10 people will stand in a circle, and each have a hat placed on their head. The hats will each be labeled with a digit between 0 and 9, with repeats allowed. They will be able to see everyone's hat but their own. After looking around, they must all simultaneously guess the digit on their hat. Before this happens, the team may agree on a strategy, but once the game begins, they may not communicate in any way. They win as long as at least one person guesses correctly. Show how the team can guarantee victory.
Let the sum of the numbers on all ten hats be $S$.
To each person, assign a unique digit between $0$ and $9$. When it comes to guessing their own hat number, each person picks the number which allows the last digit of $S$ to correspond to their assigned digit. Exactly one of them will be right.