Journal article
The Existence of Irrational Diagonally Ordered Magic Squares
Graphs and combinatorics, Vol.32(5), pp.2159-2170
09/2016
DOI: 10.1007/s00373-016-1682-2
Abstract
Let M=(mi,j) be a magic square, where 0≤mi,j≤n2−1, 0≤i,j≤n−1. M is called diagonally ordered if both the main diagonal and the back diagonal, when traversed from left to right, have strictly increasing values. Let M=nA+B, where A=(ai,j), B=(bi,j), 0≤ai,j, bi,j≤n−1 for 0≤i,j≤n−1. M is called rational if both A and B possess the property that the sums of the n numbers in every row and every column are the same; otherwise, M is said to be irrational. In this paper, a pair of weakly diagonally ordered irrational orthogonal matrices (WDOIOM for short) is introduced to construct an irrational diagonally ordered magic square (IDOMS). It is proved that there exists a WDOIOM(n) for each positive integer n≥5, and there does not exist a WDOIOM(n) for n∈{2,3,4}. Consequently, it is proved that there exists an IDOMS(n) for each positive integer n≥5, and there does not exist an IDOMS(n) for n∈{2,3,4}.
Details
- Title: Subtitle
- The Existence of Irrational Diagonally Ordered Magic Squares
- Creators
- H Yu - Department of Mathematics Guangxi Normal University Guilin 541004 People’s Republic of ChinaD Wu - Department of Mathematics Guangxi Normal University Guilin 541004 People’s Republic of ChinaH Zhang - Computer Science Department The University of Iowa Iowa City IA 52242 USA
- Resource Type
- Journal article
- Publication Details
- Graphs and combinatorics, Vol.32(5), pp.2159-2170
- Publisher
- Springer Japan; Tokyo
- DOI
- 10.1007/s00373-016-1682-2
- ISSN
- 0911-0119
- eISSN
- 1435-5914
- Grant note
- 11271089 / National Natural Science Foundation of China (http://dx.doi.org/10.13039/501100001809)
- Language
- English
- Date published
- 09/2016
- Academic Unit
- Computer Science
- Record Identifier
- 9984002404702771
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