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| calimero
| | Joined: 06 Jul 2006 | | Posts: 2 | | : | |
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 Posted: Thu Jul 06, 2006 3:17 pm Post subject: |
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| Nick70 wrote: | If you look only at the unique solutions, there are only two grids:
1234 1234
3412 3421
2143 2143
4321 4312
every other grid can be obtained from one of these two by permutating the digits, swapping rows or columns common to a box, rotating or flipping the whole grid. |
Interesting theorem, Nick70... Which transformations would you apply to:
1 2 3 4
3 4 1 2
2 3 4 1
4 1 2 3
to make it one of the two unique solutions above? |
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| Ruud
Site Admin
 | | Joined: 17 Sep 2005 | | Posts: 708 | | : | | Location: Netherlands |
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| Posted: Thu Jul 06, 2006 4:09 pm Post subject: |
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Since Nick has posted his theorem more than a year ago and is unlikely to answer, I'll reply instead:
1 2 3 4
3 4 1 2
2 3 4 1
4 1 2 3
Step 1: rotate 270 degrees
4 2 1 3
3 1 4 2
2 4 3 1
1 3 2 4
Step 2 swap upper - lower bands
2 4 3 1
1 3 2 4
4 2 1 3
3 1 4 2
step 3: relabel digits
1 2 3 4
4 3 1 2
2 1 4 3
3 4 2 1
Ruud. |
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| calimero
| | Joined: 06 Jul 2006 | | Posts: 2 | | : | |
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| Posted: Thu Jul 06, 2006 4:17 pm Post subject: |
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| Thanks Ruud... this looks very interesting. I'm reading through the rest of the posts to find more useful information, but was this generalized to n^2 x n^2? |
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