Unified Construction of Normal Bimagic Squares of Doubly Even Orders Based on Quasi Bimagic Pairs

• Published in: Journal of physics. Conference series Vol. 1575; no. 1; pp. 12172 - 12178
• Main Authors: Pan, Fengchu, Huang, Xuejun
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 01.06.2020
• Subjects: 05C05; 05C50; 15A03; 15A18; 15B35; Arrays; Integers; Matrices (mathematics); Physics
• ISSN: 1742-6588; 1742-6596
• DOI: 10.1088/1742-6596/1575/1/012172

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 2n is normal if its entries are 4n2 consecutive odd integers 1−4n2, 3−4n2,...,4n2−3, 4n2−1. A normal bimagic square of order 4n 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 2n × 2n matrices consisting of odd integers, we give a unified and very simple construction of normal bimagic square H of order 4n for all n ≥ 2 : H=4n( A AR −RA −RAR )+( B −BR RB −RBR ) . 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.