fourth powers.
The search for squared squares has, however, been successful for larger
squares. Magic 4×4 and 5×5 squared squares do exist. In 1770 Euler produced
an example without showing his method of construction. Whole families have
since been found linked to the study of the algebra of quaternions, the four-
dimensional imaginary numbers.
Exotic magic squares
Large magic squares may have spectacular properties. A 32×32 array has
been produced by magic square expert William Benson in which the numbers,
their squares, and their cubes all form magic squares. In 2001 a 1024×1024
square was produced in which all powers of elements up to the fifth power make
magic squares. There are many results like these.
We can create a whole variety of other magic squares if the requirements are
relaxed. The normal magic squares are in the mainstream. Removing the
condition that the sum of the diagonal elements must equal the sums of the
rows, and of the columns, ushers in a plethora of specialized results. We can
search for squares whose elements consist only of prime numbers, or we may
consider shapes other than squares which have ‘magic properties’. By going into
higher dimensions we are led to consider magic cubes and hypercubes.
But the prize for the most remarkable magic square of all, certainly for
curiosity value, must go to a humble 3×3 square produced by the Dutch
electronic engineer and wordsmith Lee Sallows:
What is so remarkable about this? First write out the numbers in words:Then count the number of letters making up each word to obtain: