MathDB

There are

n{}

wise men in a hall and everyone sees each other. Each man will wear a black or white hat. The wise men should simultaneously write down on their piece of paper a guess about the color of their hat. If at least one does not guess, they will all be executed.
The wise men can discuss a strategy before the test and they know that the layout of the hats will be chosen randomly from the set of all

2^n

layouts. They want to choose their strategy so that the number of layouts for which everyone guesses correctly is as high as possible. What is this number equal to?

Problem Statement