Can a knight jump into a square in both an odd and even number of moves?

Question

For example, a knight can go from b1 to d5 in 2 moves and in many other routes by an even number of moves. However, can it go to d5 in an odd number of moves?

I was wondering this because it would give me a little bit of insight into some theory.

Any help would be appreciated. Thank you.

score 9 · Accepted Answer · answered Dec 22 '16 at 04:32

9

No. There's a fairly simple proof by looking at the square colors: they alternate each move. That means the parity of the move count to travel between two squares is fixed.

answered Dec 22 '16 at 04:32

Cleveland

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Oh haha, I should have thought about that. Well, thank you for clearing. – user2999870 Dec 22 '16 at 04:43
2

This is actually relevant to actual play, e.g. from http://www.janko.at/Retros/d.php?ff=8/8/8/8/8/8/p1pK1n2/k7 White draws with Kc1! (but not Kc2? Nd3 0-1). – Noam D. Elkies Dec 23 '16 at 04:47

Can a knight jump into a square in both an odd and even number of moves?

1 Answers1