we were required to introduce four different colours, we could not have used less. + +## Extension + +We can ask the question, is there a smallest number of colours which will be able to colour in any given map? The answer is yes, and it is 4. Students may notice this pattern from trying lots of different maps, however showing it conclusively requires us to use graph theory. + +Specifically, we are going to translate the maps into mathematical *graphs* (likely not the type of +graph that you are used to). To do this we will: +- Replace the different regions on the map with circles +- Connect up the circles representing regions which share edges using lines. + +(We usually refer to these circles as *vertices*, and the lines as *edges*. This is somewhat confusing, but "edges" is used with two different meanings in this question. It can refer to the boundaries of a region, or a connection between vertices.) + +Let us take a portion of the example map and turn it into a graph. For the part in the bottom-right corner: + + + +Notice how our previous rule, that regions which share an edge cannot be the same colour, is maintained in the graph as no two connected vertices can be of the same colour. + +Also notice that the graph does not have any intersecting edges. In fact, any graph which represents a valid map can be drawn without any edges intersecting. + +Knowing these two things will allow us to prove that we can always colour a map using four colours or less. Let us try to contrsuct a graph which needs a minimum of five colours to fill. To do this, we need to contruct a graph of five vertices where each is connected to all the others and none of the edges cross. + +From the example map, we can make a graphy with four vertices where all are connected to each other and none of the lines intersect: + + + +If we now try to add a fifth vertex, there are 4 areas where we could try to do so: +- In any of the three small triangles formed by connecting the Yellow vertex with two others. +- Outside the large trianle formed by the Red, Green and Blue vertices. + + + +However, in each case, we can only connect up the additional vertex to three others without any edges crossing. This means that they do not, in fact, have to take a new colour and we can still colour the graph with four colours. + + + +So, we have shown that we cannot construct a graph requiring five different colours to fill. + +Equivalently, we cannot draw a map which needs different five colours to fill. We can fill any map with a maximum of different four colours. From 14709b9d594482b956bdd8f032d9f8ffe29a2952 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 21:55:39 +0000 Subject: [PATCH 048/112] Create colouring-maps.md --- content/student/colouring-maps.md | 22 ++++++++++++++++++++++ 1 file changed, 22 insertions(+) create mode 100644 content/student/colouring-maps.md diff --git a/content/student/colouring-maps.md b/content/student/colouring-maps.md new file mode 100644 index 00000000..b87bea62 --- /dev/null +++ b/content/student/colouring-maps.md @@ -0,0 +1,22 @@ +--- +title: Colouring Maps +type: problem +printOrder: n/a +added: 2021-01 +--- + +# Colouring Maps + +Start by drawing an empty square: + + + +Create a *map* by dividing the square into seperate regions. For example: + + + +Now colour in your map such that there are no regions of the same colour which share an edge. + +Try to find the smallest number of different colours which you need to be able to do this. + +What about other maps, can you find a certain number of colours which are able to colour in any map? From 4d0dd4ca95ca4b04356992f2ef4074a2b1a4ad64 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 21:58:24 +0000 Subject: [PATCH 049/112] Update colouring-maps.md --- content/student/colouring-maps.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/student/colouring-maps.md b/content/student/colouring-maps.md index b87bea62..d3daf3ca 100644 --- a/content/student/colouring-maps.md +++ b/content/student/colouring-maps.md @@ -9,11 +9,11 @@ added: 2021-01 Start by drawing an empty square: - + Create a *map* by dividing the square into seperate regions. For example: - + Now colour in your map such that there are no regions of the same colour which share an edge. 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From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:14:32 +0000 Subject: [PATCH 051/112] Create hanoi-tower.md --- content/facilitator/hanoi-tower.md | 46 ++++++++++++++++++++++++++++++ 1 file changed, 46 insertions(+) create mode 100644 content/facilitator/hanoi-tower.md diff --git a/content/facilitator/hanoi-tower.md b/content/facilitator/hanoi-tower.md new file mode 100644 index 00000000..cbd0388d --- /dev/null +++ b/content/facilitator/hanoi-tower.md @@ -0,0 +1,46 @@ +# Tower of Hanoi + +## Introduction + +This game is a classic recreational maths puzzle. It is, in fact, possible to move the tower abiding by the rules for any number of blocks (see Extension section). + +For students to try out the problem, stacked blocks are likely the most readily available objects. Often, the problem is shown using three pegs and different sized disks with holes in the middle which can be slid over the pegs. Alternatively, the puzzle can be drawn out at each stage to represent the moves made (as in the Solution section). + +## Solution + +For three blocks, the tower can be moved to the rightmost platform in a minimum of 7 moves. The process is as follows: + + + +Notice how, after the fourth move, we make very similar moves to 1, 2 and 3 but in reverse order and reflected horizontally. So, instead of dismantling a tower on the left (as in 1, 2 and 3), we are rebuilding the tower on the right. + +Using this may help you to work out how to move a tower of four or more blocks. + +## Extension + +For four blocks, the tower can be moved to the rightmost platform in a minimum of 15 moves: + + + +You may notice a pattern emerging. If we continue adding more blocks, the minimum number of +moves required is as follows: + + + +You may notice that for some number of blocks $n$, if the minimum number of moves is $m$. Then the +minimum number of moves for $n + 1$ blocks is $2m + 1$. +ie. $15 = 2 \times 7 + 1$, $31 = 2 \times 15 + 1$, etc. + +Thinking about why this is, in the case of four blocks: +- The first stage is to move the top three blocks to the central position, which takes 7 moves. +- Next, the largest block is moved to the rightmost position, taking 1 move. +- Finally, the top three blocks are moved from the central position to the rightmost position, taking 7 moves. + +So overall this takes $7 + 1 + 7 = 2 × 7 + 1 = 15$ moves. + +Generally, in the case of $n + 1$ blocks: +- The first stage is to move the top $n$ blocks to the central position, which takes $m$ moves. +- Next, the largest block is moved to the rightmost position, taking 1 move. +- Finally, the top $n$ blocks are moved from the central position to the rightmost position, taking $m$ moves. + +So overall this takes $m + 1 + m = 2m + 1$ moves From 4bb47ee3dc7181cb1ba6cdd4671bf35290cca838 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:15:24 +0000 Subject: [PATCH 052/112] Update hanoi-tower.md --- content/facilitator/hanoi-tower.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/content/facilitator/hanoi-tower.md b/content/facilitator/hanoi-tower.md index cbd0388d..569eba65 100644 --- a/content/facilitator/hanoi-tower.md +++ b/content/facilitator/hanoi-tower.md @@ -10,7 +10,7 @@ For students to try out the problem, stacked blocks are likely the most readily For three blocks, the tower can be moved to the rightmost platform in a minimum of 7 moves. The process is as follows: - + Notice how, after the fourth move, we make very similar moves to 1, 2 and 3 but in reverse order and reflected horizontally. So, instead of dismantling a tower on the left (as in 1, 2 and 3), we are rebuilding the tower on the right. @@ -20,12 +20,12 @@ Using this may help you to work out how to move a tower of four or more blocks. For four blocks, the tower can be moved to the rightmost platform in a minimum of 15 moves: - + You may notice a pattern emerging. If we continue adding more blocks, the minimum number of moves required is as follows: - + You may notice that for some number of blocks $n$, if the minimum number of moves is $m$. Then the minimum number of moves for $n + 1$ blocks is $2m + 1$. From 1b8d099c69a21887ac3deecb636511ef02fbc931 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:15:57 +0000 Subject: [PATCH 053/112] Update hanoi-tower.md --- content/facilitator/hanoi-tower.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/facilitator/hanoi-tower.md b/content/facilitator/hanoi-tower.md index 569eba65..573b72c5 100644 --- a/content/facilitator/hanoi-tower.md +++ b/content/facilitator/hanoi-tower.md @@ -20,12 +20,12 @@ Using this may help you to work out how to move a tower of four or more blocks. For four blocks, the tower can be moved to the rightmost platform in a minimum of 15 moves: - + You may notice a pattern emerging. If we continue adding more blocks, the minimum number of moves required is as follows: - + You may notice that for some number of blocks $n$, if the minimum number of moves is $m$. Then the minimum number of moves for $n + 1$ blocks is $2m + 1$. From d354c4657bba4fc054d74e32f25c25802a14af4d Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:17:01 +0000 Subject: [PATCH 054/112] Create hanoi-tower.md --- content/student/hanoi-tower.md | 22 ++++++++++++++++++++++ 1 file changed, 22 insertions(+) create mode 100644 content/student/hanoi-tower.md diff --git a/content/student/hanoi-tower.md b/content/student/hanoi-tower.md new file mode 100644 index 00000000..78218e2c --- /dev/null +++ b/content/student/hanoi-tower.md @@ -0,0 +1,22 @@ +--- +title: Tower of Hanoi +type: problem +printOrder: n/a +added: 2021-01 +--- + +# Tower of Hanoi + +Three blocks are stacked in size order to form a tower, with the largest block at the base. + + + +The blocks are positioned on the leftmost of three platforms. + +Can you find a way to restack the blocks in the same order on the rightmost platform, following these rules? +- You may only move one block at a time. +- A larger block can never be placed on top of a smaller block. + +What is the smallest number of moves in which you can move the tower to the other platform? + +What about if there are four blocks, or more? From c61f55ca3bb4e966b977f946f109d667db338d6b Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:19:57 +0000 Subject: [PATCH 055/112] Add files via upload --- images/minimax-1.png | Bin 0 -> 9056 bytes images/minimax-2.png | Bin 0 -> 5111 bytes images/minimax-3.png | Bin 0 -> 11234 bytes images/minimax-4.png | Bin 0 -> 21272 bytes images/minimax-5.png | Bin 0 -> 35141 bytes images/minimax-6.png | Bin 0 -> 13560 bytes images/minimax-7.png | Bin 0 -> 14530 bytes 7 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 images/minimax-1.png create mode 100644 images/minimax-2.png create mode 100644 images/minimax-3.png create mode 100644 images/minimax-4.png create mode 100644 images/minimax-5.png create mode 100644 images/minimax-6.png create mode 100644 images/minimax-7.png diff --git a/images/minimax-1.png b/images/minimax-1.png new file mode 100644 index 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+--- + +# MiniMax Algorithm + +Here is a *map* for a game: + + + +Starting at the top point, you can choose to move down the map either to the left or the right. From here, your opponent then chooses to move down left or right to one of the numbers. +How should you play if: +a. You want to end on the highest number possible and your opponent wants to end on the smallest number possible? +b. You want to end on the smallest number possible and your opponent want to end on the largest number possible? + +Here is another map: + + + +Again starting at the top, you and your opponent take turns to move either left or right down the map, you get the first move. Starting at 0, add or subtract each number that you or you opponent lands on for a combined total. You should try to score the highest total possible, and your opponent will try to score the lowest. + +Assuming you and your opponent both play optimally, what route will be taken and what will the final score be? From 497d45c9bb60245f61f76850d566687cfa10e9c7 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:24:23 +0000 Subject: [PATCH 057/112] Create minimax.md --- content/facilitator/minimax.md | 56 ++++++++++++++++++++++++++++++++++ 1 file changed, 56 insertions(+) create mode 100644 content/facilitator/minimax.md diff --git a/content/facilitator/minimax.md b/content/facilitator/minimax.md new file mode 100644 index 00000000..6cbf1e9e --- /dev/null +++ b/content/facilitator/minimax.md @@ -0,0 +1,56 @@ +# MiniMax Algorithm + +## Introduction + +These games provide a introduction to the *MiniMax Algorithm*, an important tool in computer science and “game theory”. The Minimax Algorithm is a method to work out the best way to act in order to minimise loses, or to achieve the *best outcome in a worstcase scenario* (like if our opponent plays optimally). + +The first game is deliberately simpler, and hopefully students will be able to apply the same principles to a more complicated example in the second game (see Extension Section). + +## Solution + +For the first game, say we are trying to achieve the highest score possible and our opponent the lowest (as in part a). + +We know that our opponent will get the second move, and so they will choose the lowest of the two options available to them. As such, we can predict how they will move from each possible point. + + + +So, if we move to the left, our opponent will move right to score 3. If we move right, our opponent will move left to score 4. This means that we will get a higher score from moving right, so that is what we should do. + +This is despite the fact that, ideally, we would like to land on the 9, the highest result. However, since we know that our opponent is going to choose the lower of the two options, by moving right we *achieve the best outcome from the worst-case scenario*. + +This process of working backwards, considering the result of being in each position, is how the MiniMax Algorithm works. + +By the same logic, for part b, we should more right to obtain the lowest score possible. This way we will get a 6, whereas moving left we would get a 9. + +## Extension + +For the second game, there are many more possible routes through the map, making the situation more complicated. But by working backwards and applying the same logic, we can work out how to play optimally. + +Starting by considering our opponents move on the second to bottom row, from which they will always choose the minimise the total by moving to the smaller of their two options: + +- If they are moving from the 0, they will go right to the −1. +- If they are moving from the leftmost +2, they will go left to the −1. +- If they are moving from the rightmost +2, they will go left to the 0. +- If they are moving from the leftmost −1, they will move right to the +1. +- If they are moving from the −2, they will move left to the +1. +- If they are moving from the rightmost −1, they will move right to the −1. + +So, when choosing our move from the third to bottom row, we must consider the total result of each direction including how our opponent will move subsequently (in blue below). + + + +Wishing to achieve the highest total possible, we should choose to move to the largest total (blue number) possible from any position: + + + +Notice how, from the 0, even though the choice is between two ‘+2’s the total achieved from each move is different. We should move right for a better overall score. + +From this point, our opponent (playing optimally) will move from any point in the fourth to bottom row to minimise the total achieved. And we can continue this process of deduction upwards through each row: + + + +And so, knowing how each player should move from any given position, we can deduce the route which the game will take if each player plays optimally: + + + +Counter-intuitively, to score the highest total, we should move to the +1 on our first turn as opposed to the +2! This will allow us to get an overall total of +1, whereas by moving to the +2 the best we can get is 0. 