Bimagic Square
The first known bimagic square has order 8 and magic constant 260 and a bimagic constant of 11180.
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 . Similarly, . Therefore . It follows that . 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 |
Nontrivial bimagic squares are now (2010) known for any order from eight to 64. Li Wen of China created the first known bimagic squares of orders 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62 filling the gaps of the last unknown orders.
Read more about this topic: Multimagic Square
Famous quotes containing the word square:
“If the physicians had not their cassocks and their mules, if the doctors had not their square caps and their robes four times too wide, they would never had duped the world, which cannot resist so original an appearance.”
—Blaise Pascal (1623–1662)