Mathematical_Recreations-Kraitchik-2e

176 Mathematical Recreations
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FIGURE 53b. Variations of the Constellations of Figure liSa.
10. MULTIMAGIC SQUARES
A magic square is multimagic of degree p if the square
formed by replacing each element by its kth power is magic,
for every k from 1 to p. Similarly we may define multimagic
series. The magic constant of degree 1 is the nth part of the
sum of the first n2 integers, !n(n2 + 1). The magic constant
of degree 2 is the nth part of the sum of the first n2 squares,
in(n2 + 1) (2n2 + 1). The magic constant of degree 3 is the
nth part of the sum of the first n2 cubes, tna(n2 + 1)2; and so
on. n numbers form a magic series of degree p if the sum of
their kth powers is the magic constant of degree k, for every
k from 1 to p.
Here are some statistics on multimagic series. Nl is the
number of magic series of order n, N2 is the number of bimagic
series of order n, Na is the number of trimagic series of
order n.
n 2 3 4 5 6 7 8 9 10 11
Nl 2 8 86 1,394 - - - - - -
N2 2 8 98 1,844 38,039 - - -
N3 2 2 - - 115 41 - 961
Tarry was the first to give a method of forming a trimagic