One pile misère bounded Nim with two alliances

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• 作者：Xiao Zhao¹ ; ^{xiaozhao199055@126.com" class="auth_mail" title="E-mail the corresponding author} ; Wen An Liu ; ^{liuwenan@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Impartial combinatorial game ; Nim ; Alliance ; Misè ; re play convention ; Unsafe position
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：11 December 2016
• 年：2016
• 卷：214
• 期：Complete
• 页码：16-33
• 全文大小：654 K

文摘

The game of n$n$-person one-pile bounded Nim with two alliances   is investigated: Given an integer m≥1$m\ge 1$ and a pile of counters, each player is allowed to remove ℓ$\ell$ counters from the pile, where ℓ∈{1,2,…,m}$\ell \in \{1,2,\dots ,m\}$. Suppose that n≥2$n\ge 2$ players form two alliances and that each player is in exactly one alliance. Also assume that each player will support his alliance’s interests.

Under misère play convention, all unsafe positions of two alliances are determined for some structures of two alliances. We also point out that some conclusions given by A.R. Kelly are not correct. Moreover, we present a possible explanation for Kelly’s inaccurate conclusions.