Mathematical puzzles have long been employed by parents and teachers to augment the standard mathematics curriculum. This paper reports on a study of urban elementary students engaged in the solution of mathematical puzzles. The work confirms that, given the opportunity, these students will construct their own, logically consistent, interpretations of the puzzle clues. In particular, these students used self-generated rules about alignment and orientation to construct meaning in ambiguous clues. Such exercises in logic bring to light the value of student approaches to problem solving, and the possibility of using these approaches as building blocks from which students might construct knowledge in the standard curriculum.