N-Queens Solver

The interest is in how few arrangements need checking

Placing eight queens so none attacks another has 92 solutions, and finding them by examining every way to put eight queens on sixty-four squares would mean checking more than four billion arrangements. Backtracking checks about two thousand. The saving comes from abandoning a branch the moment it becomes impossible rather than completing it and testing at the end, and that single idea is the whole of constraint search. Watching the ratio grow with board size is the clearest demonstration of why it matters.

The counts have no formula

There are 92 solutions for eight queens, 724 for ten, 14200 for twelve, and 2.28 times ten to the fifteen for twenty-seven, which took a distributed computation years to establish. No closed form is known and none is expected; the sequence is computed rather than derived. That is unusual for a problem this easy to state, and it is why the counts are still being pushed forward one board size at a time.

Two and three queens are impossible

There is no solution for a two by two or three by three board, and there is at least one for every size from four upward. The small impossible cases are worth seeing because they are the only sizes where exhaustive search is small enough to check by hand, and they make the constraint concrete: on three squares a side, the diagonals reach everywhere the rows and columns do not.

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