Abstract: We show that register allocation can be viewed as solving a collection of puzzles. We model the register file as a puzzle board and the program variables as puzzle pieces. We model pre-coloring by letting some of the puzzle pieces be already immovably placed on the puzzle board, and we model register aliasing by letting pieces have a plurality widths. For a wide variety of computer architectures, we can solve the puzzles in polynomial time. Puzzle solving is independent of spilling, that is, puzzle solving can be combined with a wide variety of approaches to spilling.