# Not the Same

Level pending

Two players are playing Not the Same. They alternate turns removing stones from a pile with $$N>0$$ stones. On the first turn, the first player removes $$1$$, $$2$$, or $$3$$ stones. On each subsequent turn, the player removes $$1$$, $$2$$, or $$3$$ stones from the pile, but they cannot remove the same number of stones as the previous player removed in the previous turn. If a player cannot go, they forfeit their turn. The player who removes the last stone wins.

For how many positive integer initial pile sizes $$N\le 1000$$ does the first player have a winning strategy?

Details and Assumption

If a player forfeits their turn, on the next turn the next player can remove 1, 2, or 3 stones.

A player cannot remove more stones than are left in the pile.

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