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Do a Knight's Tour under the following conditions:

  • 8*8 board
  • You have to alternate between a knight move and a pawn move (advance one square)
  • No pawn advancing of two squares at any time
  • You can not land on the last rank (from the pawn's direction) with a knight move because the pawn could not advance forward then.
  • You can start on any square and with any of the two moves, but if you start on the last rank, you can't start with a pawn move.
  • It is ok to land at the first rank with a knight's move.

Note:

I don't know if this is possible or not. I just like to think about it..

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It is not possible. Such a tour would essentially dissect the board into vertical dominoes, where the pawn moves go from the bottom square of a domino to its top square, and the knight moves go from one domino to another. Note also that the checkerboard colouring means that all the dominoes have the same colouring, e.g. a black square at the bottom. It is not possible to dissect the board into 32 vertical dominoes, all with the same colouring (nor even 31 such dominoes and two single squares) so no such knight-pawn tour (closed or open) is possible.

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  • $\begingroup$ Ah, you are right. Can I still change the rules to "one or two squares" pawn moves? $\endgroup$ – Tweakimp Dec 28 '17 at 21:27
  • $\begingroup$ @Tweakimp I’m not sure that would make much difference in the outcome, though it would need a new argument. The difficulty for the knight is that it is very hard to move backwards to the lower ranks, and allowing long pawn moves seems to only make it harder. $\endgroup$ – Jaap Scherphuis Dec 28 '17 at 21:43
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I don’t think it’s doable.

You can only get to the first rank by a knight’s move, and that move has to start at the second or third rank. The next move will then visit that file’s second square.

You can also never jump to the second rank, since it would leave the first square of the file unvisitable (except if that first rank square is the starting or ending square of the tour.) Therefore, once you leave the first two ranks, there’s no coming back. Also, a single visit can only get you half of the first rank squares.

There are a couple of special cases involving the start and end sequences that are too complex to write in using my phone, but all of those seem to lead to the same conclusion: you can’t get all the first rank squares.

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