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z6Gz6iMU9^D(J+X^jUruOyfJT|o!!6B6|8WP8m>&~ec14%s^pdvl4A|=I0uG;MSkWQ z02ZCEc`kWADpg=kX9kG?>~`lne%K1&0FX)%5QazOyZnQRATE<6ngt*(|DVO-|1-z= zzrs{;P$*w$W8lBp9JxeD21rReEq7*u4h@s#Rs>acze2&#p*J8Op-@e`JyS~p&_Hlh z6pn78e!wO2#2oZ}&TUFZL6VKwXW);ZxHtkpvSj;TQZcPT7|eo#ACLn0UEM>U+2E6< zXv+{(T3UT_o*jY~`Kp>i%e~$F<@Ys+M z)SlMH@t$=q1AVcNlUKHwGX7?9d|mWXUs_l&$jF$8e~M2Nq{0JbQ%Bd0$%x&czem`l zuX(nC?=t;;3D5yd`sdXh4Pffuw|$iVfBe(-<~4nL6HQF@wfD0d;HC6fUB38{Y2g0= D08+az literal 0 HcmV?d00001 From 9eb244337a29eba489d13c7f010e6e8662fbb75a Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:29:02 +0000 Subject: [PATCH 059/112] Create josephus-prob.md --- content/student/josephus-prob.md | 22 ++++++++++++++++++++++ 1 file changed, 22 insertions(+) create mode 100644 content/student/josephus-prob.md diff --git a/content/student/josephus-prob.md b/content/student/josephus-prob.md new file mode 100644 index 00000000..f247bf50 --- /dev/null +++ b/content/student/josephus-prob.md @@ -0,0 +1,22 @@ +--- +title: The Josephus Problem +type: problem +printOrder: n/a +added: 2021-01 +--- + +# The Josephus Problem + +You and 20 other people are standing in a circle to play a game: + + + +Starting with the first person, each player still in the game declares the next person still in to the left of them as ‘out’. So player 1 declares player 2 out, player 3 declares player 4 out and so on. + + + +After player 19 declares player 20 as out, player 21 will declare player 1 as out and then player 3 will declare player 5 out. This continues until just one player remains. + +Can you work out the number of the last player remaining? + +What about for different numbers of players, can you find a way to predict which player will be left? From 53503abbf73e096b7e2a4fdfa3e74600a486f2d0 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:29:46 +0000 Subject: [PATCH 060/112] Update josephus-prob.md --- content/student/josephus-prob.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/student/josephus-prob.md b/content/student/josephus-prob.md index f247bf50..1e982490 100644 --- a/content/student/josephus-prob.md +++ b/content/student/josephus-prob.md @@ -9,11 +9,11 @@ added: 2021-01 You and 20 other people are standing in a circle to play a game: - + Starting with the first person, each player still in the game declares the next person still in to the left of them as ‘out’. So player 1 declares player 2 out, player 3 declares player 4 out and so on. - + After player 19 declares player 20 as out, player 21 will declare player 1 as out and then player 3 will declare player 5 out. This continues until just one player remains. 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z6Gz6iMU9^D(J+X^jUruOyfJT|o!!6B6|8WP8m>&~ec14%s^pdvl4A|=I0uG;MSkWQ z02ZCEc`kWADpg=kX9kG?>~`lne%K1&0FX)%5QazOyZnQRATE<6ngt*(|DVO-|1-z= zzrs{;P$*w$W8lBp9JxeD21rReEq7*u4h@s#Rs>acze2&#p*J8Op-@e`JyS~p&_Hlh z6pn78e!wO2#2oZ}&TUFZL6VKwXW);ZxHtkpvSj;TQZcPT7|eo#ACLn0UEM>U+2E6< zXv+{(T3UT_o*jY~`Kp>i%e~$F<@Ys+M z)SlMH@t$=q1AVcNlUKHwGX7?9d|mWXUs_l&$jF$8e~M2Nq{0JbQ%Bd0$%x&czem`l zuX(nC?=t;;3D5yd`sdXh4Pffuw|$iVfBe(-<~4nL6HQF@wfD0d;HC6fUB38{Y2g0= D08+az literal 0 HcmV?d00001 From 02c20094441d68be6443e38efd59ce220bcd3945 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:34:43 +0000 Subject: [PATCH 066/112] Create josephus-prob.md --- content/facilitator/josephus-prob.md | 60 ++++++++++++++++++++++++++++ 1 file changed, 60 insertions(+) create mode 100644 content/facilitator/josephus-prob.md diff --git a/content/facilitator/josephus-prob.md b/content/facilitator/josephus-prob.md new file mode 100644 index 00000000..3753dba0 --- /dev/null +++ b/content/facilitator/josephus-prob.md @@ -0,0 +1,60 @@ +# The Josephus Problem + +## Introduction + +The Josephus Problem comes from an account by the first century historian and soldier Flavius Josephus, where he narrates his capture by Roman soldiers. To read about the origins of this story, you can view the... + +The problem can be generalised with different numbers of people in the circle, a nice activity is to play out the game with a class of students. + +## Solution + +For 21 people the ‘winning’ position is 11. This can be seen by working through the problem: + + + +For different numbers of players, the winning position is as follows: + + + +Students might notice a pattern, that the number of the winning position seems to increase through consecutive odd numbers, starting back at 1 at each power of 2. + +To show this pattern as a mathematical expression, we can work out the winning position from the number of players by subtracting the largest possible power of two and then multiplying by two and adding one. + +For example, for 21: + +$21 - 2^4 = 5$ +$(2 \times 5) + 1 = 11$ + +See the extension section for why this relationship is true. + +## Extension + +Consider how the game evolves when the number of players is a power of two, say, in general $2^n$: + +Player 1 declares Player 2 as out. +Player 3 declares Player 4 as out. +In general, every Player $2n − 1$ declares Player $2n$ as out. + +So, when it is the first players turn again, all of the remaining players are the odd numbers. If we renumber all of the players 1, 2, 3, ... we are essentially playing a game with $2^(n-1)$ players +(if you need to convince yourself of this, try drawing out a game with 16 players). + +So, a game with $2^n$ player will have the same winner as a game with $2^(n-1)$ players. + +Continuing with this argument, we can say that that any game with $2^(n-1)$ players will have the same winner as a game with $2^(n-2)$ players, and so on until we show that the winner will be the same as a game with just two players. Since the first player will win a game with two players, we can see that the first player will win any game where the number of players is a power of two. This conclusion is supported by the examples in the Solution section. + +Now, thinking about a general game, we can begin getting players out until the number of players left is a power of two. For example, in a game with 21 players: + +Player 1 declares Player 2 as out. +Player 3 declares Player 4 as out. +Player 5 declares Player 6 as out. +Player 7 declares Player 8 as out. +Player 9 declares Player 10 as out. + +At this point, there are 16 ($2^4$) players left and since player 11 has the next move we know that they will be the winner (we saw previously that in a game with $2^n$ players, the player taking +the first turn will win). + +So, for a general game: +- We subtract the highest power of two possible from the number of players, after this many turns the number of players remaining will be a power of two. +- At this point, if $k$ is the difference between the number of players and the highest power of two, it will be the turn of Player $2k + 1$ (or the $(𝑘 + 1)$th odd number). + +This explains the pattern we noticed in the Solution section. From acff969bf43e3f59d58491caea6d2073e57be8a8 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 25 Jan 2021 22:42:31 +0000 Subject: [PATCH 067/112] Add files via upload --- cover_images/colouring-maps.svg | 78 +++++++++++++++++++++++++++++++++ cover_images/hanoi-tower.svg | 1 + cover_images/josephus-prob.svg | 50 +++++++++++++++++++++ cover_images/minimax.svg | 70 +++++++++++++++++++++++++++++ 4 files changed, 199 insertions(+) create mode 100644 cover_images/colouring-maps.svg create mode 100644 cover_images/hanoi-tower.svg create mode 100644 cover_images/josephus-prob.svg create mode 100644 cover_images/minimax.svg diff --git a/cover_images/colouring-maps.svg b/cover_images/colouring-maps.svg new file mode 100644 index 00000000..2d76d1e8 --- /dev/null +++ b/cover_images/colouring-maps.svg @@ -0,0 +1,78 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/cover_images/hanoi-tower.svg b/cover_images/hanoi-tower.svg new file mode 100644 index 00000000..47d6a3fa --- /dev/null +++ b/cover_images/hanoi-tower.svg @@ -0,0 +1 @@ + \ No newline at end of file diff --git a/cover_images/josephus-prob.svg b/cover_images/josephus-prob.svg new file mode 100644 index 00000000..1d652587 --- /dev/null +++ b/cover_images/josephus-prob.svg @@ -0,0 +1,50 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/cover_images/minimax.svg b/cover_images/minimax.svg new file mode 100644 index 00000000..30b93847 --- /dev/null +++ b/cover_images/minimax.svg @@ -0,0 +1,70 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From 9657c8fe3775a2db3ae90bc200b4cfb24379adba Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:05:49 +0000 Subject: [PATCH 068/112] Update josephus-prob.md --- content/facilitator/josephus-prob.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/facilitator/josephus-prob.md b/content/facilitator/josephus-prob.md index 3753dba0..4455f978 100644 --- a/content/facilitator/josephus-prob.md +++ b/content/facilitator/josephus-prob.md @@ -55,6 +55,6 @@ the first turn will win). So, for a general game: - We subtract the highest power of two possible from the number of players, after this many turns the number of players remaining will be a power of two. -- At this point, if $k$ is the difference between the number of players and the highest power of two, it will be the turn of Player $2k + 1$ (or the $(𝑘 + 1)$th odd number). +- At this point, if $k$ is the difference between the number of players and the highest power of two, it will be the turn of Player $2k + 1$ (or the $(k + 1)$th odd number). This explains the pattern we noticed in the Solution section. From cdde6767e7ecd1814dd8f9a9b2e34be139463626 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:29:56 +0000 Subject: [PATCH 069/112] Delete colouring-maps.md --- content/student/colouring-maps.md | 22 ---------------------- 1 file changed, 22 deletions(-) delete mode 100644 content/student/colouring-maps.md diff --git a/content/student/colouring-maps.md b/content/student/colouring-maps.md deleted file mode 100644 index d3daf3ca..00000000 --- a/content/student/colouring-maps.md +++ /dev/null @@ -1,22 +0,0 @@ ---- -title: Colouring Maps -type: problem -printOrder: n/a -added: 2021-01 ---- - -# Colouring Maps - -Start by drawing an empty square: - - - -Create a *map* by dividing the square into seperate regions. For example: - - - -Now colour in your map such that there are no regions of the same colour which share an edge. - -Try to find the smallest number of different colours which you need to be able to do this. - -What about other maps, can you find a certain number of colours which are able to colour in any map? From 5344fab1552cdccc76e81ba01ef81416c9d37047 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:30:23 +0000 Subject: [PATCH 070/112] Delete hanoi-tower.md --- content/student/hanoi-tower.md | 22 ---------------------- 1 file changed, 22 deletions(-) delete mode 100644 content/student/hanoi-tower.md diff --git a/content/student/hanoi-tower.md b/content/student/hanoi-tower.md deleted file mode 100644 index 78218e2c..00000000 --- a/content/student/hanoi-tower.md +++ /dev/null @@ -1,22 +0,0 @@ ---- -title: Tower of Hanoi -type: problem -printOrder: n/a -added: 2021-01 ---- - -# Tower of Hanoi - -Three blocks are stacked in size order to form a tower, with the largest block at the base. - - - -The blocks are positioned on the leftmost of three platforms. - -Can you find a way to restack the blocks in the same order on the rightmost platform, following these rules? -- You may only move one block at a time. -- A larger block can never be placed on top of a smaller block. - -What is the smallest number of moves in which you can move the tower to the other platform? - -What about if there are four blocks, or more? From db6deba84bda43e1958ab02ab3e46a5e8e069cf2 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:30:31 +0000 Subject: [PATCH 071/112] Delete josephus-prob.md --- content/student/josephus-prob.md | 22 ---------------------- 1 file changed, 22 deletions(-) delete mode 100644 content/student/josephus-prob.md diff --git a/content/student/josephus-prob.md b/content/student/josephus-prob.md deleted file mode 100644 index 1e982490..00000000 --- a/content/student/josephus-prob.md +++ /dev/null @@ -1,22 +0,0 @@ ---- -title: The Josephus Problem -type: problem -printOrder: n/a -added: 2021-01 ---- - -# The Josephus Problem - -You and 20 other people are standing in a circle to play a game: - - - -Starting with the first person, each player still in the game declares the next person still in to the left of them as ‘out’. So player 1 declares player 2 out, player 3 declares player 4 out and so on. - - - -After player 19 declares player 20 as out, player 21 will declare player 1 as out and then player 3 will declare player 5 out. This continues until just one player remains. - -Can you work out the number of the last player remaining? - -What about for different numbers of players, can you find a