## 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 [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. References: [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, 2021. [5] M. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play: An Introduction to Combinatorial Game Theory, A. K. Peters, 2007. |