Let’s work through the problem of the knight’s tour problem.

A knight’s tour is a sequence of moves by a knight on a chessboard such that all squares are visited once.

Given N, write a function to return the number of knight’s tours on an N by N chessboard.

Solution

The brute force solution is here to try every possible permutation of moves and see if they’re valid. That would be pretty much computationally infeasible, since we have N * N possible spots. That would be N^2!

We can improve the performance on this using backtracking, similar to the N queens problem (#38) or the flight itinerary problem (#41). The basic idea is this:

• For every possible square, initialize a knight there, and then:
• Try every valid move from that square.
• Once we’ve hit every single square, we can add to our count.

We’ll represent the tour as just a sequence of tuples (r, c). To speed things up and to avoid having to look at the whole tour to check whether a space has been used before, we can create an N by N board to mark whether we’ve seen it already.

This takes O(N * N) space and potentially O(8^(N^2)) time, since at each step we have potentially 8 moves to check, and we have to do this for each square.

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