way to predict which player will be left? From 9fcc276b223b0b75bad45bb1e8dd41eb859d8cb5 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:30:47 +0000 Subject: [PATCH 072/112] Delete minimax.md --- content/student/minimax.md | 25 ------------------------- 1 file changed, 25 deletions(-) delete mode 100644 content/student/minimax.md diff --git a/content/student/minimax.md b/content/student/minimax.md deleted file mode 100644 index c0da4b2b..00000000 --- a/content/student/minimax.md +++ /dev/null @@ -1,25 +0,0 @@ ---- -title: MiniMax Algorithm -type: problem -printOrder: n/a -added: 2021-01 ---- - -# MiniMax Algorithm - -Here is a *map* for a game: - - - -Starting at the top point, you can choose to move down the map either to the left or the right. From here, your opponent then chooses to move down left or right to one of the numbers. -How should you play if: -a. You want to end on the highest number possible and your opponent wants to end on the smallest number possible? -b. You want to end on the smallest number possible and your opponent want to end on the largest number possible? - -Here is another map: - - - -Again starting at the top, you and your opponent take turns to move either left or right down the map, you get the first move. Starting at 0, add or subtract each number that you or you opponent lands on for a combined total. You should try to score the highest total possible, and your opponent will try to score the lowest. - -Assuming you and your opponent both play optimally, what route will be taken and what will the final score be? From 9a3961a9a9f5c30661dbd0f5e33026e909a8c925 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:30:59 +0000 Subject: [PATCH 073/112] Delete colouring-maps.md --- content/facilitator/colouring-maps.md | 64 --------------------------- 1 file changed, 64 deletions(-) delete mode 100644 content/facilitator/colouring-maps.md diff --git a/content/facilitator/colouring-maps.md b/content/facilitator/colouring-maps.md deleted file mode 100644 index 8f18cacd..00000000 --- a/content/facilitator/colouring-maps.md +++ /dev/null @@ -1,64 +0,0 @@ -# Colouring Maps - -## Introduction - -This seemingly simple problem manages to disguise some very nice mathematics related to the field of graph theory. - -Students may design their own maps or use the one given in the question, they should not need more than four colours to fill their map, although they will not know this initially (see Extension section for proof). - -## Solution - -For the map given in the question, we can show that the map can be coloured using four colours. In fact, we will se that it cannot be done in any fewer. - -The majority of the map can be coloured using just two colours, take blue and green: - - - -At this point, we wish to colour the region in the top left corner. Since it shares edges with both blue and green regions, we must introduce a third colour, say red. This is also true for the region in the bottom right corner. - - - -Now we want to colour the star-shaped region. However, the star shares edges will a blue, a green and a red region so we must introduce a fourth colour. Yellow seems appropriate. - - - -And so, we have coloured this map using exactly four colours. Clearly, we could have used more colours (we could, for example, use a different colour for each of the 29 section). But, having seen that we were required to introduce four different colours, we could not have used less. - -## Extension - -We can ask the question, is there a smallest number of colours which will be able to colour in any given map? The answer is yes, and it is 4. Students may notice this pattern from trying lots of different maps, however showing it conclusively requires us to use graph theory. - -Specifically, we are going to translate the maps into mathematical *graphs* (likely not the type of -graph that you are used to). To do this we will: -- Replace the different regions on the map with circles -- Connect up the circles representing regions which share edges using lines. - -(We usually refer to these circles as *vertices*, and the lines as *edges*. This is somewhat confusing, but "edges" is used with two different meanings in this question. It can refer to the boundaries of a region, or a connection between vertices.) - -Let us take a portion of the example map and turn it into a graph. For the part in the bottom-right corner: - - - -Notice how our previous rule, that regions which share an edge cannot be the same colour, is maintained in the graph as no two connected vertices can be of the same colour. - -Also notice that the graph does not have any intersecting edges. In fact, any graph which represents a valid map can be drawn without any edges intersecting. - -Knowing these two things will allow us to prove that we can always colour a map using four colours or less. Let us try to contrsuct a graph which needs a minimum of five colours to fill. To do this, we need to contruct a graph of five vertices where each is connected to all the others and none of the edges cross. - -From the example map, we can make a graphy with four vertices where all are connected to each other and none of the lines intersect: - - - -If we now try to add a fifth vertex, there are 4 areas where we could try to do so: -- In any of the three small triangles formed by connecting the Yellow vertex with two others. -- Outside the large trianle formed by the Red, Green and Blue vertices. - - - -However, in each case, we can only connect up the additional vertex to three others without any edges crossing. This means that they do not, in fact, have to take a new colour and we can still colour the graph with four colours. - - - -So, we have shown that we cannot construct a graph requiring five different colours to fill. - -Equivalently, we cannot draw a map which needs different five colours to fill. We can fill any map with a maximum of different four colours. From 2035ad17960accf81788086254ecae509db97ef7 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:31:08 +0000 Subject: [PATCH 074/112] Delete hanoi-tower.md --- content/facilitator/hanoi-tower.md | 46 ------------------------------ 1 file changed, 46 deletions(-) delete mode 100644 content/facilitator/hanoi-tower.md diff --git a/content/facilitator/hanoi-tower.md b/content/facilitator/hanoi-tower.md deleted file mode 100644 index 573b72c5..00000000 --- a/content/facilitator/hanoi-tower.md +++ /dev/null @@ -1,46 +0,0 @@ -# Tower of Hanoi - -## Introduction - -This game is a classic recreational maths puzzle. It is, in fact, possible to move the tower abiding by the rules for any number of blocks (see Extension section). - -For students to try out the problem, stacked blocks are likely the most readily available objects. Often, the problem is shown using three pegs and different sized disks with holes in the middle which can be slid over the pegs. Alternatively, the puzzle can be drawn out at each stage to represent the moves made (as in the Solution section). - -## Solution - -For three blocks, the tower can be moved to the rightmost platform in a minimum of 7 moves. The process is as follows: - - - -Notice how, after the fourth move, we make very similar moves to 1, 2 and 3 but in reverse order and reflected horizontally. So, instead of dismantling a tower on the left (as in 1, 2 and 3), we are rebuilding the tower on the right. - -Using this may help you to work out how to move a tower of four or more blocks. - -## Extension - -For four blocks, the tower can be moved to the rightmost platform in a minimum of 15 moves: - - - -You may notice a pattern emerging. If we continue adding more blocks, the minimum number of -moves required is as follows: - - - -You may notice that for some number of blocks $n$, if the minimum number of moves is $m$. Then the -minimum number of moves for $n + 1$ blocks is $2m + 1$. -ie. $15 = 2 \times 7 + 1$, $31 = 2 \times 15 + 1$, etc. - -Thinking about why this is, in the case of four blocks: -- The first stage is to move the top three blocks to the central position, which takes 7 moves. -- Next, the largest block is moved to the rightmost position, taking 1 move. -- Finally, the top three blocks are moved from the central position to the rightmost position, taking 7 moves. - -So overall this takes $7 + 1 + 7 = 2 × 7 + 1 = 15$ moves. - -Generally, in the case of $n + 1$ blocks: -- The first stage is to move the top $n$ blocks to the central position, which takes $m$ moves. -- Next, the largest block is moved to the rightmost position, taking 1 move. -- Finally, the top $n$ blocks are moved from the central position to the rightmost position, taking $m$ moves. - -So overall this takes $m + 1 + m = 2m + 1$ moves From ed657d10008b9d6a6b0e4e522fbfa1fe775d0343 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:31:23 +0000 Subject: [PATCH 075/112] Delete josephus-prob.md --- content/facilitator/josephus-prob.md | 60 ---------------------------- 1 file changed, 60 deletions(-) delete mode 100644 content/facilitator/josephus-prob.md diff --git a/content/facilitator/josephus-prob.md b/content/facilitator/josephus-prob.md deleted file mode 100644 index 4455f978..00000000 --- a/content/facilitator/josephus-prob.md +++ /dev/null @@ -1,60 +0,0 @@ -# The Josephus Problem - -## Introduction - -The Josephus Problem comes from an account by the first century historian and soldier Flavius Josephus, where he narrates his capture by Roman soldiers. To read about the origins of this story, you can view the... - -The problem can be generalised with different numbers of people in the circle, a nice activity is to play out the game with a class of students. - -## Solution - -For 21 people the ‘winning’ position is 11. This can be seen by working through the problem: - - - -For different numbers of players, the winning position is as follows: - - - -Students might notice a pattern, that the number of the winning position seems to increase through consecutive odd numbers, starting back at 1 at each power of 2. - -To show this pattern as a mathematical expression, we can work out the winning position from the number of players by subtracting the largest possible power of two and then multiplying by two and adding one. - -For example, for 21: - -$21 - 2^4 = 5$ -$(2 \times 5) + 1 = 11$ - -See the extension section for why this relationship is true. - -## Extension - -Consider how the game evolves when the number of players is a power of two, say, in general $2^n$: - -Player 1 declares Player 2 as out. -Player 3 declares Player 4 as out. -In general, every Player $2n − 1$ declares Player $2n$ as out. - -So, when it is the first players turn again, all of the remaining players are the odd numbers. If we renumber all of the players 1, 2, 3, ... we are essentially playing a game with $2^(n-1)$ players -(if you need to convince yourself of this, try drawing out a game with 16 players). - -So, a game with $2^n$ player will have the same winner as a game with $2^(n-1)$ players. - -Continuing with this argument, we can say that that any game with $2^(n-1)$ players will have the same winner as a game with $2^(n-2)$ players, and so on until we show that the winner will be the same as a game with just two players. Since the first player will win a game with two players, we can see that the first player will win any game where the number of players is a power of two. This conclusion is supported by the examples in the Solution section. - -Now, thinking about a general game, we can begin getting players out until the number of players left is a power of two. For example, in a game with 21 players: - -Player 1 declares Player 2 as out. -Player 3 declares Player 4 as out. -Player 5 declares Player 6 as out. -Player 7 declares Player 8 as out. -Player 9 declares Player 10 as out. - -At this point, there are 16 ($2^4$) players left and since player 11 has the next move we know that they will be the winner (we saw previously that in a game with $2^n$ players, the player taking -the first turn will win). - -So, for a general game: -- We subtract the highest power of two possible from the number of players, after this many turns the number of players remaining will be a power of two. -- At this point, if $k$ is the difference between the number of players and the highest power of two, it will be the turn of Player $2k + 1$ (or the $(k + 1)$th odd number). - -This explains the pattern we noticed in the Solution section. From d3b506b6336286117d7dc16d72e6f85d5f369c38 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:31:32 +0000 Subject: [PATCH 076/112] Delete minimax.md --- content/facilitator/minimax.md | 56 ---------------------------------- 1 file changed, 56 deletions(-) delete mode 100644 content/facilitator/minimax.md diff --git a/content/facilitator/minimax.md b/content/facilitator/minimax.md deleted file mode 100644 index 6cbf1e9e..00000000 --- a/content/facilitator/minimax.md +++ /dev/null @@ -1,56 +0,0 @@ -# MiniMax Algorithm - -## Introduction - -These games provide a introduction to the *MiniMax Algorithm*, an important tool in computer science and “game theory”. The Minimax Algorithm is a method to work out the best way to act in order to minimise loses, or to achieve the *best outcome in a worstcase scenario* (like if our opponent plays optimally). - -The first game is deliberately simpler, and hopefully students will be able to apply the same principles to a more complicated example in the second game (see Extension Section). - -## Solution - -For the first game, say we are trying to achieve the highest score possible and our opponent the lowest (as in part a). - -We know that our opponent will get the second move, and so they will choose the lowest of the two options available to them. As such, we can predict how they will move from each possible point. - - - -So, if we move to the left, our opponent will move right to score 3. If we move right, our opponent will move left to score 4. This means that we will get a higher score from moving right, so that is what we should do. - -This is despite the fact that, ideally, we would like to land on the 9, the highest result. However, since we know that our opponent is going to choose the lower of the two options, by moving right we *achieve the best outcome from the worst-case scenario*. - -This process of working backwards, considering the result of being in each position, is how the MiniMax Algorithm works. - -By the same logic, for part b, we should more right to obtain the lowest score possible. This way we will get a 6, whereas moving left we would get a 9. - -## Extension - -For the second game, there