Three years ago, professor Michel Rigo from the Discrete Mathematics Unit at the University of Liège, ventured into a universe unknown to him: that of games. At the time, he had just been joined by his first “post-doctoral” researcher, the Frenchman Eric Duchêne, who was studying combinatorial games, whereas he was working in the domain of discrete mathematics, or in simplified terms, the mathematics of 0’s and 1’s. But Professor Rigo latched on very quickly and the fruit of this mix of expertise significantly exceeded the fun side…

When games and mathematics are mentioned together, we immediately think of economic games where several players interact to find a strategy to maximise their winnings. As for Professor Michel Rigo and Eric Duchêne, they are more focused on combinatorial games for two players: “The questions we ask each other are quite natural”, Professor Rigo informs us. “If we play a certain game together and if I begin the round by choosing to make such or such a move, am I certain of winning? Is there a winning strategy that should be adopted so that whatever move you make, an appropriate answer will always ensure I win? Combinatorial games are a way to exercise good mathematics: on the one hand, they are a source of new phenomena and problems, and on the other hand, they are the area of application for other branches of mathematics.”

The duo Michel Rigo and Eric Duchêne decided to examine invariant combinatorial games in particular, in other words, combinatorial games whose rules do not vary according to the position in which you find yourself. A publication (1) endeavours to find a particular family of invariant games as well as their common characteristics.

The mathematicians are looking beyond the fun side of the game to examine the reasoning capable of revealing a possible strategy that can be adopted to ensure a win. “In a simplified game, we can precisely characterise the losing positions and hence the winning strategy: if you manage to lead your opponent into one of these losing positions and if you adopt this winning strategy, you are sure to win… providing that the strategy can be calculated within a reasonable lapse of time. Of course, in this case, the fun side of the game disappears in favour of the interest of the mathematical reasoning.!”

(1) E. Duchêne , M. Rigo, Invariant games, Proceedings of WORDS 2009, Univ. of Salerno, 14-18 Sept. 2009)