A symmetry approach to number tricks

We generalize the classical “1089-number trick”, which states that a certain combination of addition, subtraction and swapping the digits of a three-digit number will always output 1089. More precisely, we show that any pair of zero divisors $fg=0$ in the group ring $\mathbb{Z}[{\mathrm{\Sigma }}_n]$ on the n-th symmetric group gives rise to a partition of the set of n-digit numbers into subsets $U_{\mathbf{e}}$ defined by linear inequalities, such that the zero divisors act constantly on each $U_{\mathbf{e}}$ and hence define a number trick.