Ten people take part in a challenge. Each is given a hat, either black or white. Everyone can see the other hats, but not their own.
They speak one at a time. On your turn, you must say black or white. One special person may instead say a whole number between \(0\) and \(777\), and then also guess their colour.
Before the hats are placed, the group may agree on a strategy.
What is the largest number of people who can guarantee a correct guess?