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

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
Full Text:

### References:

  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  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
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