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I know what a 3x10 looks like, but I cannot seem to find a distinguishable pattern to extend it to a 3x14.

The 3x10 pattern I'm using looks like the one at the top right of figure 6 of this paper.

Any help would be greatly appreciated.

fossdeep
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  • Welcome to Math.SE! This site will support including images in your post (there's a button in the Edit dialog to assist uploading). Perhaps you or someone else will be able to include the closed $3\times 10$ pattern. – hardmath Jun 02 '15 at 20:00

2 Answers2

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Scwhenk's proof of necessary and sufficient conditions on $m,n$ for when there is a knight's tour on an $m \times n$ board is based on decomposing the board into a union of smaller boards, and has 9 base cases. You seem to want a way to "extend" a $3 \times 10$ tour into a $3 \times 14$ tour. The only obvious straightforward was to do this with Schwenk decomposition is to first tour the $3 \times 10$ and then tour the remaining $3 \times 4$. But $3 \times 4$ has no tour according to Schwenk's theorem. Thus it is unlikely, at least using the standard Schwenk decomposition approach naively, that you will be able to easily figure out how to do $3 \times 14$ by examining a single $3 \times 10$.

user2566092
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Compare:

$3\times10:$

$3\times12:$

$3\times14:$

$3\times16:$

Note

These are not the only possible tours. There are $16, 176, 1536$ and $15424$ complete Hamiltonian cycles for $3\times10,12,14,16$ respectively.

martin
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  • I really was not expecting the remaining 3x4 at the right hand to be reversed. Where did you get these images? Would you happen to have two comparing 3x12 to 3x16? – fossdeep Jun 02 '15 at 20:53
  • @fossdeep made them in Mathematica. Hold on a sec & I'll add the other two. – martin Jun 02 '15 at 20:58
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    I actually was trying to upvote, but it says I need more reputation. And I didn't want to accept until I waited for a response on the other two diagrams. Just incase it would get buried or something once it was declared answered. – fossdeep Jun 02 '15 at 21:05
  • @fossdeep no problem ;) – martin Jun 02 '15 at 21:48