西洋棋大師問題(Chessmaster Problem, CMP)是近代離散數學在介紹鴿籠原理時經常使用的例題。在這個問題中，我們會對棋手所必須進行的棋局數加入了一些限制，目標為證明棋手在滿足棋局數限制的所有情況下，都存在某連續時段剛好下了k盤棋局。
CMP的最早版本可以追溯到1962年出版的蘇聯數學奧林匹克問題集(Shklarsky et al.)，之後也出現在組合數學、離散數學的不同教科書之中。Erdös等人在1982年發現了當在CMP上施加特定限制並且滿足特定條件的情況下，可以使用鴿籠原理證明CMP是合法可解的。
在本論文中，我們從一個新的角度來看待CMP，並且以另一種方式使用鴿籠原理，從而得到一個看起來完全不同(無法以代數操作進行轉換)但卻與Erdös等人結果等價的結果。此外，我們也發現了Erdös等人的研究成果可以為參數an提供一個無條件的下界L，使得當an小於該下界L時，CMP可以被證明是合法可解的。最後，我們提供了兩個演算法來檢查在其他情況下，CMP是否是合法可解的。
關鍵字: 西洋棋大師問題, 鴿籠原理, 組合數學, 離散數學

Chessmaster Problem (CMP) is nowadays a popular example to demonstrate the use of Pigeonhole Principle. In this problem, there are some restrictions on the number of games a chessmaster has to play, and the target is to prove that the chessmaster would have played exactly a certain number k of games in a contiguous period of time, in any
situation.
The earliest version of the problem can be traced back as a USSR mathematical olympiad problem (Shklarsky et al., 1962), and appeared in various textbooks of Combinatorics and Discrete Mathematics. Erdös et al. (1982) used Pigeonhole Principle to show that when a certain restriction is posted on CMP, and if a certain bound follows, then the CMP problem is well-defined so that the target can be proven true. In this thesis, we look at CMP from a new perspective, and apply Pigeonhole Principle in an alternative way, so that we obtain a different-looking-but-equivalent bound which may not be easy to obtain directly by algebraic manipulation. Furthermore, we show that Erdös et al.’s result for the restricted case can readily provide an unconditional lower bound L(n, k) on a parameter an for which the CMP problem is well-defined whenever an < L. Finally, we propose two efficient heuristic programs to check whether a CMP problem is well-defined.
Keywords: Chess Master Problem, Pigeonhole Principle, Combinatorics, Discrete Mathematics

摘要 i
Abstract ii
Acknowledgement iii
Contents iv
1 Introduction 1
2 Our Results 3
2.1 GCMP when n=b 3
2.2 GCMP with No Restrictions 7
3 Programs 9
3.1 The Bottom-Up Algorithm 10
3.2 Modifying the Bottom-up Algorithm 11
4 Conclusion 13
Bibliography 15

[1] R. A. Brualdi. Introductory Combinatorics (5th edition), Pearson, 2010.
[2] P. Erd ̈os, R. L. Hemminger, D. A. Holton, and B. D. McKay. On the Chessmaster Problem. Progress in Graph Theory, pages 532–536, 1982.
[3] K. Rosen. Discrete Mathematics and Its Applications (8th edition), McGraw-Hill, 2018.
[4] D. O. Shklarsky, N. N. Chentzov, and I. M. Yaglom. The USSR Olympiad Problem Book, San Francisco, Freeman, 1962.
[5] Wolfram MathWorld. https://mathworld.wolfram.com/FerrersDiagram.html