论文标题
轻松的Wythoff拥有所有的Beatty解决方案
Relaxed Wythoff has All Beatty Solutions
论文作者
论文摘要
我们发现，在三个互补的比蒂序列成对的情况下，出现了三个减法游戏的p位。第一场比赛归因于Fraenkel，第二款是将第一款游戏扩展到非单调设置的。我们表明，如果满足某些不平等现象，则可以从弗朗克尔（Fraenkel）纸的复发中推断出第二款游戏的p位。如果已知P位置是互补的Beatty序列成对，并且明确给出了这种不平等的非理性家族，则表明这种不平等是必要的。我们在文献中重点介绍了几个具有P位置的游戏，这些游戏是与这个家庭中斜坡的一对互补的Beatty序列。我们提出的第三场游戏是新颖的，我们表明可以在任何情况下从同一复发中推断出P位。结果表明，任何一对互补的比蒂序列都是这个家庭中某些游戏的p位点出现的。我们还提供了过去几年中出现在该领域的某些反问题的背景，尤其是Duchêne-Rigo猜想。本文为2011年BIRS研讨会所提出的Fraenkel问题提供了解决方案，这是对Duchêne-Rigo猜想的修改。
We find conditions under which the P-positions of three subtraction games arise as pairs of complementary Beatty sequences. The first game is due to Fraenkel and the second is an extension of the first game to non-monotone settings. We show that the P-positions of the second game can be inferred from the recurrence of Fraenkel's paper if a certain inequality is satisfied. This inequality is shown to be necessary if the P-positions are known to be pairs of complementary Beatty sequences, and the family of irrationals for which this inequality holds is explicitly given. We highlight several games in the literature that have P-positions as pairs of complementary Beatty sequences with slope in this family. The third game we present is novel, and we show that the P-positions can be inferred from the same recurrence in any setting. It is shown that any pair of complementary Beatty sequences arises as the P-positions of some game in this family. We also provide background on some inverse problems which have appeared in the field over the last several years, in particular the Duchêne-Rigo conjecture. This paper presents a solution to the Fraenkel problem posed at the 2011 BIRS workshop, a modification of the Duchêne-Rigo conjecture.
