Unified Construction of Normal Bimagic Squares of Doubly Even Orders Based on Quasi Bimagic Pairs. (June 2020)

• Record Type: Journal Article
• Authors: Pan, Fengchu; Huang, Xuejun
• Is Part Of: Journal of physics. Volume 1575(2020)
• Publication Date: 2020-06
• Subjects: 05C50 -- 15B35 -- 05C05 -- 15A03 -- 15A18; Physics -- Congresses; 530.5
• DOI: 10.1088/1742-6596/1575/1/012172
• Languages: English
• ISSNs: 1742-6588

Abstract

Abstract: A general magic square of order n is an n × n matrix consisting of integers in such a way that the sum of all elements in each row, each column, main diagonal and back diagonal is the same number called the magic sum of this matrix. A general magic square of order 2 n is normal if its entries are 4 n 2 consecutive odd integers 1 − 4 n 2, 3 − 4 n 2, …, 4 n 2 − 3, 4 n 2 − 1. A normal bimagic square of order 4 n is a normal magic square such that the sum of squares of all elements in each row, each column, main diagonal and back diagonal is the same number. Using the reflection matrix R and a quasi bimagic pair ( A, B ) where A and B are two special 2 n × 2 n matrices consisting of odd integers, we give a unified and very simple construction of normal bimagic square H of order 4 n for all n ≥ 2 : H = 4 n ( A A R − R A − R A R ) + ( B − B R R B − R B R ) . We construct a quasi bimagic pair by means of orthogonal diagonal latin squares for n ≠ 3 and by means of the computer seeking for n = 3.