are many more possible routes through the map, making the situation more complicated. But by working backwards and applying the same logic, we can work out how to play optimally. - -Starting by considering our opponents move on the second to bottom row, from which they will always choose the minimise the total by moving to the smaller of their two options: - -- If they are moving from the 0, they will go right to the −1. -- If they are moving from the leftmost +2, they will go left to the −1. -- If they are moving from the rightmost +2, they will go left to the 0. -- If they are moving from the leftmost −1, they will move right to the +1. -- If they are moving from the −2, they will move left to the +1. -- If they are moving from the rightmost −1, they will move right to the −1. - -So, when choosing our move from the third to bottom row, we must consider the total result of each direction including how our opponent will move subsequently (in blue below). - - - -Wishing to achieve the highest total possible, we should choose to move to the largest total (blue number) possible from any position: - - - -Notice how, from the 0, even though the choice is between two ‘+2’s the total achieved from each move is different. We should move right for a better overall score. - -From this point, our opponent (playing optimally) will move from any point in the fourth to bottom row to minimise the total achieved. And we can continue this process of deduction upwards through each row: - - - -And so, knowing how each player should move from any given position, we can deduce the route which the game will take if each player plays optimally: - - - -Counter-intuitively, to score the highest total, we should move to the +1 on our first turn as opposed to the +2! This will allow us to get an overall total of +1, whereas by moving to the +2 the best we can get is 0. From 4664fdbac0776ce3db74fc49f5bcdd845eb3823e Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:31:52 +0000 Subject: [PATCH 077/112] Delete colouring-maps.svg --- cover_images/colouring-maps.svg | 78 --------------------------------- 1 file changed, 78 deletions(-) delete mode 100644 cover_images/colouring-maps.svg diff --git a/cover_images/colouring-maps.svg b/cover_images/colouring-maps.svg deleted file mode 100644 index 2d76d1e8..00000000 --- a/cover_images/colouring-maps.svg +++ /dev/null @@ -1,78 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - From 651731d4858cba58d4c3ef34f74a8111c3a7cebd Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:31:58 +0000 Subject: [PATCH 078/112] Delete hanoi-tower.svg --- cover_images/hanoi-tower.svg | 1 - 1 file changed, 1 deletion(-) delete mode 100644 cover_images/hanoi-tower.svg diff --git a/cover_images/hanoi-tower.svg b/cover_images/hanoi-tower.svg deleted file mode 100644 index 47d6a3fa..00000000 --- a/cover_images/hanoi-tower.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file From 17bf23946f9a8904dec423e28402954a19449e55 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:32:04 +0000 Subject: [PATCH 079/112] Delete josephus-prob.svg --- cover_images/josephus-prob.svg | 50 ---------------------------------- 1 file changed, 50 deletions(-) delete mode 100644 cover_images/josephus-prob.svg diff --git a/cover_images/josephus-prob.svg b/cover_images/josephus-prob.svg deleted file mode 100644 index 1d652587..00000000 --- a/cover_images/josephus-prob.svg +++ /dev/null @@ -1,50 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - From 0db810f0d5347ab98c9e2592a7cc0d8f82d5edc2 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:32:11 +0000 Subject: [PATCH 080/112] Delete minimax.svg --- cover_images/minimax.svg | 70 ---------------------------------------- 1 file changed, 70 deletions(-) delete mode 100644 cover_images/minimax.svg diff --git a/cover_images/minimax.svg b/cover_images/minimax.svg deleted file mode 100644 index 30b93847..00000000 --- a/cover_images/minimax.svg +++ /dev/null @@ -1,70 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - From b679f8d90debc10fbf30add0062f67542e004dc5 Mon Sep 17 00:00:00 2001 From: dankelly01 <77119345+dankelly01@users.noreply.github.com> Date: Mon, 1 Feb 2021 11:32:24 +0000 Subject: [PATCH 081/112] Delete colouring-maps-1.png --- images/colouring-maps-1.png | Bin 2614 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 images/colouring-maps-1.png diff --git a/images/colouring-maps-1.png b/images/colouring-maps-1.png deleted file 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b/maths-club-pack/content/student/konigsberg-bridges.md similarity index 100% rename from content/student/konigsberg-bridges.md rename to maths-club-pack/content/student/konigsberg-bridges.md diff --git a/content/student/tetris-shapes.md b/maths-club-pack/content/student/tetris-shapes.md similarity index 100% rename from content/student/tetris-shapes.md rename to maths-club-pack/content/student/tetris-shapes.md diff --git a/cover_images/diffy-squares.svg b/maths-club-pack/cover_images/diffy-squares.svg similarity index 98% rename from cover_images/diffy-squares.svg rename to maths-club-pack/cover_images/diffy-squares.svg index dfc97f51..624a45ab 100644 --- a/cover_images/diffy-squares.svg +++ b/maths-club-pack/cover_images/diffy-squares.svg @@ -1,95 +1,95 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/cover_images/kaprekars-number.svg b/maths-club-pack/cover_images/kaprekars-number.svg similarity index 100% rename from cover_images/kaprekars-number.svg rename to maths-club-pack/cover_images/kaprekars-number.svg diff --git a/cover_images/konigsberg-bridges.svg b/maths-club-pack/cover_images/konigsberg-bridges.svg similarity index 98% rename from cover_images/konigsberg-bridges.svg rename to maths-club-pack/cover_images/konigsberg-bridges.svg index 90760f8f..02cf40a8 100644 --- a/cover_images/konigsberg-bridges.svg +++ b/maths-club-pack/cover_images/konigsberg-bridges.svg @@ -1,81 +1,81 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/cover_images/tetris-shapes.svg b/maths-club-pack/cover_images/tetris-shapes.svg similarity index 97% rename from cover_images/tetris-shapes.svg rename to maths-club-pack/cover_images/tetris-shapes.svg index 54d080a3..f19ee36b 100644 --- a/cover_images/tetris-shapes.svg +++ b/maths-club-pack/cover_images/tetris-shapes.svg @@ -1,46 +1,46 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/images/diffy-squares-1.png b/maths-club-pack/images/diffy-squares-1.png similarity index 100% rename from images/diffy-squares-1.png rename to maths-club-pack/images/diffy-squares-1.png diff --git a/images/diffy-squares-2.png b/maths-club-pack/images/diffy-squares-2.png similarity index 100% rename from images/diffy-squares-2.png rename to maths-club-pack/images/diffy-squares-2.png diff --git a/images/diffy-squares-3.png b/maths-club-pack/images/diffy-squares-3.png similarity index 100% rename from images/diffy-squares-3.png rename to maths-club-pack/images/diffy-squares-3.png diff --git a/images/diffy-squares-4.png b/maths-club-pack/images/diffy-squares-4.png similarity index 100% rename from images/diffy-squares-4.png rename to maths-club-pack/images/diffy-squares-4.png diff --git a/images/kaprekars-number.png b/maths-club-pack/images/kaprekars-number.png similarity index 100% rename from images/kaprekars-number.png rename to maths-club-pack/images/kaprekars-number.png diff --git a/images/konigsberg-bridges-1.png b/maths-club-pack/images/konigsberg-bridges-1.png similarity index 100% rename from images/konigsberg-bridges-1.png rename to maths-club-pack/images/konigsberg-bridges-1.png diff --git a/images/konigsberg-bridges-2.png b/maths-club-pack/images/konigsberg-bridges-2.png similarity index 100% rename from images/konigsberg-bridges-2.png rename to maths-club-pack/images/konigsberg-bridges-2.png diff --git a/images/konigsberg-bridges-3.png b/maths-club-pack/images/konigsberg-bridges-3.png similarity index 100% rename from images/konigsberg-bridges-3.png rename to maths-club-pack/images/konigsberg-bridges-3.png diff --git a/images/konigsberg-bridges-4.png b/maths-club-pack/images/konigsberg-bridges-4.png similarity index 100% rename from images/konigsberg-bridges-4.png rename to maths-club-pack/images/konigsberg-bridges-4.png diff --git a/images/tetris-shapes-1.png b/maths-club-pack/images/tetris-shapes-1.png similarity index 100% rename from images/tetris-shapes-1.png rename to maths-club-pack/images/tetris-shapes-1.png diff --git a/images/tetris-shapes-2.png b/maths-club-pack/images/tetris-shapes-2.png similarity index 100% rename from images/tetris-shapes-2.png rename to maths-club-pack/images/tetris-shapes-2.png diff --git a/images/tetris-shapes-3.png b/maths-club-pack/images/tetris-shapes-3.png similarity index 100% rename from images/tetris-shapes-3.png rename to maths-club-pack/images/tetris-shapes-3.png diff --git a/images/tetris-shapes-4.png b/maths-club-pack/images/tetris-shapes-4.png similarity index 100% rename from images/tetris-shapes-4.png rename to maths-club-pack/images/tetris-shapes-4.png From c40e09fcbe8180e6f1ab2a275fae44149048456b Mon Sep 17 00:00:00 2001 From: chrismclarke Date: Mon, 1 Feb 2021 06:32:28 -0800 Subject: [PATCH 106/112] remove external readme --- README.md | 3 --- readme.md | 27 --------------------------- 2 files changed, 30 deletions(-) delete mode 100644 README.md delete mode 100644 readme.md diff --git a/README.md b/README.md deleted file mode 100644 index 8946e95a..00000000 --- a/README.md +++ /dev/null @@ -1,3 +0,0 @@ -# MathsProblems - -Maths problems for SAMI Maths Club App diff --git a/readme.md b/readme.md deleted file mode 100644 index 3d66933c..00000000 --- a/readme.md +++ /dev/null @@ -1,27 +0,0 @@ -# SAMI Maths Club - -This repo contains source code for the SAMI Maths Club App - -![screenshots](documentation/images/app-screenshots.png) - -Download for Android from [Google Play](https://play.google.com/store/apps/details?id=io.c2dev.samimathsclub), -iOS on [App Store](https://apps.apple.com/app/sami-maths-club/id1527293312), -or [Use Online](http://mathsclub.samicharity.co.uk/) - -# Get Involved - -## Contributing Problems - -Have a favourite maths puzzle or problem? - -See the [Authors Documentation](./documentation/authors.md) for ways to include it in pack. - -## Contributing Translations - -Want to translate problems into more languages? - -See the [Translators Documentation](./documentation/translators.md) to learn how to join our translations project. - -## App Development - -See the [Developer Documentation](./maths-club-app/README.md) for more information about running locally or on android. From 08caa03ef0802c790b70cbd5a9b814b9ed169825 Mon Sep 17 00:00:00 2001 From: chrismclarke Date: Mon, 1 Feb 2021 06:33:02 -0800 Subject: [PATCH 107/112] revert readme --- README.md | 27 +++++++++++++++++++++++++++ 1 file changed, 27 insertions(+) create mode 100644 README.md diff --git a/README.md b/README.md new file mode 100644 index 00000000..3d66933c --- /dev/null +++ b/README.md @@ -0,0 +1,27 @@ +# SAMI Maths Club + +This repo contains source code for the SAMI Maths Club App + +![screenshots](documentation/images/app-screenshots.png) + +Download for Android from [Google Play](https://play.google.com/store/apps/details?id=io.c2dev.samimathsclub), +iOS on [App Store](https://apps.apple.com/app/sami-maths-club/id1527293312), +or [Use Online](http://mathsclub.samicharity.co.uk/) + +# Get Involved + +## Contributing Problems + +Have a favourite maths puzzle or problem? + +See the [Authors Documentation](./documentation/authors.md) for ways to include it in pack. + +## Contributing Translations + +Want to translate problems into more languages? + +See the [Translators Documentation](./documentation/translators.md) to learn how to join our translations project. + +## App Development + +See the [Developer Documentation](./maths-club-app/README.md) for more information about running locally or on android. From a3d2fdd69b027ad7f1162ff290983f0132bc3823 Mon Sep 17 00:00:00 2001 From: chrismclarke Date: Mon, 1 Feb 2021 06:37:49 -0800 Subject: [PATCH 108/112] update weekly problems list --- maths-club-app/src/app/data/weeklyProblems.ts | 298 +++++++++--------- 1 file changed, 149 insertions(+), 149 deletions(-) diff --git a/maths-club-app/src/app/data/weeklyProblems.ts b/maths-club-app/src/app/data/weeklyProblems.ts index d0e56c11..c3d20bdb 100644 --- a/maths-club-app/src/app/data/weeklyProblems.ts +++ b/maths-club-app/src/app/data/weeklyProblems.ts @@ -50,153 +50,153 @@ export const WEEKLY_PROBLEMS = { "2021-08-01": "letter-matching", "2021-08-08": "countdown", "2021-08-15": "15-game", - "2021-08-22": "pell-numbers", - "2021-08-29": "paths", - "2021-09-05": "number-challenge-1", - "2021-09-12": "making-squares", - "2021-09-19": "logic-puzzles-1", - "2021-09-26": "iq-challenge-1", - "2021-10-03": "find-the-path", - "2021-10-10": "counting-squares", - "2021-10-17": "buffons-needle", - "2021-10-24": "balls-and-books", - "2021-10-31": "addition-squares", - "2021-11-07": "adding-to-15", - "2021-11-14": "9-dots", - "2021-11-21": "treasure-hunt", - "2021-11-28": "russian-multiplication", - "2021-12-05": "river-crossing", - "2021-12-12": "poisoned-wine-puzzle", - "2021-12-19": "matchstick-puzzles", - "2021-12-26": "magic-cards", - "2022-01-02": "langfords-problem", - "2022-01-09": "handshake-puzzles", - "2022-01-16": "geometry-puzzle-1", - "2022-01-23": "gabriels-problem", - "2022-01-30": "fence-around-a-field", - "2022-02-06": "domino-tilings", - "2022-02-13": "collatz-conjecture", - "2022-02-20": "coin-game", - "2022-02-27": "coconut-trader", - "2022-03-06": "circumference", - "2022-03-13": "apple-teaser", - "2022-03-20": "two-eggs", - "2022-03-27": "special-numbers", - "2022-04-03": "scales-and-weights", - "2022-04-10": "picture-puzzles-1", - "2022-04-17": "monkey-business", - "2022-04-24": "locks-and-keys", - "2022-05-01": "frog-party", - "2022-05-08": "counting-chickens", - "2022-05-15": "tic-tac-toe", - "2022-05-22": "tic-tac-toe-with-levels", - "2022-05-29": "tangrams", - "2022-06-05": "secret-santa", - "2022-06-12": "patience", - "2022-06-19": "paper-pieces", - "2022-06-26": "nim-related-games", - "2022-07-03": "nim-21", - "2022-07-10": "mastermind", - "2022-07-17": "make-many", - "2022-07-24": "letter-matching", - "2022-07-31": "countdown", - "2022-08-07": "15-game", - "2022-08-14": "pell-numbers", - "2022-08-21": "paths", - "2022-08-28": "number-challenge-1", - "2022-09-04": "making-squares", - "2022-09-11": "logic-puzzles-1", - "2022-09-18": "iq-challenge-1", - "2022-09-25": "find-the-path", - "2022-10-02": "counting-squares", - "2022-10-09": "buffons-needle", - "2022-10-16": "balls-and-books", - "2022-10-23": "addition-squares", - "2022-10-30": "adding-to-15", - "2022-11-06": "9-dots", - "2022-11-13": "treasure-hunt", - "2022-11-20": "russian-multiplication", - "2022-11-27": "river-crossing", - "2022-12-04": "poisoned-wine-puzzle", - "2022-12-11": "matchstick-puzzles", - "2022-12-18": "magic-cards", - "2022-12-25": "langfords-problem", - "2023-01-01": "handshake-puzzles", - "2023-01-08": "geometry-puzzle-1", - "2023-01-15": "gabriels-problem", - "2023-01-22": "fence-around-a-field", - "2023-01-29": "domino-tilings", - "2023-02-05": "collatz-conjecture", - "2023-02-12": "coin-game", - "2023-02-19": "coconut-trader", - "2023-02-26": "circumference", - "2023-03-05": "apple-teaser", - "2023-03-12": "two-eggs", - "2023-03-19": "special-numbers", - "2023-03-26": "scales-and-weights", - "2023-04-02": "picture-puzzles-1", - "2023-04-09": "monkey-business", - "2023-04-16": "locks-and-keys", - "2023-04-23": "frog-party", - "2023-04-30": "counting-chickens", - "2023-05-07": "tic-tac-toe", - "2023-05-14": "tic-tac-toe-with-levels", - "2023-05-21": "tangrams", - "2023-05-28": "secret-santa", - "2023-06-04": "patience", - "2023-06-11": "paper-pieces", - "2023-06-18": "nim-related-games", - "2023-06-25": "nim-21", - "2023-07-02": "mastermind", - "2023-07-09": "make-many", - "2023-07-16": "letter-matching", - "2023-07-23": "countdown", - "2023-07-30": "15-game", - "2023-08-06": "pell-numbers", - "2023-08-13": "paths", - "2023-08-20": "number-challenge-1", - "2023-08-27": "making-squares", - "2023-09-03": "logic-puzzles-1", - "2023-09-10": "iq-challenge-1", - "2023-09-17": "find-the-path", - "2023-09-24": "counting-squares", - "2023-10-01": "buffons-needle", - "2023-10-08": "balls-and-books", - "2023-10-15": "addition-squares", - "2023-10-22": "adding-to-15", - "2023-10-29": "9-dots", - "2023-11-05": "treasure-hunt", - "2023-11-12": "russian-multiplication", - "2023-11-19": "river-crossing", - "2023-11-26": "poisoned-wine-puzzle", - "2023-12-03": "matchstick-puzzles", - "2023-12-10": "magic-cards", - "2023-12-17": "langfords-problem", - "2023-12-24": "handshake-puzzles", - "2023-12-31": "geometry-puzzle-1", - "2024-01-07": "gabriels-problem", - "2024-01-14": "fence-around-a-field", - "2024-01-21": "domino-tilings", - "2024-01-28": "collatz-conjecture", - "2024-02-04": "coin-game", - "2024-02-11": "coconut-trader", - "2024-02-18": "circumference", - "2024-02-25": "apple-teaser", - "2024-03-03": "two-eggs", - "2024-03-10": "special-numbers", - "2024-03-17": "scales-and-weights", - "2024-03-24": "picture-puzzles-1", - "2024-03-31": "monkey-business", - "2024-04-07": "locks-and-keys", - "2024-04-14": "frog-party", - "2024-04-21": "counting-chickens", - "2024-04-28": "tic-tac-toe", - "2024-05-05": "tic-tac-toe-with-levels", - "2024-05-12": "tangrams", - "2024-05-19": "secret-santa", - "2024-05-26": "patience", - "2024-06-02": "paper-pieces", - "2024-06-09": "nim-related-games", - "2024-06-16": "nim-21", - "2024-06-23": "mastermind" + "2021-08-22": "tetris-shapes", + "2021-08-29": "kaprekars-number", + "2021-09-05": "diffy-squares", + "2021-09-12": "pell-numbers", + "2021-09-19": "paths", + "2021-09-26": "number-challenge-1", + "2021-10-03": "making-squares", + "2021-10-10": "logic-puzzles-1", + "2021-10-17": "iq-challenge-1", + "2021-10-24": "find-the-path", + "2021-10-31": "counting-squares", + "2021-11-07": "buffons-needle", + "2021-11-14": "balls-and-books", + "2021-11-21": "addition-squares", + "2021-11-28": "adding-to-15", + "2021-12-05": "9-dots", + "2021-12-12": "treasure-hunt", + "2021-12-19": "russian-multiplication", + "2021-12-26": "river-crossing", + "2022-01-02": "poisoned-wine-puzzle", + "2022-01-09": "matchstick-puzzles", + "2022-01-16": "magic-cards", + "2022-01-23": "langfords-problem", + "2022-01-30": "handshake-puzzles", + "2022-02-06": "geometry-puzzle-1", + "2022-02-13": "gabriels-problem", + "2022-02-20": "fence-around-a-field", + "2022-02-27": "domino-tilings", + "2022-03-06": "collatz-conjecture", + "2022-03-13": "coin-game", + "2022-03-20": "coconut-trader", + "2022-03-27": "circumference", + "2022-04-03": "apple-teaser", + "2022-04-10": "two-eggs", + "2022-04-17": "special-numbers", + "2022-04-24": "scales-and-weights", + "2022-05-01": "picture-puzzles-1", + "2022-05-08": "monkey-business", + "2022-05-15": "locks-and-keys", + "2022-05-22": "frog-party", + "2022-05-29": "counting-chickens", + "2022-06-05": "tic-tac-toe", + "2022-06-12": "tic-tac-toe-with-levels", + "2022-06-19": "tangrams", + "2022-06-26": "secret-santa", + "2022-07-03": "patience", + "2022-07-10": "paper-pieces", + "2022-07-17": "nim-related-games", + "2022-07-24": "nim-21", + "2022-07-31": "mastermind", + "2022-08-07": "make-many", + "2022-08-14": "letter-matching", + "2022-08-21": "countdown", + "2022-08-28": "15-game", + "2022-09-04": "tetris-shapes", + "2022-09-11": "kaprekars-number", + "2022-09-18": "diffy-squares", + "2022-09-25": "pell-numbers", + "2022-10-02": "paths", + "2022-10-09": "number-challenge-1", + "2022-10-16": "making-squares", + "2022-10-23": "logic-puzzles-1", + "2022-10-30": "iq-challenge-1", + "2022-11-06": "find-the-path", + "2022-11-13": "counting-squares", + "2022-11-20": "buffons-needle", + "2022-11-27": "balls-and-books", + "2022-12-04": "addition-squares", + "2022-12-11": "adding-to-15", + "2022-12-18": "9-dots", + "2022-12-25": "treasure-hunt", + "2023-01-01": "russian-multiplication", + "2023-01-08": "river-crossing", + "2023-01-15": "poisoned-wine-puzzle", + "2023-01-22": "matchstick-puzzles", + "2023-01-29": "magic-cards", + "2023-02-05": "langfords-problem", + "2023-02-12": "handshake-puzzles", + "2023-02-19": "geometry-puzzle-1", + "2023-02-26": "gabriels-problem", + "2023-03-05": "fence-around-a-field", + "2023-03-12": "domino-tilings", + "2023-03-19": "collatz-conjecture", + "2023-03-26": "coin-game", + "2023-04-02": "coconut-trader", + "2023-04-09": "circumference", + "2023-04-16": "apple-teaser", + "2023-04-23": "two-eggs", + "2023-04-30": "special-numbers", + "2023-05-07": "scales-and-weights", + "2023-05-14": "picture-puzzles-1", + "2023-05-21": "monkey-business", + "2023-05-28": "locks-and-keys", + "2023-06-04": "frog-party", + "2023-06-11": "counting-chickens", + "2023-06-18": "tic-tac-toe", + "2023-06-25": "tic-tac-toe-with-levels", + "2023-07-02": "tangrams", + "2023-07-09": "secret-santa", + "2023-07-16": "patience", + "2023-07-23": "paper-pieces", + "2023-07-30": "nim-related-games", + "2023-08-06": "nim-21", + "2023-08-13": "mastermind", + "2023-08-20": "make-many", + "2023-08-27": "letter-matching", + "2023-09-03": "countdown", + "2023-09-10": "15-game", + "2023-09-17": "tetris-shapes", + "2023-09-24": "kaprekars-number", + "2023-10-01": "diffy-squares", + "2023-10-08": "pell-numbers", + "2023-10-15": "paths", + "2023-10-22": "number-challenge-1", + "2023-10-29": "making-squares", + "2023-11-05": "logic-puzzles-1", + "2023-11-12": "iq-challenge-1", + "2023-11-19": "find-the-path", + "2023-11-26": "counting-squares", + "2023-12-03": "buffons-needle", + "2023-12-10": "balls-and-books", + "2023-12-17": "addition-squares", + "2023-12-24": "adding-to-15", + "2023-12-31": "9-dots", + "2024-01-07": "treasure-hunt", + "2024-01-14": "russian-multiplication", + "2024-01-21": "river-crossing", + "2024-01-28": "poisoned-wine-puzzle", + "2024-02-04": "matchstick-puzzles", + "2024-02-11": "magic-cards", + "2024-02-18": "langfords-problem", + "2024-02-25": "handshake-puzzles", + "2024-03-03": "geometry-puzzle-1", + "2024-03-10": "gabriels-problem", + "2024-03-17": "fence-around-a-field", + "2024-03-24": "domino-tilings", + "2024-03-31": "collatz-conjecture", + "2024-04-07": "coin-game", + "2024-04-14": "coconut-trader", + "2024-04-21": "circumference", + "2024-04-28": "apple-teaser", + "2024-05-05": "two-eggs", + "2024-05-12": "special-numbers", + "2024-05-19": "scales-and-weights", + "2024-05-26": "picture-puzzles-1", + "2024-06-02": "monkey-business", + "2024-06-09": "locks-and-keys", + "2024-06-16": "frog-party", + "2024-06-23": "counting-chickens" } \ No newline at end of file From 646e878d8b2cf2d80da1dbd455d8bbc807011404 Mon Sep 17 00:00:00 2001 From: chrismclarke Date: Mon, 1 Feb 2021 06:38:07 -0800 Subject: [PATCH 109/112] update top meta --- maths-club-pack/content/student/diffy-squares.md | 2 +- maths-club-pack/content/student/kaprekars-number.md | 2 +- maths-club-pack/content/student/konigsberg-bridges.md | 2 +- maths-club-pack/content/student/tetris-shapes.md | 4 ++-- 4 files changed, 5 insertions(+), 5 deletions(-) diff --git a/maths-club-pack/content/student/diffy-squares.md b/maths-club-pack/content/student/diffy-squares.md index 9b5fdd8e..b2ba9289 100644 --- a/maths-club-pack/content/student/diffy-squares.md +++ b/maths-club-pack/content/student/diffy-squares.md @@ -1,7 +1,7 @@ --- title: Diffy Squares type: problem -printOrder: n/a +printOrder: 999 added: 2021-01 --- diff --git a/maths-club-pack/content/student/kaprekars-number.md b/maths-club-pack/content/student/kaprekars-number.md index 6cd779ab..e3c64ba8 100644 --- a/maths-club-pack/content/student/kaprekars-number.md +++ b/maths-club-pack/content/student/kaprekars-number.md @@ -1,7 +1,7 @@ --- title: Kaprekar's Number type: problem -printOrder: n/a +printOrder: 999 added: 2021-01 --- diff --git a/maths-club-pack/content/student/konigsberg-bridges.md b/maths-club-pack/content/student/konigsberg-bridges.md index bcff2f84..a4aeb20b 100644 --- a/maths-club-pack/content/student/konigsberg-bridges.md +++ b/maths-club-pack/content/student/konigsberg-bridges.md @@ -1,7 +1,7 @@ --- title: Bridges of Konigsberg type: problem -printOrder: n/a +printOrder: 999 added: 2021-01 --- diff --git a/maths-club-pack/content/student/tetris-shapes.md b/maths-club-pack/content/student/tetris-shapes.md index 1ddce730..d806525e 100644 --- a/maths-club-pack/content/student/tetris-shapes.md +++ b/maths-club-pack/content/student/tetris-shapes.md @@ -1,7 +1,7 @@ --- title: Tetris Shapes -type: problem/game -printOrder: n/a +type: problem +printOrder: 999 added: 2021-01 --- From 8ffbb75284758f5662faa03a0f949aacdf5c20b5 Mon Sep 17 00:00:00 2001 From: chrismclarke Date: Mon, 1 Feb 2021 06:38:24 -0800 Subject: [PATCH 110/112] run scripts to add translation version --- translations/en/facilitator/diffy-squares.md | 103 ++++++++++++++++++ .../en/facilitator/kaprekars-number.md | 54 +++++++++ .../en/facilitator/konigsberg-number.md | 41 +++++++ translations/en/facilitator/tetris-shapes.md | 50 +++++++++ translations/en/student/diffy-squares.md | 30 +++++ translations/en/student/kaprekars-number.md | 19 ++++ translations/en/student/konigsberg-bridges.md | 13 +++ translations/en/student/tetris-shapes.md | 16 +++ 8 files changed, 326 insertions(+) create mode 100644 translations/en/facilitator/diffy-squares.md create mode 100644 translations/en/facilitator/kaprekars-number.md create mode 100644 translations/en/facilitator/konigsberg-number.md create mode 100644 translations/en/facilitator/tetris-shapes.md create mode 100644 translations/en/student/diffy-squares.md create mode 100644 translations/en/student/kaprekars-number.md create mode 100644 translations/en/student/konigsberg-bridges.md create mode 100644 translations/en/student/tetris-shapes.md diff --git a/translations/en/facilitator/diffy-squares.md b/translations/en/facilitator/diffy-squares.md new file mode 100644 index 00000000..dcb32bac --- /dev/null +++ b/translations/en/facilitator/diffy-squares.md @@ -0,0 +1,103 @@ +# Diffy Squares + +## Introduction + +This problem, at the simplest level, is a great way to practice subtraction and, for higher level students, can reveal some interesting mathematics. + +With regards to the question asked, it is quite difficult to find particularly long seqences and not at all obvious when they will occur. In the solutions section, you fill find an example of the longest sequence possible under certain assumptions for the starting numbers. However, allowing any starting numbers to be used, this challenge can be as difficult as you want it to be. + +It is technically possible to choose four numbers that make an infinite sequence where you never end up at $0, 0, 0, 0$. However, demonstrating this is tricky. [Here](https://mathforlove.com/2011/10/squares-of-difference-iii-a-surprising-solution/) is a link to a blog discussing the problem. + +## Solution + +Using just single-digit integers, the longest sequence length possible is 7. +Here is an example of such a sequence (using the column notation): + +$0, 1, 4, 9$ +$1, 3, 5, 9$ +$2, 2, 4, 8$ +$0, 2, 4, 6$ +$2, 2, 2, 6$ +$0, 0, 4, 4$ +$0, 4, 0, 4$ +$4, 4, 4, 4$ +$0, 0, 0, 0$ + +It is interesting to point out that the same four starting numbers will not always give the same length of sequence. Taking $0, 4, 1, 9$ as the starting line for example: + +$0, 4, 1, 9$ +$4, 3, 8, 9$ +$1, 5, 1, 5$ +$4, 4, 4, 4$ +$0, 0, 0, 0$ + +We managed to make this sequence quite a bit shorter just by swapping the position of the $1$ and the $4$. You might like to think about how many different lengths of sequence it is possible to get from the same four starting numbers (see Extention section for solution). + +If we allow any integers less that 100 to be used, the longest sequence length possible is 12. +Here is an example of such a sequence: + +$0, 7, 20, 44$ +$7, 13, 24, 44$ +$6, 11, 20, 37$ +$5, 9, 17, 31$ +$4, 8, 14, 26$ +$4, 6, 12, 22$ +$2, 6, 10, 18$ +$4, 4, 8, 16$ +$0, 4, 8, 12$ +$4, 4, 4, 12$ +$0, 0, 8, 8$ +$0, 8, 0, 0$ +$8, 8, 8, 8$ +$0, 0, 0, 0$ + +Introducing fractions is an effective way to increase the length of sequences which we can obtain. Using fractions we can find sequences with numbers less that 10 which are longer than the longest sequence possible using integers up to 100. +Here is an example: + +$0, \frac{17}{12}, 4, \frac{35}{4}$ +$\frac{17}{12}, \frac{31}{12}, \frac{19}{4}, \frac{35}{4}$ +$\frac{7}{6}, \frac{13}{6}, 4, \frac{22}{3}$ +$1, \frac{11}{6}, \frac{10}{3}, \frac{37}{6}$ +$\frac{5}{6}, \frac{3}{2}, \frac{17}{6}, \frac{31}{6}$ +$\frac{2}{3}, \frac{4}{3}, \frac{7}{3}, \frac{13}{3}$ +$\frac{1}{3}, 1, 2, \frac{11}{3}$ +$\frac{1}{3}, 1, \frac{5}{3}, 3$ +$\frac{2}{3}, \frac{2}{3}, \frac{4}{3}, \frac{8}{3}$ +$0, \frac{2}{3}, \frac{4}{3}, 2$ +$\frac{2}{3}, \frac{2}{3}, \frac{2}{3}, 2$ +$0, 0, \frac{4}{3}, \frac{4}{3}$ +$0, \frac{4}{3}, 0, \frac{4}{3}$ +$\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}$ +$0, 0, 0, 0$ + +(In particular, 13 is the longest a sequence can be using numbers which are integer multiples of $\frac{1}{12}$ and less than 10, above is such an example) + +## Extension + +How many sequences of different length are possible from the