Unsolved problems in Combinatorial Games

Abstract

Abstract. This periodically updated reference resource is intended to put eager researchers on the path to fame and (perhaps) fortune.As in our earlier articles, WW stands for Winning Ways [Berlekamp et al. 1982]. We say that the nim-value of a position is n when its value is the nimber * n.

Games played by Boole and Galois

“…By Proposition 4 we thus see that (14) holds also for G 6 . In conclusion, S = P for G 6 , where the first few entries of S are displayed in Table 6.…”

“…for all 0 m < n. As we pointed out, 2 5 = 7 > 2 6 = 6, so without monotonicity, we do not seem to be able to get (14). We use an auxiliary result.…”

From graceful labellings of paths to cyclic solutions of the Oberwolfach problem

“…Consider the 5-pendant graceful (4, 5)-labelling (4,9,3,7,10,2,11,1,8,6,5) for P 11 . Using this along with the 5-pendant graceful labelling (5, 1, 4, 2, 3) for P 5 leads to the following terrace for Z 16 , which has match list (4, 5):…”

“…Reversing this and lifting to Z 32 can give the symmetrically sectionable directed terrace (0, 1,31,18,30,9,24,6,13,3,12,20,23,27,21,26,10,5,11,7,4,28,19,29,22,8,25,14,2,15,17,16) and hence a cyclic solution to OP (4,4,5,5,15) from the 2-factor [6,13,3,12,20], [23,27,21,26], [10,5,11,7], [4,28,19,29,22], [8, 25, 14, 2, 15, 17, 16, ∞, 0, 1, 31, 18, 30, 9, 24].…”

On ?-Fold Equipartite Oberwolfach Problem with Uniform Table Sizes

“…The well-known Oberwolfach problem [3] was first formulated by Ringel in 1967: "Is it possible to seat an odd number n of people at s round tables T 1 , T 2 ,... , T s (where each T i can accommodate t i ≥ 3 people and ∑ti = n) for k different meals so that each person has every other person for a neighbor exactly once?" In terms of graph theory, this problem is equivalent to asking for an odd integer n, is it possible for K n to have a 2-factorization in which each 2-factor consists of cycles of lengths t 1 , t 2 ,... , t s ?…”