A magic square of squares is a magic square whose entries are distinct positive square numbers. The entries need not be consecutive, and the array of their positive square roots need not be a magic square. If that array of square roots is magic as well, the root array is a bimagic square (Rome and Yamagishi 2025).
For order three, the equal-line-sum conditions give the general integer array
 |
where 
,
,
and 
are integers, and every row, column, and main diagonal
sums to 
.
A magic square of squares would require all nine entries in this array to be distinct
positive square numbers, including its central entry
(Bremner 2001, p. 290). The existence of such a 
square remains open (Rome and Yamagishi 2025, p. 2).
For every integer order 
, a magic square of squares exists (Rome and Yamagishi
2025, p. 2). An example of order four is
 |
whose magic constant is 8515 (Rome and Yamagishi 2025, p. 2). The Parker square is a partial attempt at the order-three problem, with repeated entries and one incorrect diagonal sum.
