| 引用本文: | 黄振高,黄受安.一个组合对策问题的解.[J].国防科技大学学报,1990,12(3):15-20.[点击复制] |
| Huang Zhengao,Huang Shouan.Solution to a Combinatorial Game[J].Journal of National University of Defense Technology,1990,12(3):15-20[点击复制] |
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| 一个组合对策问题的解 |
| 黄振高, 黄受安 |
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(系统工程与应用数学系)
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| 摘要: |
| 本文求解如下的组合对策问题:设有一堆棋子，总数N是奇数，甲乙两人轮流取子，每人每次可取一颗、二颗，最多可取 s 颗，但不能不取，直至取完后分别来数甲乙两人所取棋子的总数，总数为奇数者获胜。站在甲的立场上考虑获胜的策略，文中解决了如下两个问题：(Ⅰ)总数N应是什么样的奇数，甲才有获胜策略；(II)当 N 一定时，甲应采取什么样的策略取子，才能获胜。 |
| 关键词: 运筹学，对策论，策略，组合，获胜策略 |
| DOI： |
| 投稿日期：1989-08-16 |
| 基金项目:国家自然科学基金资助课题 |
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| Solution to a Combinatorial Game |
| Huang Zhengao, Huang Shouan |
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(Department of Applied Mathematics and System Engineering)
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| Abstract: |
| Let there be a heap of beans with total number N(odd) and an integer s，we difine a 2-person game Γ(N，s) as follows: The first player PⅠ takes some beans from the heap, at least one bean and at most s beans. Player two，PII now picks some beans under the same constraint. The play then reverts to PII and continues in the same way until all beans have been removed. The player with odd number of beans at hand wins. In this paper，We completely solve the game. At first，we give the
winning strategy (if exists) for PⅠ. Theorem Let r be the number of beans PⅠ leaving to PII at any step，and q be the number pⅠ have at the end of the step. Then pⅠ will win if r=1 or 0(mod(2s+2)) if q is odd ,and r=s+1 or s+2(mod(2s+2)) if q is even when s is odd and r=1 or 0 (mod(s+2)) if q is odd,and r=s+1(mod(s+2)) if q is even when s is even. The theorem can be proved by induction. As a by-product of the theorem we have Corollary Γ(n，s) is a win for PII iff n=S十1 (mod(2s+2)) when s is odd and n=s+1 (mod(s+2)) when s is even. |
| Keywords: operations research,game theory,strategy,combination,winning strategy |
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