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exercises(sieve): sync docs #572

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40 changes: 27 additions & 13 deletions exercises/practice/sieve/.docs/instructions.md
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# Instructions

Your task is to create a program that implements the Sieve of Eratosthenes algorithm to find prime numbers.
Your task is to create a program that implements the Sieve of Eratosthenes algorithm to find all prime numbers less than or equal to a given number.

A prime number is a number that is only divisible by 1 and itself.
A prime number is a number larger than 1 that is only divisible by 1 and itself.
For example, 2, 3, 5, 7, 11, and 13 are prime numbers.

The Sieve of Eratosthenes is an ancient algorithm that works by taking a list of numbers and crossing out all the numbers that aren't prime.

A number that is **not** prime is called a "composite number".
By contrast, 6 is _not_ a prime number as it not only divisible by 1 and itself, but also by 2 and 3.

To use the Sieve of Eratosthenes, you first create a list of all the numbers between 2 and your given number.
Then you repeat the following steps:

1. Find the next unmarked number in your list. This is a prime number.
2. Mark all the multiples of that prime number as composite (not prime).
1. Find the next unmarked number in your list (skipping over marked numbers).
This is a prime number.
2. Mark all the multiples of that prime number as **not** prime.

You keep repeating these steps until you've gone through every number in your list.
At the end, all the unmarked numbers are prime.

~~~~exercism/note
[Wikipedia's Sieve of Eratosthenes article][eratosthenes] has a useful graphic that explains the algorithm.

The tests don't check that you've implemented the algorithm, only that you've come up with the correct list of primes.
A good first test is to check that you do not use division or remainder operations.

[eratosthenes]: https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
To check you are implementing the Sieve correctly, a good first test is to check that you do not use division or remainder operations.
~~~~

## Example

Let's say you're finding the primes less than or equal to 10.

- List out 2, 3, 4, 5, 6, 7, 8, 9, 10, leaving them all unmarked.
- 2 is unmarked and is therefore a prime.
Mark 4, 6, 8 and 10 as "not prime".
- 3 is unmarked and is therefore a prime.
Mark 6 and 9 as not prime _(marking 6 is optional - as it's already been marked)_.
- 4 is marked as "not prime", so we skip over it.
- 5 is unmarked and is therefore a prime.
Mark 10 as not prime _(optional - as it's already been marked)_.
- 6 is marked as "not prime", so we skip over it.
- 7 is unmarked and is therefore a prime.
- 8 is marked as "not prime", so we skip over it.
- 9 is marked as "not prime", so we skip over it.
- 10 is marked as "not prime", so we stop as there are no more numbers to check.

You've examined all numbers and found 2, 3, 5, and 7 are still unmarked, which means they're the primes less than or equal to 10.
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