Research projects

Project CEMAPRE internal

TitleA note on numbers: the ruleset PUSH
ParticipantsAlda Carvalho (Principal Investigator), Carlos Santos
SummaryIt is well known that there are rulesets whose positions only have numbers as game-values and
rulesets that may admit values other than numbers [1]. A notable example of the first class is
blue-red-hackenbush [1, 2, 3]. However, most rulesets belong to the second class. When analyzing
games, an early question is: is it possible that all the positions are numbers? The problem is how
to recogn ize when all the positions are numbers.
A general method, based on two fundamental properties, was recently proposed [4]. One of the
rulesets that illustrates these properties well is SHOVE. This ruleset is already solved with a
closed formula based on the simplest number between Left and Right options. However, a closed
variant called PUSH has not been solved [5]. We propose to analyse this ruleset in order to find a
formula to easily compute the value of any PUSH position.

[1] J. Conway. On Numbers and Games. Academic Press, 1976.
[2] E. Berlekamp, J. Conway, R. Guy. Winning Ways. Academic Press, London, 1982.
[3] A. N. Siegel. Combinatorial Game Theory, American Math. Soc., 2013.
[4] A. Carvalho, M. A. Huggan, R. J. Nowakowski, C. P. Santos, «A Note on Numbers», Integers, 21B,
[5] M. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play: An Introduction to Combinatorial
Game Theory, A. K. Peters, 2007.