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It is well known that a lone knight can move all around the board covering each square once and only once. Is this possible when the lone knight starts only on certain squares ? How many unique solutions does this problem have now that powerful engines exist?

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Is this possible when the lone knight starts only on certain squares ?

If you stop and think about this for a second you will realize that the answer is obviously "Yes". If you have a "Knight's Tour" (that's what it is called) starting on square X then by definition it also goes through square Y (for all squares Y on the chessboard). Hence the exact same moves, just starting on Y, will also be a Knight's Tour.

How many unique solutions does this problem have

According to Wikipedia:

On an 8 × 8 board, there are exactly 26,534,728,821,064 directed closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are rotations and reflections). The number of undirected closed tours is half this number, since every tour can be traced in reverse.

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  • I don't follow. What do you mean "the exact same moves, just starting on Y". Wouldn't that just be the same tour, but shifted to start at Y instead of X? Which would be invalid as it would go off the board? And to the second part: you quoted the number of closed tours, but the question seems to ask about open tours.
    – Kevin Wang
    Jul 2 at 15:30
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    @Kevin, imagine for simplicity we are talking a rook tour of a 2 x 2 board. This could be a1-a2-b2-b1-a1. Now, if we start at b2, we can just follow this path again b2-b1-a1; then pick up the start a1-a2-b2 and we're done
    – sjb
    Jul 4 at 9:03
  • @sjb again, that's only for closed tours, no?
    – Kevin Wang
    Jul 5 at 13:04

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