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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. Mathematics is the study of quantity, structure, space and change. ...
In mathematics, a magic square of order n is an arrangement of n² numbers in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. ...
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. J. R. Hendricks, born September 4, 1929; mathematician specializing in magic squares and hypercubes. ...
1998 is a common year starting on Thursday of the Gregorian calendar, and was designated the International Year of the Ocean. ...
It is well know 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 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 | or | 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
In mathematics, a magic square of order n is an arrangement of n² numbers in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. ...
In mathematics, a trimagic square is a magic square that also remains magic if all of the numbers it contains are squared or cubed. ...
In mathematics, a P-multimagic square is a magic square that remains magic even if all its numbers are replaced by their kth power for 1 ≤ k ≤ P. Thus, a magic square is bimagic if it is 2-multimagic, and trimagic if it is 3-multimagic. ...
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is...
In mathematics, a bimagic cube is a magic cube that also remains magic if all of the numbers it contains are squared. ...
In mathematics, a trimagic cube is a magic cube that also remains magic if all of the numbers it contains are squared or cubed. ...
In mathematics, a P-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their k-th power for 1 <= k <= P. Thus, a magic cube is bimagic iff it is 2-multimagic, and trimagic iff it is 3-multimagic. ...
External links - Aale de Winkel's listing of all 80 bimagic squares of order 8.
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