The Silver Dollar Game With No Silver Dollar
(Bogus Nim)

This game is played very much like the games of Nimble and Scoring. Counters are placed on strip of squares and are moved (dragged) leftward. Counters are not allowed to move over each other, and no two counters may be placed on the same square.


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

The game has a very close association with Nim and is often called Bogus Nim. Why Nim and why bogus?

Start with the rightmost counter and count the number of empty squares up to the next counter. This number is the size of the first heap. Skip one gap between the counters. The length of the next gap is the size of the second heap, and so on. If the number of counters is odd, the size of the last heap is determined by the position of the leftmost counter. Note that any eligible move may only affect the size of a single heap.

The reason the game is known as Bogus Nim is that some moves may increase the size of one of the heaps. If this happens, such an increase may be immediately countered by reduction to the previous size. The winning player has no need in those moves. As the foregoing argument shows, the losing player gains nothing by increasing one of the heaps. The heap increasing moves are known as reversible. They are harmless because they may be only made a finite number of times, so that, like Nim, Bogus Nim is liable to terminate.

References

  1. J. H. Conway, On Numbers And Games, A K Peters, 2001
  2. R. K. Guy, fair game, Comap's Explorations in Mathematics, 1989

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