same four starting numbers? + +If we take a general starting square, using the numbers $a, b, c, d$. + +Say that if we start with line $(a, b, c, d)$, this means that $a$ is in the top-left corner, $b$ is in the top-right, $c$ the bottom-right and $d$ the bottom-left. + +By rotating the square, we will not change the length of the sequence generated, and we can see therefore that the starting lines $(b, c, d, a)$, $(c, d, a, b)$ and $(d, a, b, c)$ will give sequences of the same length as that given by $(a, b, c, d)$. + + + +We could also reflect the square in any of its four lines of symmetry without changing the length of the resulting square. So the starting lines $(a, d, c, b)$, $(d, c, b, a)$, $(c, b, a, d)$ and $(b, a, d, c)$ will all give sequences of the same length as $(a, b, c, d)$. + + + +So for each starting line $(a, b, c, d)$ there are 8 lines in total using the numbers $a$, $b$, $c$ and $d$ which give sequences of the same length. + +But how many starting lines are possible in total from the numbers $a$, $b$, $c$ and $d$? + +If we choose the number in the top-left corner first, we have four choices $a$, $b$, $c$ or $d$. Next, if we choose the number in the top-right we have a choice of the three remaining number. Likewise, for the bottom-right, we have a choice from two and the bottom-left must be the one remaining number. + +So, the total number of possible lines is $4 \times 3 \times 2 \times 1 = 24$ + +But since we know that these lines come in groups of 8, all of which have the same length. The largest number of different lengths possible must be $\frac{24}{3}$ which is 3. + +Another way to see this would be to fix the position of $a$ to the top-left, which we can do since rotation does not change the length of the sequence as we have seen. We can then ask, of the remaining numbers $b$, $c$ and $d$, how many different ways can we choose the two numbers which $a$ will share an edge with? + +The answer is 3. Either $b$ and $c$, $b$ and $d$ or $c$ and $d$. diff --git a/translations/en/facilitator/kaprekars-number.md b/translations/en/facilitator/kaprekars-number.md new file mode 100644 index 00000000..fe1f99eb --- /dev/null +++ b/translations/en/facilitator/kaprekars-number.md @@ -0,0 +1,54 @@ +# Kaprekar's Number + +## Introduction + +This problem is a good way to practice subtraction, it also demonstrates an interesting result. + +After trying a few examples, it becomes quite apparent that all four-digit numbers seem to lead to the same number under repeated application of this opperation, namely 6174 or *Kaprekar's Number*. + +Students will notice a similar pattern for three-digit numbers, with the resulting number being 495. They may also be able to demonstrate that this will happen for any three-digit number (see Extension section), although this is a difficult task for the four-digit problem. + +## Solution + +Starting with 4252, as in the question, and applying the operation defined in the question (known as the *Kaprekar Mapping*) the process would be as follows: + +$5422 - 2245 = 3177$ +$7731 - 1377 = 6354$ +$6543 - 3456 = 3087$ +$8730 - 0378 = 8352$ +$8532 - 2358 = 6174$ +$7641 - 1467 = 6174$ +... + +So we can see that the sequence eventually reaches 6174 at which point, if we continued, it would repeat 6174 indefinately. + +## Extension + +For three-digit numbers, we can demonstrate that repeated application of the Kaprekar mapping will always lead to 495. + +Consider a general three-digit number $*abc* = 100a + 10b + c$ +where $9 \geq a \geq b \geq c \geq 0$ + +Applying the Kaprekar mapping gives the second number in our sequence: + +$(100a + 10b + c) - (100c + 10b + a) = 100(a - c) + (c - a)$ + +We can simplify this to: $99(a - c)$ + +So, after one application of the Kaprekar mapping, the only possible results are the multiples of 99: + +099, 198, 297, 396, 495, 594, 693, 792, 891, 990 + +In fact, each of these numbers uses the same digits as exactly one other, and will give the same result under the Kaprekar mapping. So, we only need to consider how half of them will be changed by a second application. + +$990 - 099 = 891$ +$981 - 189 = 792$ +$972 - 279 = 693$ +$963 - 369 = 594$ +$954 - 459 = 495$ + +We could express this information as a flow diagram: + + + +After one application of the Kaprekar mapping, the sequence will be one of the numbers on the flow diagram and then will follow it to reach 495. In total, any number will reach 495 in a maximum of 5 applications of the Kaprekar mapping. For example, 545 will require 5 applications. diff --git a/translations/en/facilitator/konigsberg-number.md b/translations/en/facilitator/konigsberg-number.md new file mode 100644 index 00000000..2fe823ad --- /dev/null +++ b/translations/en/facilitator/konigsberg-number.md @@ -0,0 +1,41 @@ +# Bridges of Konigsberg + +## Introduction + +This is a classic mathematics problem. Its resolution in 1736 by Leonhard Euler is considered a foundation of the mathematical field of graph theory. + +Without knowing the problem, it is initially quite enjoyable to try any find a solution. However, there is in fact no way to cross every bridge once, as Euler showed (see Solution section). + +You could say that the ‘solution’ to this problem is to wait for the second world war, where two of the bridges were destroyed, making it possible to find a route crossing each remaining bridge once. + +## Solution + +As mentioned, there is no solution to the problem as it is outlined in the question. To demonstrate this, it is helpful to reduce the situation to a mathematical structure called a *graph* (a different type of graph to what you are probably used to). + + + +We have replaced the sections of land with circles and the bridges with connecting lines. We usually refer to these circles as *vertices*, and the lines as *edges*. + +Say we start our walk on the middle vertex and walk to the top one, and then to the vertex on the right. Our path might look like this: + + + +Notice how, since we have walked both to and from the top vertex, we have used two of the edges connected to it. Similarly, once we leave the rightmost vertex, we will use another edge and so will have removed two of its connected edges: + + + +We can see that, as we walk, we always use pairs of edges connected to a particular vertex. The only time that this is not true is for the vertices that we begin and end on. + +Because of this, after walking we will always have used an even number of edges connected to a given vertex (unless they are one that we start or end on). So, if we want to remove all the edges, the vertices in the graph must have an even number of edges connected to them. The only two vertices that can have an odd number of edges are the starting and ending ones. + +This was Euler's discovery, that to be able to pass through all the edges in a graph exactly once, there can be no more than two vertices which have an odd number of edges connected to them. + +In the case of Konigsberg, three of the vertices have 3 edges and one (the central island) has 5. Since all four of the vertices have an odd number of edges, it is impossible to find a path which passes over each edge exactly once. + +## Extension + +Test this result by creating other graphs and trying to find a route passing over each edge once, so-called *Eulerian paths*. See if you can predict whether a given graph will have a Eulerian path. + +Where a Eulerian path is possible, do you notice anything else about the number of vertices which can have an odd number of edges? Try finding a path for a graph which only has one vertex with an odd number of edges, if you cannot, can you see why? + +*Hint: we said that only the starting and ending vertices can have an odd number of connected edges, what happens in the two possible cases where these are different vertices and where these are the same vertex?* diff --git a/translations/en/facilitator/tetris-shapes.md b/translations/en/facilitator/tetris-shapes.md new file mode 100644 index 00000000..ac187110 --- /dev/null +++ b/translations/en/facilitator/tetris-shapes.md @@ -0,0 +1,50 @@ +# Tetris Shapes + +## Introduction + +This creative puzzle is a good exercise in spatial visualtisation and deduction. For most of the +shapes, there is a clear starting point where a certain space can only be filled by one of the +blocks, and a solution can then be logically worked through. + +It should be noted that the choice of blocks is arbitrary and only done in reference to the +game ‘Tetris’. The essence of this puzzle can be maintained using blocks of smaller or larger +size, depending on the age and ability of students. + +## Solution + +The shape which cannot be made by fitting together the blocks given is the Star. It may be +interesting for students to try and reason this logically (see Extension). + +The other four solutions, for the Tree, Boat, Spaceship, and Flower, are as follows: + + + +Note that there may be other solutions not given here, any which uses each block once is equally valid. + +## Extension + +In the case of the star, we can reason as to why is cannot be created from the blocks given: + + + +Firstly, looking at the bottom two points, there is no way to fill this space other than to use +the **S** and **Z** blocks. + +If we fill these bottom points, we can consider the remaining three, which have a similar +three-square line structure. We cannot fit the **O** or **T** blocks in these spaces, so we need to +use the **I**, **J** and **L** blocks to fill these three points. + +We could attempt to do this in a few ways: + +Firstly, we could fill the top point with the **I** block and use the **J** and **L** blocks in the two side points (left on the diagram). At this point, we will not be able to fit the **O** block in the remaining space, so this way will not work. + +If we instead fill the top point with either the **J** or **L** block and use the remaining one to fill the side points, along with the **I** block, we will get a situation like on the right of the +diagram. Notice that, had we used the **L** block at the top, rather that the **J**, this would have +given a situation which is a reflection of the case shown in the diagram. In either case, there +is only one space where we can fit the **O** block and after this the remaining space will not fit +the **T** block, so this way will also not work. + +Therefore, we have exhausted all possibilities and can conclude that the star shape is +impossible to make from the blocks given. + + diff --git a/translations/en/student/diffy-squares.md b/translations/en/student/diffy-squares.md new file mode 100644 index 00000000..db4be8cb --- /dev/null +++ b/translations/en/student/diffy-squares.md @@ -0,0 +1,30 @@ +# Diffy Squares + +Create a Diffy Square by following this process: + +1. Draw a square and choose four numbers to place on each vertex. + +2. Along each edge, write the positive difference of the numbers at the vertices that the edge connects. Join these four numbers up to form a new square. + +3. Now repeat this process with the new square. Find the positive difference between each pair of numbers which share an edge, write these differences on the edge and connect up the new numbers. + +4. At some point, you will reach a square with four zeros on the edges. At this point, you should stop. + +5. That's it. You should now count the number of squares which you drew, not including the largest square on the outside. This is the *length* of the sequence of squares. + +Here is an example, starting with the numbers 7, 15, 9 and 13: + + + +We drew two smaller squares, so this sequence has length 2. + +Try following this process with different starting numbers. + +What is the longest sequence of squares that you can find? + + +Tip: For longer sequences, it may be easier to write your sequences as a column of numbers, where each row represents a square. + +For our example of length 2, we could have written: + + diff --git a/translations/en/student/kaprekars-number.md b/translations/en/student/kaprekars-number.md new file mode 100644 index 00000000..0b7c675b --- /dev/null +++ b/translations/en/student/kaprekars-number.md @@ -0,0 +1,19 @@ +# Kaprekar's Number + +Choose a four digit number where the digits are not all the same (so not 1111, 2222, ...). + +Rearrange the digits to get the largest and smallest numbers that these digits can make. + +Subtract the smallest number from the largest to get a new number, and carry on repeating this operation for each new number. + +For example, starting with 4252: + +5422 - 2245 = 3177 +7731 - 1377 = 6354 +6543 - 3456 = 3087 +... + +What do you notice as you continue this process? +What happens when you use different starting numbers? + +Does something similar happen for three digit numbers? diff --git a/translations/en/student/konigsberg-bridges.md b/translations/en/student/konigsberg-bridges.md new file mode 100644 index 00000000..bb2b89f9 --- /dev/null +++ b/translations/en/student/konigsberg-bridges.md @@ -0,0 +1,13 @@ +# Bridges of Konigsberg + +The Prussian city of Konigsberg (now Kaliningrad, Russia) is divided into four landmasses by the river Pregel. + + + +You can see that there are seven bridges connecting these sections of the city. + +Can you find a way to walk through the city crossing each bridge exactly once? + +Do you think that it is possible? + +Assume that there are no other ways to cross the river than by walking over one of these bridges. You may begin and end anywhere in the city. diff --git a/translations/en/student/tetris-shapes.md b/translations/en/student/tetris-shapes.md new file mode 100644 index 00000000..ea68d932 --- /dev/null +++ b/translations/en/student/tetris-shapes.md @@ -0,0 +1,16 @@ +# Tetris Shapes + +Using these seven blocks (from the arcade game *Tetris*), can you create the shapes at the bottom of this page? + + + +You are allowed to rotate the blocks but not to reflect them, so you cannot change the 'L-block' into a 'J-block' by flipping it over. + +Four of these five shapes can be created using exactly one of each block. +(left to right: Tree, Boat, Spaceship, Star, Flower) + + + +Can you spot which one is not possible? + +What other shapes can you make using these blocks? From 473d5b686f4a1e66681f76a28d9ed2e9f2001a4d Mon Sep 17 00:00:00 2001 From: chrismclarke Date: Mon, 1 Feb 2021 07:09:21 -0800 Subject: [PATCH 111/112] fix notes --- .../content/facilitator/diffy-squares.md | 136 ++++++++++-------- translations/en/facilitator/diffy-squares.md | 136 ++++++++++-------- 2 files changed, 158 insertions(+), 114 deletions(-) diff --git a/maths-club-pack/content/facilitator/diffy-squares.md