A multigrade is a set of equations in which the sums of powers of two different sets of numbers are the same for several different exponents. The simplest example is:

 

1 + 6 + 8 = 2 + 4 + 9
1² + 6² + 8² = 2² + 4² + 9².

 

Another multigrade is:

 

1 + 8 + 10 + 17 = 36 = 2 + 5 + 13 + 16
1² + 8² + 10² + 17² = 454 = 2² + 5² + 13² + 16²
1³ + 8³ + 10³ + 17³ = 6426 = 2³ + 5³ + 13³ + 16³.

 

Remarkably, if any integer is added to all the terms of a multigrade it will still hold. Adding 1 to the example above, gives the multigrade (2, 9, 11, 18); (3, 6, 14, 17) (n = 1, 2, 3).

 

Some high-order multigrades include:

 

(1, 50, 57, 15, 22, 71); (2, 45, 61, 11, 27, 70); (5, 37, 66, 6, 35, 67) (n = 1, 2, 3, 4, 5)
and
(1, 9, 25, 51, 75, 79, 107, 129, 131, 157, 159, 173); (3, 15, 19, 43, 89, 93, 97, 137, 139, 141, 167, 171) (n = 1, 3, 5, 7, 9, 11, 13).