Interpreting a Sum of Third Powers by using a Geometric Assembly in Four Dimensions

Abstract

We present new combinatorial proofs of Nicomachus's Theorem for the sum of the third powers of the first n natural numbers. The key step is that we define a 4-dimensional block which comprises unit hyper-cubes. In our first proof we assemble 4 copies of this block to construct a rectangular solid. In our second proof we partition the block into two parts, then map and reassemble them into a solid whose shape is a step triangle along two coordinate axes, and is likewise on the other two. For corollaries we present a q-analogue of this identity using taxicab distance, and we interpret q-analogues from 4 different (sets of) authors, using taxicab distances from different starting points.