理论数学

Vol.3 No.4 (July 2013)

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:

在本文中,我们研究六角数独的计数问题。首先,我们用多项式的Grobner基理论方法,给出计算六角数独的总数的方法,并给出了总数的一个估计值。其次,我们考虑六角数独关于旋转群的对称性,利用群论著名的Burnside引理,给出了旋转对称的等价意义下的六角数独的总数。最后,我们研究六角数独拼接成可无限延展的圆形几何图形的设计方案,并提出了拼图效率的概念,给出了拼图效率的变化规律。

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

 

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