Handbook of Constraint Programming 329 Edited by F. Rossi, P. van Beek and T. Walsh c 2006 Elsevier All rights reserved Chapter 10 Symmetry in Constraint Programming Ian P. Gent, Karen E. Petrie, Jean-Franc ¸ois Puget Symmetry in constraints has always been important but in recent years has become a ma- jor research area in its own right. A key problem in constraint programming has long been recognised: search can revisit equivalent states over and over again. In principle this prob- lem has been solved, with a number of different techniques. As we write, research remains very active for two reasons. First, there are many difficulties in the practical application of the techniques that are known for symmetry exclusion, and overcoming these remain important research problems. Second, the successes achieved in the area so far have en- couraged researchers to find new ways to exploit symmetry. In this chapter we cover both these issues, and the details of the symmetry exclusion methods that have been conceived. Figure 10.1: The solution to the puzzle of finding a chess position containing nine queens and a king of each colour, with the rule that no piece is on the same line (row, column or diagonal) as any queen of the opposite colour. Up to symmetry, the solution is unique.