In the stylish 1961 nouvelle vague film Last Year at Marienbad two men vie for the attention of a woman. This spa town was the venue for the famous 1925 tournament won by Nimzowitsch. However, the film protagonists decide the matter with a game of Nim. In the crucial scene, the players nervously remove sticks from four piles: the last one to remove a stick is the loser in both senses. Their nervousness defines the message of the film – casting doubt where there should be knowledge, chance where there should be strategy. Nowadays there is no prospect of a Nim challenge because the players would discover that the game had been cracked.
Let’s look at a very simple game of Nim with four counters in one pile.
You can remove one or two counters on your move. The last person to take a counter loses. The game can be resolved into the question “Should you go first or second.”
The second player has a winning strategy: to leave one counter whatever the first player does. For example, if the first player removes two counters, the second player will remove one counter. The next move for the first player loses immediately.
We now play this same game of Nim on a chessboard.
The diagram shows a pawn joust in which you win by capturing your opponent’s pawn. Normal rules of chess apply. There are four empty squares for each pawn. If White moves two squares ♙d2-d4, then Black only moves one square. White only has bad moves and must lose the pawn next move.
NimChess may be defined as a game in which each move reduces the subsequent number of moves available. Pawns fulfil this condition because they only move forwards. This Nim game is equivalent to the Pawn Joust Game.