Dissecting brick into bars

Abstract

An $N$-dimensional parallelepiped will be called a bar if and only if there are no more than $k$ different numbers among the lengths of its sides (the definition of bar depends on $k$). We prove that a parallelepiped can be dissected into finite number of bars iff the lengths of sides of the parallelepiped span a linear space of dimension no more than $k$ over $\QQ$. This extends and generalizes a well-known theorem of Max Dehn about partition of rectangles into squares. Several other results about dissections of parallelepipeds are obtained.