The number of nodes in the search tree can often be reduced by using the symmetry or property-preserving operations. If A is the incidence matrix of a projective plane, then the operations of permuting the rows and permuting the columns of A correspond only to reordering the lines and points of the plane. These operations preserve the property of being a projective plane. To see how they reduce the size of a search tree, consider the plane of order 10. There are choices for the first column, corresponding to the number of ways of placing 11 ones in the 111 rows. By using the row permutations, we can assume that all these ones are placed on the first 11 rows, reducing the number of partial solutions from to 1. Mathematicians love to use the phrase ``without loss of generality'' to indicate a simplification by symmetry operations. So without loss of generality, the second column has only one choice --- with a one in the first row and the remaining 10 ones in rows 12 to 21. Now, row 1 has nine remaining ones. By permuting columns, we can assume that these remaining ones of row 1 are in columns 3 to 11. Next, by row permutation, the remaining 10 ones of column 3 can be placed in rows 22 to 31. Continuing in this manner, it is not difficult to show that there is only one choice up to column 21. Beyond this point, symmetry operations are difficult to visualize, because they often involve combinations of row and column permutations.
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