数独的计数、分类与图案设计
Counting, Classification and Graphic Design of Sudoku
作者: 杨一超 ：浙江大学数学系; 李梦鸽 ：浙江大学社会科学学部;
关键词: 六角数独； Grobner基； Burnside引理； 拼图效率； Hexagonal Sudoku； Grobner Basis； Burnside’s Lemma； Splicing Efficiency
摘要:Abstract: The aim of this paper is to study how to count the number of hexagonal sudokus. Firstly, using the method of Grobner basis theory of polynomials, we show the way to count the number of hexagonal sudokus and give an estimate of the number. Secondly, we consider the symmetry properties of hexagonal sudokus under the action of the cyclic group of order 6. Using the famous Burnside lemma in group theory, the number of hexagonal sudokus under the symmetry of rotation group is obtained. Lastly, we discuss the design project of the circular disc with any radius via spelling hexagonal sudokus and introduce the concept of spelling efficiency whilst showing its changing rules.
文章引用: 杨一超 , 李梦鸽 (2013) 数独的计数、分类与图案设计。 理论数学， 3， 257-269. doi: 10.12677/PM.2013.34040
参考文献
[1] 于雷. 北大清华学生爱做的数独游戏[M]. 北京: 中央编译出版社, 2009.
[2] B. Felgenhauer, F. Jarvis. Mathematics of sudoku (I). Mathematical Spectrum, 2006, 39(1): 15-22.
[3] E. Arnold, S. Lucas and L. Taalman. Groebner basis representations of sudoku. College Mathematics Journal, 2010, 41: 101-111.
[4] 王东明. 多项式代数[M]. 北京: 高等教育出版社, 2010.
[5] D. Cox, J. Little and D. O. Shea. Using algebraic geometry. New York: Springer-Verlag, 1998.
[6] W. W. Adams, P. Loustaunau. An Introduction to grobner bases. In: Graduate Studies in Mathematics, Vol. 3, Providence: American Mathematical Society, 1994.
[7] 陈辉. 群的结构与对称性[M]. 杭州: 浙江大学出版社, 2008.