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Compound Games with Counters

There is a well-known game played by two players (Abel and Baker) which runs as follows [1]: Let M be any given integer, greater than 1. There is a pile of counters on a table in front of the players. Abel moves by taking from the heap any number x 1 of counters he pleases, subject to the restriction that this number is greater than 0 and less than M. If Abel removes the last counter, he wins; if not, Baker can now take from the heap any number x 2 he chooses, subject to the same restriction as above. If he removes the last counter, he wins; otherwise it is Abel’s move again, and so on alternately until the last counter is taken, and the player taking it wins. This is an old game, but seems to have no distinctive name. I will call it Na(M). Now the object in a mathematical game is to find the “best strategy” which will guarantee a win to a player, provided he starts off from a favorable initial position. (Unfortunately a knowledge of the best strategy often kills the interest of a game, since an uninitiated opponent will not be happy about losing repeatedly, and an initiated opponent will know what is coming. But let us ignore this objection.) In Na(M) the best strategy is to move to a pile containing a multiple of M counters, if one can. Suppose, for example, that Abel moves to such a pile. If it is empty, Abel wins immediately. If not, it is Baker’s turn, and he must move to a pile which is not a multiple of M. Abel now moves back to a multiple of M, and the argument repeats; sooner or later the pile which Abel moves to must be empty, and he wins. For example, if M = 3 and the game begins with a pile of 7 counters, Abel will take 1 away, leaving 6. Baker can now take 1 or 2, leaving 4 or 5. Abel will diminish the pile to 3, and Baker to 1 or 2, Abel to 0, and win.