b/maths-club-pack/content/facilitator/diffy-squares.md index 29e3f12e..10a1ee2d 100644 --- a/maths-club-pack/content/facilitator/diffy-squares.md +++ b/maths-club-pack/content/facilitator/diffy-squares.md @@ -6,69 +6,91 @@ This problem, at the simplest level, is a great way to practice subtraction and, With regards to the question asked, it is quite difficult to find particularly long seqences and not at all obvious when they will occur. In the solutions section, you fill find an example of the longest sequence possible under certain assumptions for the starting numbers. However, allowing any starting numbers to be used, this challenge can be as difficult as you want it to be. -It is technically possible to choose four numbers that make an infinite sequence where you never end up at $0, 0, 0, 0$. However, demonstrating this is tricky. [Here](https://mathforlove.com/2011/10/squares-of-difference-iii-a-surprising-solution/) is a link to a blog discussing the problem. +It is technically possible to choose four numbers that make an infinite sequence where you never end up at `0, 0, 0, 0`. However, demonstrating this is tricky. [Here](https://mathforlove.com/2011/10/squares-of-difference-iii-a-surprising-solution/) is a link to a blog discussing the problem. ## Solution Using just single-digit integers, the longest sequence length possible is 7. Here is an example of such a sequence (using the column notation): -$0, 1, 4, 9$ -$1, 3, 5, 9$ -$2, 2, 4, 8$ -$0, 2, 4, 6$ -$2, 2, 2, 6$ -$0, 0, 4, 4$ -$0, 4, 0, 4$ -$4, 4, 4, 4$ -$0, 0, 0, 0$ - -It is interesting to point out that the same four starting numbers will not always give the same length of sequence. Taking $0, 4, 1, 9$ as the starting line for example: - -$0, 4, 1, 9$ -$4, 3, 8, 9$ -$1, 5, 1, 5$ -$4, 4, 4, 4$ -$0, 0, 0, 0$ +``` +0, 1, 4, 9 +1, 3, 5, 9 +2, 2, 4, 8 +0, 2, 4, 6 +2, 2, 2, 6 +0, 0, 4, 4 +0, 4, 0, 4 +4, 4, 4, 4 +0, 0, 0, 0 +``` + +It is interesting to point out that the same four starting numbers will not always give the same length of sequence. Taking `0, 4, 1, 9` as the starting line for example: + +``` +0, 4, 1, 9 +4, 3, 8, 9 +1, 5, 1, 5 +4, 4, 4, 4 +0, 0, 0, 0 +``` We managed to make this sequence quite a bit shorter just by swapping the position of the $1$ and the $4$. You might like to think about how many different lengths of sequence it is possible to get from the same four starting numbers (see Extention section for solution). If we allow any integers less that 100 to be used, the longest sequence length possible is 12. Here is an example of such a sequence: -$0, 7, 20, 44$ -$7, 13, 24, 44$ -$6, 11, 20, 37$ -$5, 9, 17, 31$ -$4, 8, 14, 26$ -$4, 6, 12, 22$ -$2, 6, 10, 18$ -$4, 4, 8, 16$ -$0, 4, 8, 12$ -$4, 4, 4, 12$ -$0, 0, 8, 8$ -$0, 8, 0, 0$ -$8, 8, 8, 8$ -$0, 0, 0, 0$ +``` +0, 7, 20, 44 +7, 13, 24, 44 +6, 11, 20, 37 +5, 9, 17, 31 +4, 8, 14, 26 +4, 6, 12, 22 +2, 6, 10, 18 +4, 4, 8, 16 +0, 4, 8, 12 +4, 4, 4, 12 +0, 0, 8, 8 +0, 8, 0, 0 +8, 8, 8, 8 +0, 0, 0, 0 +``` Introducing fractions is an effective way to increase the length of sequences which we can obtain. Using fractions we can find sequences with numbers less that 10 which are longer than the longest sequence possible using integers up to 100. Here is an example: -$0, \frac{17}{12}, 4, \frac{35}{4}$ -$\frac{17}{12}, \frac{31}{12}, \frac{19}{4}, \frac{35}{4}$ -$\frac{7}{6}, \frac{13}{6}, 4, \frac{22}{3}$ -$1, \frac{11}{6}, \frac{10}{3}, \frac{37}{6}$ -$\frac{5}{6}, \frac{3}{2}, \frac{17}{6}, \frac{31}{6}$ -$\frac{2}{3}, \frac{4}{3}, \frac{7}{3}, \frac{13}{3}$ -$\frac{1}{3}, 1, 2, \frac{11}{3}$ -$\frac{1}{3}, 1, \frac{5}{3}, 3$ -$\frac{2}{3}, \frac{2}{3}, \frac{4}{3}, \frac{8}{3}$ -$0, \frac{2}{3}, \frac{4}{3}, 2$ -$\frac{2}{3}, \frac{2}{3}, \frac{2}{3}, 2$ -$0, 0, \frac{4}{3}, \frac{4}{3}$ -$0, \frac{4}{3}, 0, \frac{4}{3}$ -$\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}$ -$0, 0, 0, 0$ +``` +0, $\frac{17}{12}$, 4, $\frac{35}{4}$ + +$\frac{17}{12}$, $\frac{31}{12}$, $\frac{19}{4}$, $\frac{35}{4}$ + +$\frac{7}{6}$, $\frac{13}{6}$, 4, $\frac{22}{3}$ + +1, $\frac{11}{6}$, $\frac{10}{3}$, $\frac{37}{6}$ + +$\frac{5}{6}$, $\frac{3}{2}$, $\frac{17}{6}$, $\frac{31}{6}$ + +$\frac{2}{3}$, $\frac{4}{3}$, $\frac{7}{3}$, $\frac{13}{3}$ + +$\frac{1}{3}$, 1, 2, $\frac{11}{3}$ + +$\frac{1}{3}$, 1, $\frac{5}{3}$, 3 + +$\frac{2}{3}$, $\frac{2}{3}$, $\frac{4}{3}$, $\frac{8}{3}$ + +0, $\frac{2}{3}$, $\frac{4}{3}$, 2 + +$\frac{2}{3}$, $\frac{2}{3}$, $\frac{2}{3}$, 2 + +0, 0, $\frac{4}{3}$, $\frac{4}{3}$ + +0, $\frac{4}{3}$, 0, $\frac{4}{3}$ + +$\frac{4}{3}$, $\frac{4}{3}$, $\frac{4}{3}$, $\frac{4}{3}$ + +0, 0, 0, 0 +``` (In particular, 13 is the longest a sequence can be using numbers which are integer multiples of $\frac{1}{12}$ and less than 10, above is such an example) @@ -76,28 +98,28 @@ $0, 0, 0, 0$ How many sequences of different length are possible from the same four starting numbers? -If we take a general starting square, using the numbers $a, b, c, d$. +If we take a general starting square, using the numbers `a, b, c, d`. -Say that if we start with line $(a, b, c, d)$, this means that $a$ is in the top-left corner, $b$ is in the top-right, $c$ the bottom-right and $d$ the bottom-left. +Say that if we start with line `(a, b, c, d)`, this means that `a` is in the top-left corner, `b` is in the top-right, `c` the bottom-right and `d` the bottom-left. -By rotating the square, we will not change the length of the sequence generated, and we can see therefore that the starting lines $(b, c, d, a)$, $(c, d, a, b)$ and $(d, a, b, c)$ will give sequences of the same length as that given by $(a, b, c, d)$. +By rotating the square, we will not change the length of the sequence generated, and we can see therefore that the starting lines `(b, c, d, a)`, `(c, d, a, b)` and `(d, a, b, c)` will give sequences of the same length as that given by `(a, b, c, d)`. -We could also reflect the square in any of its four lines of symmetry without changing the length of the resulting square. So the starting lines $(a, d, c, b)$, $(d, c, b, a)$, $(c, b, a, d)$ and $(b, a, d, c)$ will all give sequences of the same length as $(a, b, c, d)$. +We could also reflect the square in any of its four lines of symmetry without changing the length of the resulting square. So the starting lines `(a, d, c, b)`, `(d, c, b, a)`, `(c, b, a, d)` and `(b, a, d, c)` will all give sequences of the same length as `(a, b, c, d)`. -So for each starting line $(a, b, c, d)$ there are 8 lines in total using the numbers $a$, $b$, $c$ and $d$ which give sequences of the same length. +So for each starting line `(a, b, c, d)` there are 8 lines in total using the numbers `a`, `b`, `c` and `d` which give sequences of the same length. -But how many starting lines are possible in total from the numbers $a$, $b$, $c$ and $d$? +But how many starting lines are possible in total from the numbers `a`, `b`, `c` and `d`? -If we choose the number in the top-left corner first, we have four choices $a$, $b$, $c$ or $d$. Next, if we choose the number in the top-right we have a choice of the three remaining number. Likewise, for the bottom-right, we have a choice from two and the bottom-left must be the one remaining number. +If we choose the number in the top-left corner first, we have four choices `a`, `b`, `c` or `d`. Next, if we choose the number in the top-right we have a choice of the three remaining number. Likewise, for the bottom-right, we have a choice from two and the bottom-left must be the one remaining number. -So, the total number of possible lines is $4 \times 3 \times 2 \times 1 = 24$ +So, the total number of possible lines is `4 x 3 x 2 x 1 = 24` But since we know that these lines come in groups of 8, all of which have the same length. The largest number of different lengths possible must be $\frac{24}{3}$ which is 3. -Another way to see this would be to fix the position of $a$ to the top-left, which we can do since rotation does not change the length of the sequence as we have seen. We can then ask, of the remaining numbers $b$, $c$ and $d$, how many different ways can we choose the two numbers which $a$ will share an edge with? +Another way to see this would be to fix the position of `a` to the top-left, which we can do since rotation does not change the length of the sequence as we have seen. We can then ask, of the remaining numbers `b`, `c` and `d`, how many different ways can we choose the two numbers which `a` will share an edge with? -The answer is 3. Either $b$ and $c$, $b$ and $d$ or $c$ and $d$. +The answer is 3. Either `b` and `c`, `b` and `d` or `c` and `d`. diff --git a/translations/en/facilitator/diffy-squares.md b/translations/en/facilitator/diffy-squares.md index dcb32bac..3125a658 100644 --- a/translations/en/facilitator/diffy-squares.md +++ b/translations/en/facilitator/diffy-squares.md @@ -6,69 +6,91 @@ This problem, at the simplest level, is a great way to practice subtraction and, With regards to the question asked, it is quite difficult to find particularly long seqences and not at all obvious when they will occur. In the solutions section, you fill find an example of the longest sequence possible under certain assumptions for the starting numbers. However, allowing any starting numbers to be used, this challenge can be as difficult as you want it to be. -It is technically possible to choose four numbers that make an infinite sequence where you never end up at $0, 0, 0, 0$. However, demonstrating this is tricky. [Here](https://mathforlove.com/2011/10/squares-of-difference-iii-a-surprising-solution/) is a link to a blog discussing the problem. +It is technically possible to choose four numbers that make an infinite sequence where you never end up at `0, 0, 0, 0`. However, demonstrating this is tricky. [Here](https://mathforlove.com/2011/10/squares-of-difference-iii-a-surprising-solution/) is a link to a blog discussing the problem. ## Solution Using just single-digit integers, the longest sequence length possible is 7. Here is an example of such a sequence (using the column notation): -$0, 1, 4, 9$ -$1, 3, 5, 9$ -$2, 2, 4, 8$ -$0, 2, 4, 6$ -$2, 2, 2, 6$ -$0, 0, 4, 4$ -$0, 4, 0, 4$ -$4, 4, 4, 4$ -$0, 0, 0, 0$ - -It is interesting to point out that the same four starting numbers will not always give the same length of sequence. Taking $0, 4, 1, 9$ as the starting line for example: - -$0, 4, 1, 9$ -$4, 3, 8, 9$ -$1, 5, 1, 5$ -$4, 4, 4, 4$ -$0, 0, 0, 0$ +``` +0, 1, 4, 9 +1, 3, 5, 9 +2, 2, 4, 8 +0, 2, 4, 6 +2, 2, 2, 6 +0, 0, 4, 4 +0, 4, 0, 4 +4, 4, 4, 4 +0, 0, 0, 0 +``` + +It is interesting to point out that the same four starting numbers will not always give the same length of sequence. Taking `0, 4, 1, 9` as the starting line for example: + +``` +0, 4, 1, 9 +4, 3, 8, 9 +1, 5, 1, 5 +4, 4, 4, 4 +0, 0, 0, 0 +``` We managed to make this sequence quite a bit shorter just by swapping the position of the $1$ and the $4$. You might like to think about how many different lengths of sequence it is possible to get from the same four starting numbers (see Extention section for solution). If we allow any integers less that 100 to be used, the longest sequence length possible is 12. Here is an example of such a sequence: -$0, 7, 20, 44$ -$7, 13, 24, 44$ -$6, 11, 20, 37$ -$5, 9, 17, 31$ -$4, 8, 14, 26$ -$4, 6, 12, 22$ -$2, 6, 10, 18$ -$4, 4, 8, 16$ -$0, 4, 8, 12$ -$4, 4, 4, 12$ -$0, 0, 8, 8$ -$0, 8, 0, 0$ -$8, 8, 8, 8$ -$0, 0, 0, 0$ +``` +0, 7, 20, 44 +7, 13, 24, 44 +6, 11, 20, 37 +5, 9, 17, 31 +4, 8, 14, 26 +4, 6, 12, 22 +2, 6, 10, 18 +4, 4, 8, 16 +0, 4, 8, 12 +4, 4, 4, 12 +0, 0, 8, 8 +0, 8, 0, 0 +8, 8, 8, 8 +0, 0, 0, 0 +``` Introducing fractions is an effective way to increase the length of sequences which we can obtain. Using fractions we can find sequences with numbers less that 10 which are longer than the longest sequence possible using integers up to 100. Here is an example: -$0, \frac{17}{12}, 4, \frac{35}{4}$ -$\frac{17}{12}, \frac{31}{12}, \frac{19}{4}, \frac{35}{4}$ -$\frac{7}{6}, \frac{13}{6}, 4, \frac{22}{3}$ -$1, \frac{11}{6}, \frac{10}{3}, \frac{37}{6}$ -$\frac{5}{6}, \frac{3}{2}, \frac{17}{6}, \frac{31}{6}$ -$\frac{2}{3}, \frac{4}{3}, \frac{7}{3}, \frac{13}{3}$ -$\frac{1}{3}, 1, 2, \frac{11}{3}$ -$\frac{1}{3}, 1, \frac{5}{3}, 3$ -$\frac{2}{3}, \frac{2}{3}, \frac{4}{3}, \frac{8}{3}$ -$0, \frac{2}{3}, \frac{4}{3}, 2$ -$\frac{2}{3}, \frac{2}{3}, \frac{2}{3}, 2$ -$0, 0, \frac{4}{3}, \frac{4}{3}$ -$0, \frac{4}{3}, 0, \frac{4}{3}$ -$\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}$ -$0, 0, 0, 0$ +``` +0, $\frac{17}{12}$, 4, $\frac{35}{4}$ + +$\frac{17}{12}$, $\frac{31}{12}$, $\frac{19}{4}$, $\frac{35}{4}$ + +$\frac{7}{6}$, $\frac{13}{6}$, 4, $\frac{22}{3}$ + +1, $\frac{11}{6}$, $\frac{10}{3}$, $\frac{37}{6}$ + +$\frac{5}{6}$, $\frac{3}{2}$, $\frac{17}{6}$, $\frac{31}{6}$ + +$\frac{2}{3}$, $\frac{4}{3}$, $\frac{7}{3}$, $\frac{13}{3}$ + +$\frac{1}{3}$, 1, 2, $\frac{11}{3}$ + +$\frac{1}{3}$, 1, $\frac{5}{3}$, 3 + +$\frac{2}{3}$, $\frac{2}{3}$, $\frac{4}{3}$, $\frac{8}{3}$ + +0, $\frac{2}{3}$, $\frac{4}{3}$, 2 + +$\frac{2}{3}$, $\frac{2}{3}$, $\frac{2}{3}$, 2 + +0, 0, $\frac{4}{3}$, $\frac{4}{3}$ + +0, $\frac{4}{3}$, 0, $\frac{4}{3}$ + +$\frac{4}{3}$, $\frac{4}{3}$, $\frac{4}{3}$, $\frac{4}{3}$ + +0, 0, 0, 0 +``` (In particular, 13 is the longest a sequence can be using numbers which are integer multiples of $\frac{1}{12}$ and less than 10, above is such an example) @@ -76,28 +98,28 @@ $0, 0, 0, 0$ How many sequences of different length are possible from the same four starting numbers? -If we take a general starting square, using the numbers $a, b, c, d$. +If we take a general starting square, using the numbers `a, b, c, d`. -Say that if we start with line $(a, b, c, d)$, this means that $a$ is in the top-left corner, $b$ is in the top-right, $c$ the bottom-right and $d$ the bottom-left. +Say that if we start with line `(a, b, c, d)`, this means that `a` is in the top-left corner, `b` is in the top-right, `c` the bottom-right and `d` the bottom-left. -By rotating the square, we will not change the length of the sequence generated, and we can see therefore that the starting lines $(b, c, d, a)$, $(c, d, a, b)$ and $(d, a, b, c)$ will give sequences of the same length as that given by $(a, b, c, d)$. +By rotating the square, we will not change the length of the sequence generated, and we can see therefore that the starting lines `(b, c, d, a)`, `(c, d, a, b)` and `(d, a, b, c)` will give sequences of the same length as that given by `(a, b, c, d)`. -We could also reflect the square in any of its four lines of symmetry without changing the length of the resulting