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Sum of cubes in Archimedes way[siu-kái | kái goân-sú-bé]

First part, to decompose each cubic into a sum of squares, then compose them from different direction.

Let

so

apply it to

...

To sum these n identities, let

where


Second part, make use of ; apply it to :

...

To sum these n identities,


OK, easy part now, to merge the results of first part and second part, we can get

BTW, Archimedes would use at first part, and use at second part.