There are \(n+1\) hats, each labeled with a number from \(1\) to \(n+1\), and \(n\) people. Each person is randomly assigned exactly one hat, and each hat is assigned to at most one person (i.e., the assignment is injective). A person can see all other assigned hats but cannot see their own hat and the unassigned hat. Each person must independently guess the number on their own hat.

If everyone correctly guesses their own hat’s number, they win; otherwise, they lose. They may discuss a strategy before the hats are assigned, but no communication is allowed afterward.

Determine a strategy that maximizes their probability of winning.

The best solution was submitted by 김동훈 (수리과학과 22학번, +4). Congratulations!

Here is the best solution of problem 2025-06.

Other solutions were submitted by 김준홍 (수리과학과 석박통합과정, +3), 박기윤 (전산학부 23학번, +3), 채지석 (수리과학과 석박통합과정, +3). There were incorrect solutions submitted.