Some interesting brain teasers

Dice

1. You are given a 100-sided die. After you roll once, you can choose to either get paid the dollar amount of that roll OR pay one dollar for one more roll. What is the expected value of the game? (There is no limit on the number of rolls.)

   100 dice problem

   A more basic version -- 6 and 10 dice

   a little complex one

Number guessing problem

1. A number from 0 to 1 is put on a piece of paper. You and person X are playing a game. You must get less than the number on the paper, but above X’s guess. You know X chooses to pick a number at random. What is lowest number you can choose to maximize your probability of winning?

   Answer

Mental math

1. Calculate pow(2, 5104) mod 10

Coin tossing

1. Flip a coin repeatedly and stop whenever you want. Your payoff is the propor- tion of heads and you wish to maximize this payoff in expectation. What's your optimal strategy

   Robbin's problem