Unsolved problems in Combinatorial Games

Summary

We have sorted the problems into sections:

• A. Taking and Breaking

• B. Pushing and Placing Pieces

• C. Playing with Pencil and Paper

• D. Disturbing and Destroying

• E. Theory of Games

They have been given new numbers. The numbers in parentheses are the old numbers used in each of the lists of unsolved problems given on pp. 183–189 of AMS Proc. Sympos. Appl. Math. 43 (1991), called PSAM 43 below; on pp. 475–491 of Games of No Chance, hereafter referred to as GONC; and on pp. 457–473 of More Games of No Chance (MGONC). Missing numbers are of problems which have been solved, or for which we have nothing new to add. References [year] may be found in Fraenkel's Bibliography at the end of this volume. References [#] are at the end of this article. A useful reference for the rules and an introduction to many of the specific games mentioned below is M. Albert, R. J. Nowakowski and D. Wolfe, Lessons in Play: An Introduction to the Combinatorial Theory of Games, A K Peters, 2007 (LIP).

A. Taking and breaking games

A1 (1). Subtraction games with finite subtraction sets are known to have periodic nim-sequences. Investigate the relationship between the subtraction set and the length and structure of the period. The same question can be asked about partizan subtraction games, in which each player is assigned an individual subtraction set.