Summary: We analyze a partition problem and its inverse problem both in discrete variables and in two continuous ones through dynamic programming. We show that an inverse relation and an enveloping relation hold in each case. It is shown that an optimal solution for the discrete partition can be expressed through either an upper-semi-inverse function or a lower-semi-inverse function and that the optimal solution for the continuous partition can be expressed through the (regular) inverse function. As a result, we show that the optimal partition is to partition equally in essence any quantity into quantities of the same size of \(e\). We call this optimal policy the Euler partition rule.