The game of pebbles is a two-player game. The game starts with N stones. On a player's turn, he or she must remove either 1, a power of 2 (2,4,8,...) or a power of 3 stones (3,9,27,...). The player who does not take the last stone is the winner.
Is there a winning strategy?
Answer
The losing positions for the first player are those for which
The number of stones N is one more than a multiple of 5, that is, of the form $N = 5k+1$.
The reason for this:
Note that the players always have the option to remove 1,2,3 or 4 stones. However, they can't remove 5 or a multiple of 5. So whatever move the first player makes in one of the positions of the form $5k+1$, the second player can make a move that brings the first player back to one more than a multiple of 5, until they reach 1 stone and the first player is forced to take it.
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