Research projects

Project CEMAPRE internal

TitleHabitats of sets of combinatorial game values
ParticipantsAlda Carvalho (Principal Investigator), Carlos Santos
SummaryCombinatorial Game Theory is a branch of mathematics that studies sequential games (alternated
decisions) with perfect information (no hidden information, no chance devices, [1, 2, 3, 4]). Let S
be a set of combinatorial game values. A ruleset A is an Habitat of S if, for every G in S, there is
a legal position of A with game value equal to G. In [5], Berlekamp asked the question "What is the
habitat of *2?". A natural more general question is "For a ruleset A, what is the largest n such
that *n is a legal position of A?". That question is related to the concept of Nim dimension. A even
more general concern is the following: "Is there a natural habitat for the short Conway group?",
looking for a ruleset with a legal position for each short game value. All these questions are
important for the development of good techniques related to Atomic Weight and for the construction
of tools to measure the complexity of combinatorial games.


[1] J. Conway. On Numbers and Games. Academic Press, 1976.
[2] E. Berlekamp, J. Conway, R. Guy. Winning Ways. Academic Press, London, 1982.
[3] M. Albert, R. Nowakowski, D. Wolfe. Lessons in Play: An Introduction to Combinatorial Game
Theory. A. K. Peters, 2007.
[4] A. N. Siegel. Combinatorial Game Theory, American Math. Soc., 2013.
[5] R. K. Guy. Unsolved problems in combinatorial games, in Games of No Chance, ed. R. J.
Nowakowski MSRI Publ., 29, Cambridge University Press, 475-49, 1996.