  We investigate a stochastic process where a rectangle breaks into smaller
rectangles through a series of horizontal and vertical fragmentation events. We
focus on the case where both the vertical size and the horizontal size of a
rectangle are discrete variables. Because of this constraint, the system
reaches a jammed state where all rectangles are sticks, that is, rectangles
with minimal width. Sticks are frozen as they can not break any further. The
average number of sticks in the jammed state, $S$, grows as $S\simeq
A/\sqrt{2\pi\ln A}$ with rectangle area $A$ in the large-area limit, and
remarkably, this behavior is independent of the aspect ratio. The distribution
of stick length has a power-law tail, and further, its moments are
characterized by a nonlinear spectrum of scaling exponents. We also study an
asymmetric breakage process where vertical and horizontal fragmentation events
are realized with different probabilities. In this case, there is a phase
transition between a weakly asymmetric phase where the length distribution is
independent of system size, and a strongly asymmetric phase where this
distribution depends on system size.
