The box problem is a problem concerning a rectangular sheet of paper which is cut from each corner by an identical square and finding the size of the excised square so that the resulting paper folds into an open box of maximum volume. In [2], Hotchkiss obtained the necessary and sufficient conditions for the existence of the integral solution and the rational solution to the box problem. Our aim in this paper is to study various properties concerning the integral volume of the box. Under the existence of the integral solution, some conditions about the integral volume are obtained and the minimum volume is known. Moreover, there are finitely many box problems that have the same solution and the same integral volume.

box problem, multiplicative structure.