The Coin Board and the Key

Two prisoners, Ann and Bob, may plan together once. A warden then lays out a board with N squares, each carrying a coin whose side (heads or tails) is chosen arbitrarily by the warden. The warden hides a key under one square and shows Ann the whole board and the key's square. Ann must flip exactly one coin, and then leaves. Bob enters, sees the board in its new state (but not the original), and must name the key's square. They go free if Bob is right. For how many values of N in the range 1 ≤ N ≤ 100 does a strategy exist that always works, whatever the warden chooses?

• combinations
• partitions

Source: Generalisation of the classic 64-square chessboard-and-coins prisoner puzzle in wide circulation; no traceable original source. Statement written for AxiomIQ.