用代数方法探讨四圆相交区域填充数字问题  

Algebraic method for filling numerical problems of intersection region of four circles

作　　者：石业娇[1] 孟宪涛[2] SHI Yejiao MENG Xiantao(Applied Technology College, Dalian Ocean University, Dalian 116300, China School of Mathematics and System Science, Shenyang Normal University, Shenyang 110034, China)

机构地区：[1]大连海洋大学应用技术学院,辽宁大连116300 [2]沈阳师范大学数学与系统科学学院,沈阳110034

出　　处：《沈阳师范大学学报（自然科学版）》2017年第3期335-338,共4页Journal of Shenyang Normal University:Natural Science Edition

基　　金：国家自然科学基金资助项目(11201313)

摘　　要：受在三圆相交区域中填充数字问题以及用代数方法求解幻方问题的启发,设计了利用线性代数方法在四圆相交区域中填充数字问题。首先,建立了填充问题的约束方程组,根据需要将约束方程组变形为5种形式,即所谓5个约束条件;然后,对约束条件进行讨论,得出四圆重叠区域中心位置的数与两圆重叠区域的4个数字之和的奇偶性,以及三圆重叠区域的4个数字之和与只属于一个圆区域的4个数字之和的奇偶性,以约束条件为基础,兼顾数字的对称性与互补性,采用试验的方法,考虑3种情况下的不同取值,得到相应问题的15个解;最后,给出了相对于每一个解,每一个圆中所包含的7个数字之和的上限与下限,给出相应的证明。Inspired by the filling numbers of three circles in the intersectional region and algebraic method to solve the problem of magic square,the problem of filling numbers in intersectional regions of four circles is designed based on linear algebra methods. Firstly,the constraint equations of the filling problem are established,and they are deformed into five forms,namely,the five constraints. Then,upon discussion of the constraints,odevity of the number in intersectional region of four circles and summation of the four numbers in intersectional region of two circles as well as the odevity of summation of the four numbers in intersectional region of three circles and the four numbers in only one circle are obtained. On the basis of constraints,considering the symmetry and complementation of the numbers and different values in three cases,the 15 solutions of the problem are obtained by the method of experiment. Finally,the upper and lower bounds of the sum of 7 numbers contained in each circle are given to each solution.Proofs are provided respectively.

关 键 词：四圆相交区域 填充数字 约束方程组 求解 

分 类 号：O151.26[理学—数学]