Some theorems on construction of magic squares
Y.H. Ku, Nan‐Xian Chen · Journal of the Franklin Institute · 1986
Four theorems are given on the construction of magic squares. Theorem I proves that substituting the number k in a N×N magic square by the kth incremental square of a m×m magic square, the resultant mN×mN square is a magic square. Theorem II shows that dividing an even rank N×N magic square into four quadrants, substituting the number k in the odd-number quadrants by the kth incremental square of a type-1 simple square and substituting the number k in the even-number quadrants by the kth incremental square of a type-2 simple square, the resultant mN×mN square is a magic square. In an even rank N×N magic square, Theorem III proves that substituting the number kAij by the kth 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×N magic square with each row and each column having an equal number of odd and even numbers, substituting for odd numbers by the kth incremental simple square of type-1 and for even numbers by the kth incremental simple square of type-2, the resultant square is a magic square. Sixteen examples are given.