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A sumber is a sum of ups, downs and star. Sumbers can describe the positions of many partisan infinitesimal games. Earlier, we provided a simplification rule [6] that can determine whether a game G is a sumber or not, and if it is, determine the exact sumber of G from its left and right options, GL and GR. This article extends...
State of the art algorithms for stochastic multi-armed bandit problem are based on either one of the two principles: optimism in face of uncertainty, and probability matching. In this paper, we provide a unified approach to combine these principles. The major result is a new upper confidence bound formula, UCB-max, by which the p-value of an arm's UCB value roughly matches the arm's probability of...
Triangular Nim, one variant of the game Nim, is a common two-player game in Taiwan and China. In the past, Hsu strongly solved 7 layer Triangular Nim while some of the authors recently strongly solved 8 layer Triangular Nim. The latter required 8 gigabytes in memory and 8878 seconds. Using a retrograde method, this paper strongly solves 9 layer Triangular Nim. In our first version, the program requires...
This paper introduces a new game named Chilled Domineering, together with its mathematical analysis. Chilled Domineering is modified from the Domineering game by allowing a player placing a 1×1 domino while unable to place a 2×1 domino at a position. After modifying the rule, each position in the game becomes an infinitesimal. Many interesting infinitesimals are found in this new game. This paper...
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