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 know that a property of magic squares is that <math>a+i=2e</math>. Similarly, <math>a^2+i^2=(2e)^2</math>. Therefore <math>(a-i)^2=2(a^2+i^2)-(a+i)^2=4e^2-4e^2=0</math>. It follows that <math>a=e=i</math>. The same holds for all lines going through the center.
For 4x4 squares, Luke Pebody was able to show by similar methods that the only 4x4 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 |
| a | a | b | b |
| b | b | a | a |
| a | a | b | b |
| b | b | a | a |
| 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 |
See also
- Magic square
- Trimagic square
- Multimagic square
- Magic cube
- Bimagic cube
- Trimagic cube
- Multimagic cube
External links
- Aale de Winkel's listing of all 80 bimagic squares of order 8.fr:Carré bimagique