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Published online by Cambridge University Press: 30 May 2025
Positions of the game of TOPPLING DOMINOES exhibit many familiar combinatorial game theory values, often arranged in unusual and striking patterns. We show that for any given dyadic rational x, there is a unique TOPPLING DOMINOES position G equal to x, and that G is necessarily a palindrome. We also exhibit positions of value + x for each x > 0. We show that for each integer m ≥ 0, there are exactly m distinct LR-TOPPLING DOMINOES positions of value ∗m (modulo a trivial symmetry). Lastly, every infinitesimal TOPPLING DOMINOES position has atomic weight 0, 1 or −1.
TOPPLING DOMINOES, introduced by Albert, Nowakowski and Wolfe [1], is a combinatorial game played with a row of dominoes, such as the one pictured in Figure 1. Here each domino is colored blue or red (black or white, respectively, when color printing is unavailable). On his turn, Left selects any bLue (black) domino and topples it either east or west (his choice). This removes the toppled domino from the game, together with all other dominoes in the chosen direction. Likewise, Right’s options are to topple Red (white) dominoes east or west. For example, the Left options of are
Here A and B result from toppling the westmost domino respectively west or east, while C and D result from toppling the eastern black domino respectively west or east.
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