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Which would mean that the opponent can only do -1 (by playing a move from $S$) or +1 (by not playing a move from $S$), so he can't change the parity of our number!

Which would mean that the opponent can only do -1 (by playing a move from $S$) or +1 (by not playing a move from $S$), so he can't change the parity of our $|S|$!

This surprises me. Because if my hypothesis is true, it means that players have no influence on the result of the game. We could even play by random and if the initial $|S|$ was odd - we will win, if it was even - we will lose. No matter what. That will be a silly game, I can't believe in that. But I can't find a counterexample for that too.

I really hope that the hypothesis is untrue. Maybe you can spot some mistake I'm making or have a counterexample?

So, if all of above is true, it means that players have no influence on the result of the game. We could even play by random and if the initial $|S|$ was odd - we will win, if it was even - we will lose.

It's hard for me to believe in that. What will be the sense of making AI challenge on HackerRank with such a game? Players in the leader-board have different scores and these scores are consistent each time after recalculation (same people in the top 10), suggesting that players actually have an influence on the game result.

Maybe you can see some mistake I'm making?

You win when you make the last move. In other words, you win when after your move there are only 0 on the board.

My research so far:

I really hope that the hypothesis is untrue. Maybe you can spot some mistake I'm making or have an counterexample?

You win when after your move there are only 0 on board.

Research so far:

I really hope that the hypothesis is untrue. Maybe you can spot some mistake I'm making or have a counterexample?