If replacing each number by its square in a magic square produces another magic square, the square is said
to be a bimagic square. Bimagic squares are also called doubly magic squares, and
are 2-multimagic squares.
Lucas (1891) and later Hendricks (1998) showed that a bimagic square of order 3 is impossible for any set of numbers except the trivial case of using the same
number 9 times.
The first known bimagic square, constructed by Pfeffermann (1891a; left figure), had order 8 with magic constant 260 for the base
square and
after squaring. Another order 8 bimagic square is shown at right.
Benson and Jacoby (1976) stated their belief that no bimagic squares of order less than 8 exist, and this was subsequently proved by Boyer and Trump in 2002 (Boyer).
Pfeffermann (1891b) also published the first 9th-order bimagic square. Only a part of the first Pfeffermann's bimagic squares of both order 8 and of order 9 were published, with their completion left as puzzles to the reader and their solutions appearing two weeks later in the following issues (Boyer).
Wroblewski found the first known bimagic square using distinct (but nonconsecutive)
integers (Boyer 2006), illustrated above.
after squaring. Another order 8 bimagic square is shown at right.
bimagic square using distinct (but nonconsecutive)
integers (Boyer 2006), illustrated above.
