What's a multimagic square?
A magic square is
bimagic
(or 2multimagic) if it remains magic after each of its numbers have been
squared.
By extension, a square is Pmultimagic
if it remains magic after each of its numbers have been replaced by their kth
power (for k=1, 2, ..., to P).
A reminder about what is a magic square. It is a square n x n sized
(or nth 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 before JC! It is 3rd
ordered, so contains all the numbers from 1 to 9.
Chinese magic square 3x3

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 a nthorder 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 2multimagic). 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 
=? 
=? 
=? 
=? 
=? 
=? 
so be it:

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.
goto
http://www.multimagie.com
