Disjunctive sums of quasi-nimbers
Alexandre Ferreira da Silva, Carlos Pereira dos Santos, João Pedro Neto, Richard J. Nowakowski · Theoretical Computer Science · 2022
paint can is an example of a game whose positions are disjunctive sums, and a move in any component reduces that component to a nimber. Conway, in On Numbers and Games, partially analyzed the related game supernim, and called these components “superstars”, mentioning “There does not appear to be a complete theory”. The book contains one result about these games, and, until now, there has been no advance in finding good strategies. Here, we show that, for a human, the use of canonical forms is not a good approach. We present an algorithmic, recursive approach to the general case, based on a fundamental reduction of these positions, as well as on a Nimber Avoidance Theorem. An analysis of the computational time of the algorithm is presented.