Self-complement Magic Squares


If complementing a magic square results in another aspect of the square itself, the square is called self-complement(ary). These include the associative squares and squares in which all complement pairs are side-to-side, i.e., left-to-right or top-to-bottom, symmetric.

Odd Order

All odd order self-complement magic squares are associative.

Even Order

Doubly-even self-complement magic squares include associative and side-to-side symmetric; singly-even are only side-to-side symmetric, (there are no singly-even associative magic squares).

There are 352 order 4 self-complement magic squares, (48 associative, 304 side-to-side symmetric).

SelfComplementSquares makes side-to-side symmetric squares using the double border method:


Heinz, Harvey "Self-similar Magic Squares"

Suzuki, Mutsumi "Magic Squares"