Minimize or Maximize The NimSum at a certain step?

Question

Suppose, we are playing a nim game. And We can make say N possible moves, each generating some NimSum Value. For a Winning, Strategy, Should we chose the move with minimum nimsum?

Answer 1

In nim, and nim-like games, you should always leave a position with nim sum $0$ if possible. Then you can win no matter what your opponent does, provided you play perfectly from then on.

If you cannot make $0$, then it doesn't matter what you do, your opponent can win if he plays perfectly.

You can say that a winning strategy is to always leave the minimum nim sum, but I'm not sure that's best if your opponent is not a perfect player. Suppose you are playing against someone who does know the analysis of the game. To be "fair" you take turns starting first. Let's say that the first player always wins this game. When your opponent starts first, you may have to play from a losing position. If you want to try to win this game, I would say that the best thing to do is to try to leave as complicated a position as you can, to give your opponent the greatest opportunity to make a mistake. If he makes one mistake, you can win the game.

In short, I think the answer to your question depends on whether or not your opponent is a perfect player.