What's a multimagic square?
A magic square is bimagic (or 2-multimagic) if it remains magic after each of its numbers have been squared.
By extension, a square is P-multimagic if it remains magic after each of its numbers have been replaced by their k-th power (for k=1, 2, ..., to P).
A reminder about what a magic square is. It is a square n x n sized (or n-th order) where we have succeeded in placing all the numbers from 1 to n², such that the sums of the n rows, n columns and 2 main diagonals are equal. Here is a sample of the smallest magic square (except the trivial order 1). This square was discovered by the Chinese more than 2000 years B.C! It is order 3, so contains all the numbers from 1 to 9.
|
|
4 |
3 |
8 |
=15 |
|
|
9 |
5 |
1 |
=15 |
|
|
2 |
7 |
6 |
=15 |
|
=15 |
=15 |
=15 |
=15 |
=15 |
You can easilly verify that all the sums of the Chinese square are equal to 15. It is easy to demonstrate that the sums of an nth-order magic square have to be equal to n(n² + 1)/2, so in this case : 3(3² + 1) / 2 = 15.
The idea of multimagic squares started at the end of the XIXth century with the discovery in France of the first bimagic squares (or 2-multimagic). A magic square is bimagic if it remains magic after each of its numbers have been squared. Look at what happens to the Chinese square:
|
|
4² |
3² |
8² |
=? |
|
|
9² |
5² |
1² |
=? |
|
|
2² |
7² |
6² |
=? |
|
=? |
=? |
=? |
=? |
=? |
and so we get:
|
|
16 |
9 |
64 |
=89 |
|
|
81 |
25 |
1 |
=107 |
|
|
4 |
49 |
36 |
=89 |
|
=93 |
=101 |
=83 |
=101 |
=77 |
So the Chinese magic square is far from being bimagic, since the sums of the squared numbers vary between 77 and 107! G. Pfeffermann was the first to build a bimagic square, in 1890. As a result, this first bimagic square is also the first multimagic square in history.
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