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I had a math teacher in high school who gave us a puzzle to solve. We had to draw a chessboard with one piece that could only move in straight paths. Diagonal movement was forbidden. The goal was to draw a path which passed every square on the board only once. The piece had to start at A1 and finish at H8. Unfortunately he never gave us the answer to the puzzle and I'm getting to the point that I believe the reason for that is because there is no such answer.

What do you guys think? Can this puzzle be solved?

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    $\begingroup$ Only once? Or can it pass through each square more than once? And Welcome to Puzzling!! $\endgroup$ – Sid Nov 24 '16 at 15:21
  • $\begingroup$ @Sid yes, only once. My bad, should've been more specific. $\endgroup$ – AMG Nov 24 '16 at 15:22
  • $\begingroup$ Does the path need to be closed; i.e., to return to its starting point? $\endgroup$ – Gareth McCaughan Nov 24 '16 at 15:22
  • $\begingroup$ @GarethMcCaughan No, not necessarily. $\endgroup$ – AMG Nov 24 '16 at 15:23
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    $\begingroup$ Are we missing some rule or can't you just move along the board in a zig-zag fashion? Or am I misunderstanding the puzzle? $\endgroup$ – Alenanno Nov 24 '16 at 15:24
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The task is

impossible (and would incidentally be likewise impossible with a knight)

because

the total number of moves has to be 63, and each move changes the colour of the square the piece is on, so the piece must end on a square of colour opposite to the one it started on.

An earlier version of the question didn't have the prohibition on diagonal moves. In that case

the task is possible

because

we can use a zig-zagging path made mostly of diagonals, as follows. Start on a1. Move east to b1, and then northwest to a2. Move north to a3, and then southeast to c1. Move east to d1, and then northwest to a4. Keep going in this fashion; you end up on h8 as required.

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  • $\begingroup$ Sorry for deleting the comment, I didn't think it would be relevant anymore. I posted this link: math.stackexchange.com/questions/479204/… $\endgroup$ – user14478 Nov 24 '16 at 15:40
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    $\begingroup$ Hmm makes sense. Conclusion: my teacher was d-ck. $\endgroup$ – AMG Nov 24 '16 at 15:45

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