Bimagic square

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In mathematics, a bimagic square is a magic square that also remains magic if all of the numbers it contains are squared. The first known bimagic square has order 8 and magic constant 260; it has been conjectured by Bensen and Jacoby that no nontrivial bimagic squares of order less than 8 exist. This was shown for magic squares containing the elements 1 to n2 by Boyer and Trump.

However, J. R. Hendricks was able to show in 1998 that no bimagic square of order 3 exists, save for the trivial bimagic square containing the same number nine times. The proof is fairly simple: let the following be our bimagic square.

a b c
d e f
g h i

It is well known that a property of magic squares is that a + i = 2e. Similarly, a2 + i2 = 2e2. Therefore (a − i)2 = 2(a2 + i2) − (a + i)2 = 4e2 − 4e2 = 0. It follows that a = e = i. The same holds for all lines going through the center.

For 4×4 squares, Luke Pebody was able to show by similar methods that the only 4×4 bimagic squares (up to symmetry) are of the form

a b c d
c d a b
d c b a
b a d c

or

a a b b
b b a a
a a b b
b b a a

An 8×8 bimagic square.

16 41 36 5 27 62 55 18
26 63 54 19 13 44 33 8
1 40 45 12 22 51 58 31
23 50 59 30 4 37 48 9
38 3 10 47 49 24 29 60
52 21 32 57 39 2 11 46
43 14 7 34 64 25 20 53
61 28 17 56 42 15 6 35


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