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- Ordinal sums of impartial gamesPublication . Carvalho, Alda; Neto, João; Santos, CarlosIn an ordinal sum of two combinatorial games G and H, denoted by G : H, a player may move in either G (base) or H (subordinate), with the additional constraint that any move on G completely annihilates the component H. It is well-known that the ordinal sum does not depend on the form of its subordinate, but depends on the form of its base. In this work, we analyze g(G : H) where G and H are impartial forms, observing that the g-values are related to the concept of minimum excluded value of order k. As a case study, we introduce the ruleset OAK, a generalization of GREEN HACKENBUSH. By defining the operation gin sum, it is possible to determine the literal forms of the bases in polynomial time. (C) 2017 Elsevier B.V. All rights reserved.
- Ordinal sums, clockwise hackenbush, and domino shavePublication . Carvalho, Alda; Huggan, Melissa A.; Nowakowski, Richard; Santos, CarlosWe present two rulesets, domino shave and clockwise hackenbush . The first is somehow natural and, as special cases, includes stirling shave and Hetyei’s Bernoulli game. Clockwise hackenbush seems artificial yet it is equivalent to domino shave. From the pictorial form of the game, and a knowledge of hackenbush, the decomposition into ordinal sums is immediate. The values of clockwise blue-red hackenbush are numbers and we provide an explicit formula for the ordinal sum of numbers where the literal form of the base is { x | } or { | x }, and x is a number. That formula generalizes van Roode’s signed binary number method for blue-red hackenbush.