Some theorems on construction of magic squares. (English) Zbl 0607.05020

Ku, Y. H.; Chen, Nan-Xian

J. Franklin Inst. 322, 253-266 (1986).

Four theorems are given on the construction of magic squares. Theorem I proves that substituting the number k in an \(N\times N\) magic square by the k-th incremental square of an \(m\times m\) magic square, the resultant mN\(\times mN\) square is a magic square. Theorem II shows that dividing an even rank \(N\times N\) magic square into four quadrants, substituting the number k in the odd-number quadrants by the k-th incremental square of a type-1 simple square and substituting the number k in the even-number quadrants by the k-th incremental square of a type-2 simple square, the resultant mN\(\times mN\) square is a magic square. In an even rank \(N\times N\) magic square, Theorem III proves that substituting the number \(k=A_{ij}\) by the k-th incremental simple square of type-1 or type-2, depending on the sum of \(i+j\) even or odd, the resultant square is a magic square. Theorem IV shows that in an even rank \(N\times N\) magic square with each row and each column having an equal number of odd and even numbers, substituting for odd numbers by the k-th incremental simple square of type-1 and for even numbers by the k-th incremental simple square of type-2, the resultant square is a magic square. Sixteen examples are given.

MSC:

05B99	Designs and configurations
05B15	Orthogonal arrays, Latin squares, Room squares

Keywords:

construction of magic squares

References:

[1]	Ku, Y.H.; Chen, N.-X., On systematic procedures for constructing magic squares, J. franklin inst., Vol. 321, 337-350, (1986), (See other references given at the end of this paper.) · Zbl 0596.05020
[2]	Ku, Y.H.; Chen, N.-X., On construction of magic squares, J. southwest jiaotong univ., (1986), 90th Anniversary Issue, Emei, China, June · Zbl 0607.05020