Safe strategy games are those in which one of the two players, who can be the one who makes the first move or the one who makes the second, always wins if he makes the correct moves at all times. And there are also safe strategy games that necessarily end in a draw if both players always make the best move.
If in chess there was a sure winning strategy for Black (which is the one who makes the second move), a possibility that was raised last week, White could make a “no” move (for example, take out a knight and then return it to the starting position) to usurp the “second” role of Black, which shows that a chess game in which both players play in the best possible way can only end in a draw or with the victory of White, since that the initial move (opening) can be advantageous or indifferent, but not detrimental. Is this reasoning correct or does it have any weak points?
The alquerque and the mill, as we have seen in previous installments, are safe strategy games that end in a draw if both players play in the best possible way, and the same thing happens with checkers on the 8×8 board and with twelve checkers per side. , derived from alquerque. However, in the case of Polish checkers, on a 10×10 board and with twenty pieces per side, it has not yet been possible to prove (to my knowledge) that the game necessarily ends in a draw if the optimal strategy is followed, although consider that to be the most likely.
The neem and its variants
One of the simplest and most popular surefire strategy games is nim, of which there are many variations. The simplest consists of putting 20 small objects (coins, matches, pebbles…) in a row and taking turns removing between one and three units; The one who takes the last piece loses. In this modality there is a simple winning strategy, what is it? How can the result be generalized to n objects, from which between 1 and p can be removed each time?
To complicate things a bit, you can arrange the objects – let’s say they are matches – in several rows of different numbers; for example, in four rows of 1, 3, 5 and 7 matches respectively. The two players, in turn, remove from one to three matches, all from the same row, and the one who keeps the last one loses. In this case the winning strategy is not so simple, but it exists. Which? Who is assured of victory if he plays correctly, the first or the second to play?
Another variant consists of being able to take all the matches you want from the same row, and the number of rows and matches for each row can be varied at will. Logically, the more rows and matches, the more complicated the game becomes.
Going back to the chessboard (we always go back to it), there is a variant of nim that is played with checkers on an 8×8 board. One player places eight white checkers in the first row and the other eight black checkers in the last, as shown in the figure. Then, in turn, each player moves one of his pieces forward as many spaces as he wishes, until the white and black pieces of the same column reach contiguous squares, in which case neither of them can continue moving. The player who makes the last move wins. Why is this game a variant of nim? Which of the two players necessarily wins if he makes the correct moves and what is his winning strategy?
Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn Physics’, ‘Damn Mathematics’ or ‘The Great Game’. He was a screenwriter for ‘The Crystal Ball’.
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