Project CEMAPRE internal
|Title||A note on numbers: the ruleset PUSH|
|Participants||Alda Carvalho (Principal Investigator), Carlos Santos|
|Summary||It is well known that there are rulesets whose positions only have numbers as game-values and|
rulesets that may admit values other than numbers . 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 . 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 . We propose to analyse this ruleset in order to find a
formula to easily compute the value of any PUSH position.
 J. Conway. On Numbers and Games. Academic Press, 1976.
 E. Berlekamp, J. Conway, R. Guy. Winning Ways. Academic Press, London, 1982.
 A. N. Siegel. Combinatorial Game Theory, American Math. Soc., 2013.
 A. Carvalho, M. A. Huggan, R. J. Nowakowski, C. P. Santos, «A Note on Numbers», Integers, 21B,
 M. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play: An Introduction to Combinatorial
Game Theory, A. K. Peters, 2007.