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Talk:Knight's tour: Difference between revisions
→The 7x7 problem: more on 7x7 problem
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RothTieBreaker | 70.00% 0.00% 70.00% 0.00% 90.00% 0.00% 80.00% 90.00% 0.00% 100.00% 50.00%</pre>
:: The notable thing about the pattern of failure in 7x7 is that tours started every other square fail and this shifts by one every rank. The symmetries of the squares above hold for all tie breakers and the overall pattern of failure is a cross-hatching. Like this:
<pre> a b c d e f g
+---------------+
7 | T - T - T - T | 7
6 | - T - T - T - | 6
5 | T - T - T - T | 5
4 | - T - T - T - | 4
3 | T - T - T - T | 3
2 | - T - T - T - | 2
1 | T - T - T - T | 1
+---------------+
a b c d e f g</pre>
::: Where T indicates that Warndsdorf/Roth found a tour and - indicates a failure to find a tour. A quick estimate of the number of paths to be test for an exhaustive search confirmed that would be impossible. I tried a number of searches to find references to unsolvable knights tours on 7x7 boards and found none. I find myself wonder if there are any solutions on any of those failed squares. --[[User:Dgamey|Dgamey]] 10:59, 6 June 2011 (UTC)
::: Running tours for all squares looking at the failed 7x7 start at a2 running 48 moves with a3 empty and all symmetries found no reverse paths either. --[[User:Dgamey|Dgamey]] 10:56, 3 June 2011 (UTC)
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