From a 1960s film, a game played by players alternately picking up sticks or cards. What is the winning strategy? Some years ago, we discussed the game of nim and its winning strategy, which many readers analysed effectively. To recap, the game has two players and begins with a predesignated number of matchsticks. Each player, in turn, takes 1, 2 or 3 matches. The player who takes the last matchstick wins. If there are, say, 25 matches, the advantage lies with the first person, who takes 1 match in the first move, leaving 24 on the table. Whatever the number of matches the second player takes, the first player takes a number that ensures that a multiple of 4 remains on the table. This is maintained at every step. At the penultimate step, the second player is left facing 4 matches and can take only 1, 2 or 3, so the first player subsequently wins by taking whatever remains.I had long been planning to use the game’s strategy as fodder for a puzzle. And the ideal time is now, the 200th week of Problematics. (Unsplash)There is also a “misère version” of nim. Here, the player who takes the last match is the loser. The strategy in both games is based on the same principles, but this time the advantage lies with the second player (if starting with 25 matches). In this version, the strategy is to leave a number that is one more than a multiple of 4, i.e. 4k + 1. Since 25 is already is (4 x 6 + 1), it is a losing situation for whoever has to make their move at this stage. There will come a stage when the first player is left facing 5 matches. Whatever number of matches the first player takes at that stage, there will be 4, 3 or 2 matches remaining on the table. So, at the next step, the second player has to take all except one of the remaining matches, forcing the first player to take the solitary match.The game has generated other versions and inspired some spinoffs over the centuries. One very interesting version is the subject of our leading puzzle this week. I had long been planning to use the game’s strategy as fodder for a puzzle. And the ideal time is now, the 200th week of Problematics.#Puzzle 200.1#Puzzle 200.1