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

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 2n is normal if its entries are 4n 2 consecutive odd integers 1−4n 2, 3−4n 2,…,4n 2 −3, 4n 2 −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 : . 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.