The authors supplement the well-known upper and lower box-counting product inequalities. If we denote \(\dim_{LB} F\) and \(\dim_B F\) as the lower box-counting dimension and box-counting dimension of a set \(F\), respectively, then the authors obtain the following new formula: \[ \begin{aligned}\dim_{LB} F+\dim_{LB} G \leq \dim_{LB} (F\times G) \leq \min(\dim_{LB} F+\dim_B G, \dim_B F+\dim_{LB} G)\\ \leq \max(\dim_{LB} F+\dim_B G, \dim_B F+\dim_{LB} G) \leq \dim_B (F\times G) \leq \dim_B F+\dim_B G\end{aligned} \] for subsets of metric spaces. The authors also develop a procedure for constructing sets so that the upper and lower box-counting dimensions of these sets and their product can take arbitrary values satisfying the above product formula. In particular, the authors illustrate how badly the products of both the lower and upper box-counting dimensions can behave.