Subtract a Square
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Subtract-a-square (also referred to as take-a-square) is a two-player mathematical game of strategy starting with a positive integer and both players taking turns subtracting a non-zero square number not larger than the current value. The game is usually played as a normal play game, which means that the last person who can make a subtraction wins. The game subtract-a-square can also be played with multiple numbers. At each turn the player to make a move first selects...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Subtract-a-square (also referred to as take-a-square) is a two-player mathematical game of strategy starting with a positive integer and both players taking turns subtracting a non-zero square number not larger than the current value. The game is usually played as a normal play game, which means that the last person who can make a subtraction wins. The game subtract-a-square can also be played with multiple numbers. At each turn the player to make a move first selects one of the numbers, and then subtracts a square from it. Such a ''sum of normal games'' can be analysed using the Sprague Grundy theory. This requires the positions in the game subtract-a-square to be mapped onto equivalent nim heap sizes. This mapping is captured for the normal game in the On-Line Encyclopedia of Integer Sequences as A014586. Notice that all ''cold'' positions get mapped onto a zero heap size.
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