Misère domineering on \(2 \times n\) boards. (English) Zbl 1490.91034
Summary: Domineering is a well-studied tiling game in which one player places vertical dominoes and a second places horizontal dominoes, alternating turns until someone cannot place on their turn. Previous research has found game outcomes and values for certain rectangular boards under normal play (last move wins); however, nothing has been published about domineering under misère play (last move loses). We find optimal-play outcomes for all \(2 \times n\) boards under misère play: these games are Right-win for \(n \geqslant 12\). We also present algebraic results including sums, inverses, and comparisons in misère domineering.
References:
| [1] | M.R. Allen, Peeking at partizan misère quotients, in R.J. Nowakowski (Ed.) Games of No Chance 4, MSRI Publ. 63 (2015), 1-12. · Zbl 1380.91037 |
| [2] | E.R. Berlekamp, Blockbusting and domineering, J. Combin. Theory Ser. A 49 (1988), 67-116. · Zbl 0651.90092 |
| [3] | D.M. Breuker, J.W.H.M.Uiterwirjk, and J. van der Herik, Solving 8 × 8 domineering, Theoret. Comput. Sci. (Math Games) 230 (2000), 195-206. · Zbl 0951.91006 |
| [4] | N. Bullock, Domineering: Solving large combinatorial search spaces, ICGA J. 25 (2002), 67-84. |
| [5] | M. Lachmann, C. Moore, and I. Rapaport, Who wins Domineering on rectangular boards? in R.J. Nowakowski (Ed.) More Games of No Chance, MSRI Publ. 42 (2002), 307-315. · Zbl 1062.91525 |
| [6] | U. Larsson, R. Milley, R.J. Nowakowski, G. Renault, and C.P. Santos, Recursive comparison tests for dicots and dead-ending games under misère play, Integers, to appear. |
| [7] | G.A. Mesdal and P. Ottaway, Simplification of partizan games in misère play, Integers 7 (2007), #G6. · Zbl 1165.91008 |
| [8] | R. Milley and G. Renault, Dead ends in misère play: the misère monoid of canonical numbers, Discrete Math. 313 (2013), 2223-2231. · Zbl 1293.91035 |
| [9] | T.E. Plambeck and A.N. Siegel, Misère quotients for impartial games, J. Combin. Theory Ser. A 115 (2008), 593-622. · Zbl 1142.91022 |
| [10] | J.W.H.M. Uiterwirjk, 11 × 11 domineering is solved: the first player wins, in A. Plaat, W. Kosters, J. van den Herik (Eds.) Computers and Games, Springer, Leiden (2016), 129-136. · Zbl 1398.91134 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
