In many problems that require extensive searching, the solution can be described as satisfying two competing constraints, where satisfying each independently does not pose a challenge. As an alternative to tree-based and stochastic searching, for these problems we propose using an iterated map built from the projections to the two constraint sets. Algorithms of this kind have been the method of choice in a large variety of signal-processing applications; we show here that the scope of these algorithms is surprisingly broad, with applications as diverse as protein folding and Sudoku.Our survey of applications shows the difference map algorithm often achieves results comparable to much more sophisticated, special purpose algorithms. Efficient implementations of constraint projections usually are easier than designing the linear program solver at the heart of many optimization algorithms.