(pp. 101-102)

Multigrades consist of integers whose sums are the same for more than one power, such as\[{{1}^{k}}+{{5}^{k}}+{{8}^{k}}+{{12}^{k}}={{2}^{k}}+{{3}^{k}}+{{10}^{k}}+{{11}^{k}}\]for*k*= 1, 2, 3. We can write this as$[1,5,8,12]{{=}^{3}}[2,3,10,11]$.

It can be shown that to have agreement of the first*k*powers, a multigrade must have at least*k*+ 1 elements on each side.

Multigrades are mostly of recreational interest. Two theorems that can be verified by algebra are:

Theorem*If*\[[{{a}_{1}},{{a}_{2}},\ldots ,{{a}_{r}}]{{=}^{k}}[{{b}_{1}},{{b}_{2}},,\ldots ,{{b}_{r}}]\]*then*\[[{{a}_{1}}+c,{{a}_{2}}+c,\ldots ,{{a}_{r}}+c]{{=}^{k}}[{{b}_{1}}+c,{{b}_{2}}+c,\ldots ,{{b}_{r}}+c].\]

This allows multigrades to be normalized with smallest entry 0 or 1.

Theorem*If*\[[{{a}_{1}},{{a}_{2}},\ldots ,{{a}_{r}}]{{=}^{k}}[{{b}_{1}},{{b}_{2}},,\ldots ,{{b}_{r}}]\]*then*\[[{{a}_{1}},{{a}_{2}},\ldots ,{{a}_{r}},{{b}_{1}}+c,{{b}_{2}}+c,\ldots ,{{b}_{r}}+c]{{=}^{k+1}}[{{b}_{1}},{{b}_{2}},\ldots ,{{b}_{r}},{{a}_{1}}+c,{{a}_{2}}+c,\ldots ,{{a}_{r}}+c].\]

This allows multigrades of higher order to be constructed...