  We give a short proof of the formula $n^p=\sum_{\ell=0}^{p-1} (-1)^{\ell}
c_{p,\ell} F^{p-\ell}_n$, where $F^{p-\ell}_n$ is the figurate number and
$c_{p,\ell}$ is the number of $(p-\ell)$-dimensional facets of $p$-dimensional
simplices obtained by cutting the $p$-dimensional cube.

Number Theory