Unsolved problems on magic squares

✍ Scribed by Gakuho Abe

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 577 KB
• Volume: 127
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)00462-z

✦ Synopsis

In this paper, we collect 23 unsolved problems or conjectures on magic squares, and some updated results related to these problems are mentioned.

1. Preliminaries

In this paper we collect 23 unsolved problems or conjectures on magic squares, which come from recent research. We shall propose these problems and give updated results, in particular some new interesting magic squares are presented. Many known results obtained in magic squares can be found in the following journals, which are written in Japanese: Fascinating

Puzzles Monthly (FPM), Meeting of Puzzles (MP), Researches in Magic Squares (RMS) and Month/y Abacus (MA). Basic ideas of magic squares can be found in [3, Ch. 7; 2; 4, Ch. 33. The readers who are interested in the history of magic squares and the current development in Japan can refer to [l].

A general magic square A= (uij), 0~ i, j< n -1, is a square array of n2 distinct nonnegative integers with the property that the sum of the n numbers in every row, in every column and in each diagonal is the same. The number n of rows (and of columns) of the square is called its order. A general magic square is called a magic square if the natural numbers from 1 to n2 are used. A square array consisting of n2 distinct nonnegative integers (or natural numbers from 1 to n') is called a general semi-magic square (or a semi-magic square) if the sums of the numbers in each row and each column are the same. A magic square (Uij) is said to be symmetric if Uij+U,_1_i._1_j=n2+1, for all O,<i,j<n-1.

Furthermore, in a magic square (aij), if UOi+Uli+l+..'+Un-li+n_l and uoi+uli-l+...+un-li-n,,