Alice and Bob play a game with n coins. Starting with Alice the players remove $x_1, x_2, …$ or $x_m$ coins. The first player who can not make a move loses. Who has the winning strategy?
This is a general form of well known problems with many different adaptations, but when I encountered more difficult problems of this kind, the solutions involved seemingly random moduli.
Example: $x_1=2, x_2=5, x_3=6$
Solution of example: Losing positions are $0, 1, 4, 8 \mod 11$
Even a strategy, which is better than just guessing random moduli to find an invariant is greatly appreciated.