square. So the starting lines $(a, d, c, b)$, $(d, c, b, a)$, $(c, b, a, d)$ and $(b, a, d, c)$ will all give sequences of the same length as $(a, b, c, d)$. +We could also reflect the square in any of its four lines of symmetry without changing the length of the resulting square. So the starting lines `(a, d, c, b)`, `(d, c, b, a)`, `(c, b, a, d)` and `(b, a, d, c)` will all give sequences of the same length as `(a, b, c, d)`. -So for each starting line $(a, b, c, d)$ there are 8 lines in total using the numbers $a$, $b$, $c$ and $d$ which give sequences of the same length. +So for each starting line `(a, b, c, d)` there are 8 lines in total using the numbers `a`, `b`, `c` and `d` which give sequences of the same length. -But how many starting lines are possible in total from the numbers $a$, $b$, $c$ and $d$? +But how many starting lines are possible in total from the numbers `a`, `b`, `c` and `d`? -If we choose the number in the top-left corner first, we have four choices $a$, $b$, $c$ or $d$. Next, if we choose the number in the top-right we have a choice of the three remaining number. Likewise, for the bottom-right, we have a choice from two and the bottom-left must be the one remaining number. +If we choose the number in the top-left corner first, we have four choices `a`, `b`, `c` or `d`. Next, if we choose the number in the top-right we have a choice of the three remaining number. Likewise, for the bottom-right, we have a choice from two and the bottom-left must be the one remaining number. -So, the total number of possible lines is $4 \times 3 \times 2 \times 1 = 24$ +So, the total number of possible lines is `4 x 3 x 2 x 1 = 24` But since we know that these lines come in groups of 8, all of which have the same length. The largest number of different lengths possible must be $\frac{24}{3}$ which is 3. -Another way to see this would be to fix the position of $a$ to the top-left, which we can do since rotation does not change the length of the sequence as we have seen. We can then ask, of the remaining numbers $b$, $c$ and $d$, how many different ways can we choose the two numbers which $a$ will share an edge with? +Another way to see this would be to fix the position of `a` to the top-left, which we can do since rotation does not change the length of the sequence as we have seen. We can then ask, of the remaining numbers `b`, `c` and `d`, how many different ways can we choose the two numbers which `a` will share an edge with? -The answer is 3. Either $b$ and $c$, $b$ and $d$ or $c$ and $d$. +The answer is 3. Either `b` and `c`, `b` and `d` or `c` and `d`. From 4803a0151c7e994054dae6c6bbfc83c29d04e6a9 Mon Sep 17 00:00:00 2001 From: chrismclarke Date: Mon, 1 Feb 2021 07:17:33 -0800 Subject: [PATCH 112/112] update konigsberge bridges name --- maths-club-app/src/app/data/weeklyProblems.ts | 296 +++++++++--------- ...gsberg-number.md => konigsberg-bridges.md} | 0 .../content/student/konigsberg-bridges.md | 2 +- ...gsberg-number.md => konigsberg-bridges.md} | 0 4 files changed, 149 insertions(+), 149 deletions(-) rename maths-club-pack/content/facilitator/{konigsberg-number.md => konigsberg-bridges.md} (100%) rename translations/en/facilitator/{konigsberg-number.md => konigsberg-bridges.md} (100%) diff --git a/maths-club-app/src/app/data/weeklyProblems.ts b/maths-club-app/src/app/data/weeklyProblems.ts index c3d20bdb..72215112 100644 --- a/maths-club-app/src/app/data/weeklyProblems.ts +++ b/maths-club-app/src/app/data/weeklyProblems.ts @@ -51,152 +51,152 @@ export const WEEKLY_PROBLEMS = { "2021-08-08": "countdown", "2021-08-15": "15-game", "2021-08-22": "tetris-shapes", - "2021-08-29": "kaprekars-number", - "2021-09-05": "diffy-squares", - "2021-09-12": "pell-numbers", - "2021-09-19": "paths", - "2021-09-26": "number-challenge-1", - "2021-10-03": "making-squares", - "2021-10-10": "logic-puzzles-1", - "2021-10-17": "iq-challenge-1", - "2021-10-24": "find-the-path", - "2021-10-31": "counting-squares", - "2021-11-07": "buffons-needle", - "2021-11-14": "balls-and-books", - "2021-11-21": "addition-squares", - "2021-11-28": "adding-to-15", - "2021-12-05": "9-dots", - "2021-12-12": "treasure-hunt", - "2021-12-19": "russian-multiplication", - "2021-12-26": "river-crossing", - "2022-01-02": "poisoned-wine-puzzle", - "2022-01-09": "matchstick-puzzles", - "2022-01-16": "magic-cards", - "2022-01-23": "langfords-problem", - "2022-01-30": "handshake-puzzles", - "2022-02-06": "geometry-puzzle-1", - "2022-02-13": "gabriels-problem", - "2022-02-20": "fence-around-a-field", - "2022-02-27": "domino-tilings", - "2022-03-06": "collatz-conjecture", - "2022-03-13": "coin-game", - "2022-03-20": "coconut-trader", - "2022-03-27": "circumference", - "2022-04-03": "apple-teaser", - "2022-04-10": "two-eggs", - "2022-04-17": "special-numbers", - "2022-04-24": "scales-and-weights", - "2022-05-01": "picture-puzzles-1", - "2022-05-08": "monkey-business", - "2022-05-15": "locks-and-keys", - "2022-05-22": "frog-party", - "2022-05-29": "counting-chickens", - "2022-06-05": "tic-tac-toe", - "2022-06-12": "tic-tac-toe-with-levels", - "2022-06-19": "tangrams", - "2022-06-26": "secret-santa", - "2022-07-03": "patience", - "2022-07-10": "paper-pieces", - "2022-07-17": "nim-related-games", - "2022-07-24": "nim-21", - "2022-07-31": "mastermind", - "2022-08-07": "make-many", - "2022-08-14": "letter-matching", - "2022-08-21": "countdown", - "2022-08-28": "15-game", - "2022-09-04": "tetris-shapes", - "2022-09-11": "kaprekars-number", - "2022-09-18": "diffy-squares", - "2022-09-25": "pell-numbers", - "2022-10-02": "paths", - "2022-10-09": "number-challenge-1", - "2022-10-16": "making-squares", - "2022-10-23": "logic-puzzles-1", - "2022-10-30": "iq-challenge-1", - "2022-11-06": "find-the-path", - "2022-11-13": "counting-squares", - "2022-11-20": "buffons-needle", - "2022-11-27": "balls-and-books", - "2022-12-04": "addition-squares", - "2022-12-11": "adding-to-15", - "2022-12-18": "9-dots", - "2022-12-25": "treasure-hunt", - "2023-01-01": "russian-multiplication", - "2023-01-08": "river-crossing", - "2023-01-15": "poisoned-wine-puzzle", - "2023-01-22": "matchstick-puzzles", - "2023-01-29": "magic-cards", - "2023-02-05": "langfords-problem", - "2023-02-12": "handshake-puzzles", - "2023-02-19": "geometry-puzzle-1", - "2023-02-26": "gabriels-problem", - "2023-03-05": "fence-around-a-field", - "2023-03-12": "domino-tilings", - "2023-03-19": "collatz-conjecture", - "2023-03-26": "coin-game", - "2023-04-02": "coconut-trader", - "2023-04-09": "circumference", - "2023-04-16": "apple-teaser", - "2023-04-23": "two-eggs", - "2023-04-30": "special-numbers", - "2023-05-07": "scales-and-weights", - "2023-05-14": "picture-puzzles-1", - "2023-05-21": "monkey-business", - "2023-05-28": "locks-and-keys", - "2023-06-04": "frog-party", - "2023-06-11": "counting-chickens", - "2023-06-18": "tic-tac-toe", - "2023-06-25": "tic-tac-toe-with-levels", - "2023-07-02": "tangrams", - "2023-07-09": "secret-santa", - "2023-07-16": "patience", - "2023-07-23": "paper-pieces", - "2023-07-30": "nim-related-games", - "2023-08-06": "nim-21", - "2023-08-13": "mastermind", - "2023-08-20": "make-many", - "2023-08-27": "letter-matching", - "2023-09-03": "countdown", - "2023-09-10": "15-game", - "2023-09-17": "tetris-shapes", - "2023-09-24": "kaprekars-number", - "2023-10-01": "diffy-squares", - "2023-10-08": "pell-numbers", - "2023-10-15": "paths", - "2023-10-22": "number-challenge-1", - "2023-10-29": "making-squares", - "2023-11-05": "logic-puzzles-1", - "2023-11-12": "iq-challenge-1", - "2023-11-19": "find-the-path", - "2023-11-26": "counting-squares", - "2023-12-03": "buffons-needle", - "2023-12-10": "balls-and-books", - "2023-12-17": "addition-squares", - "2023-12-24": "adding-to-15", - "2023-12-31": "9-dots", - "2024-01-07": "treasure-hunt", - "2024-01-14": "russian-multiplication", - "2024-01-21": "river-crossing", - "2024-01-28": "poisoned-wine-puzzle", - "2024-02-04": "matchstick-puzzles", - "2024-02-11": "magic-cards", - "2024-02-18": "langfords-problem", - "2024-02-25": "handshake-puzzles", - "2024-03-03": "geometry-puzzle-1", - "2024-03-10": "gabriels-problem", - "2024-03-17": "fence-around-a-field", - "2024-03-24": "domino-tilings", - "2024-03-31": "collatz-conjecture", - "2024-04-07": "coin-game", - "2024-04-14": "coconut-trader", - "2024-04-21": "circumference", - "2024-04-28": "apple-teaser", - "2024-05-05": "two-eggs", - "2024-05-12": "special-numbers", - "2024-05-19": "scales-and-weights", - "2024-05-26": "picture-puzzles-1", - "2024-06-02": "monkey-business", - "2024-06-09": "locks-and-keys", - "2024-06-16": "frog-party", - "2024-06-23": "counting-chickens" + "2021-08-29": "konigsberg-bridges", + "2021-09-05": "kaprekars-number", + "2021-09-12": "diffy-squares", + "2021-09-19": "pell-numbers", + "2021-09-26": "paths", + "2021-10-03": "number-challenge-1", + "2021-10-10": "making-squares", + "2021-10-17": "logic-puzzles-1", + "2021-10-24": "iq-challenge-1", + "2021-10-31": "find-the-path", + "2021-11-07": "counting-squares", + "2021-11-14": "buffons-needle", + "2021-11-21": "balls-and-books", + "2021-11-28": "addition-squares", + "2021-12-05": "adding-to-15", + "2021-12-12": "9-dots", + "2021-12-19": "treasure-hunt", + "2021-12-26": "russian-multiplication", + "2022-01-02": "river-crossing", + "2022-01-09": "poisoned-wine-puzzle", + "2022-01-16": "matchstick-puzzles", + "2022-01-23": "magic-cards", + "2022-01-30": "langfords-problem", + "2022-02-06": "handshake-puzzles", + "2022-02-13": "geometry-puzzle-1", + "2022-02-20": "gabriels-problem", + "2022-02-27": "fence-around-a-field", + "2022-03-06": "domino-tilings", + "2022-03-13": "collatz-conjecture", + "2022-03-20": "coin-game", + "2022-03-27": "coconut-trader", + "2022-04-03": "circumference", + "2022-04-10": "apple-teaser", + "2022-04-17": "two-eggs", + "2022-04-24": "special-numbers", + "2022-05-01": "scales-and-weights", + "2022-05-08": "picture-puzzles-1", + "2022-05-15": "monkey-business", + "2022-05-22": "locks-and-keys", + "2022-05-29": "frog-party", + "2022-06-05": "counting-chickens", + "2022-06-12": "tic-tac-toe", + "2022-06-19": "tic-tac-toe-with-levels", + "2022-06-26": "tangrams", + "2022-07-03": "secret-santa", + "2022-07-10": "patience", + "2022-07-17": "paper-pieces", + "2022-07-24": "nim-related-games", + "2022-07-31": "nim-21", + "2022-08-07": "mastermind", + "2022-08-14": "make-many", + "2022-08-21": "letter-matching", + "2022-08-28": "countdown", + "2022-09-04": "15-game", + "2022-09-11": "tetris-shapes", + "2022-09-18": "konigsberg-bridges", + "2022-09-25": "kaprekars-number", + "2022-10-02": "diffy-squares", + "2022-10-09": "pell-numbers", + "2022-10-16": "paths", + "2022-10-23": "number-challenge-1", + "2022-10-30": "making-squares", + "2022-11-06": "logic-puzzles-1", + "2022-11-13": "iq-challenge-1", + "2022-11-20": "find-the-path", + "2022-11-27": "counting-squares", + "2022-12-04": "buffons-needle", + "2022-12-11": "balls-and-books", + "2022-12-18": "addition-squares", + "2022-12-25": "adding-to-15", + "2023-01-01": "9-dots", + "2023-01-08": "treasure-hunt", + "2023-01-15": "russian-multiplication", + "2023-01-22": "river-crossing", + "2023-01-29": "poisoned-wine-puzzle", + "2023-02-05": "matchstick-puzzles", + "2023-02-12": "magic-cards", + "2023-02-19": "langfords-problem", + "2023-02-26": "handshake-puzzles", + "2023-03-05": "geometry-puzzle-1", + "2023-03-12": "gabriels-problem", + "2023-03-19": "fence-around-a-field", + "2023-03-26": "domino-tilings", + "2023-04-02": "collatz-conjecture", + "2023-04-09": "coin-game", + "2023-04-16": "coconut-trader", + "2023-04-23": "circumference", + "2023-04-30": "apple-teaser", + "2023-05-07": "two-eggs", + "2023-05-14": "special-numbers", + "2023-05-21": "scales-and-weights", + "2023-05-28": "picture-puzzles-1", + "2023-06-04": "monkey-business", + "2023-06-11": "locks-and-keys", + "2023-06-18": "frog-party", + "2023-06-25": "counting-chickens", + "2023-07-02": "tic-tac-toe", + "2023-07-09": "tic-tac-toe-with-levels", + "2023-07-16": "tangrams", + "2023-07-23": "secret-santa", + "2023-07-30": "patience", + "2023-08-06": "paper-pieces", + "2023-08-13": "nim-related-games", + "2023-08-20": "nim-21", + "2023-08-27": "mastermind", + "2023-09-03": "make-many", + "2023-09-10": "letter-matching", + "2023-09-17": "countdown", + "2023-09-24": "15-game", + "2023-10-01": "tetris-shapes", + "2023-10-08": "konigsberg-bridges", + "2023-10-15": "kaprekars-number", + "2023-10-22": "diffy-squares", + "2023-10-29": "pell-numbers", + "2023-11-05": "paths", + "2023-11-12": "number-challenge-1", + "2023-11-19": "making-squares", + "2023-11-26": "logic-puzzles-1", + "2023-12-03": "iq-challenge-1", + "2023-12-10": "find-the-path", + "2023-12-17": "counting-squares", + "2023-12-24": "buffons-needle", + "2023-12-31": "balls-and-books", + "2024-01-07": "addition-squares", + "2024-01-14": "adding-to-15", + "2024-01-21": "9-dots", + "2024-01-28": "treasure-hunt", + "2024-02-04": "russian-multiplication", + "2024-02-11": "river-crossing", + "2024-02-18": "poisoned-wine-puzzle", + "2024-02-25": "matchstick-puzzles", + "2024-03-03": "magic-cards", + "2024-03-10": "langfords-problem", + "2024-03-17": "handshake-puzzles", + "2024-03-24": "geometry-puzzle-1", + "2024-03-31": "gabriels-problem", + "2024-04-07": "fence-around-a-field", + "2024-04-14": "domino-tilings", + "2024-04-21": "collatz-conjecture", + "2024-04-28": "coin-game", + "2024-05-05": "coconut-trader", + "2024-05-12": "circumference", + "2024-05-19": "apple-teaser", + "2024-05-26": "two-eggs", + "2024-06-02": "special-numbers", + "2024-06-09": "scales-and-weights", + "2024-06-16": "picture-puzzles-1", + "2024-06-23": "monkey-business" } \ No newline at end of file diff --git a/maths-club-pack/content/facilitator/konigsberg-number.md b/maths-club-pack/content/facilitator/konigsberg-bridges.md similarity index 100% rename from maths-club-pack/content/facilitator/konigsberg-number.md rename to maths-club-pack/content/facilitator/konigsberg-bridges.md diff --git a/maths-club-pack/content/student/konigsberg-bridges.md b/maths-club-pack/content/student/konigsberg-bridges.md index a4aeb20b..d50a41c5 100644 --- a/maths-club-pack/content/student/konigsberg-bridges.md +++ b/maths-club-pack/content/student/konigsberg-bridges.md @@ -1,5 +1,5 @@ --- -title: Bridges of Konigsberg +title: Konigsberg Bridges type: problem printOrder: 999 added: 2021-01 diff --git a/translations/en/facilitator/konigsberg-number.md b/translations/en/facilitator/konigsberg-bridges.md similarity index 100% rename from translations/en/facilitator/konigsberg-number.md rename to translations/en/facilitator/konigsberg-bridges.md