Prisoners Problem We have an infinite number of prisoners enumerated $\\{1, 2, \dots\\}$, and on each prisoner there is a hat of either blue or red color. The $n$th prisoner sees the hats of prisoners $\\{n+1, n+2, \dots\\}$. A warden asks each prisoner in succession, starting with prisoner $n=1$, "What is the color of your hat?" If any prisoner fails, then the warden executes that prisoner. Prisoners can not communicate. What strategy should the prisoners use (they discussed it before the moment that hats are put on them) to end up with a finite number of executions?

This is a classic riddle, whose solution surprisingly requires the axiom of choice as discussed here.

The solution, however, is short and clever. Encode the colors into $0$ and $1$, and define the equivalence relation on $2^\Bbb N$, $\langle x_i\rangle\sim\langle y_i\rangle$ if and only if there is some $k$ such that for all $n\geq k$, $x_n=y_n$. Using the axiom of choice the prisoners pick a representative from each equivalence class.

In his turn, then $n$-th prisoner looks for the representative class fitting the string of hats he sees ahead, assuming that all hats up to him are blue. Since all the prisoners follow the same representative to guess their own color, it is guaranteed that after finitely many deaths, the representative and the fashionable selection of hats by the warden will agree, and everyone else will survive.

\-- "The needs of the many, outweigh the needs of the few."