https://www.hexwiki.net/api.php?action=feedcontributions&feedformat=atom&user=Dvd+AvinsHexWiki - User contributions [en]2026-04-16T19:49:44ZUser contributionsMediaWiki 1.23.15https://www.hexwiki.net/index.php/Sixth_row_template_problemSixth row template problem2009-01-10T22:25:17Z<p>Dvd Avins: include no higher stone in intro, clarify language</p>
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<div>As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|open]]:<br />
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Is there a one stone sixth row [[template]] that uses no stones higher than the sixth row?<br />
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More generally, it is still unknown whether one stone edge templates that use no cell higher than the initial stone) can be found for all heights. Such [[Edge templates with one stone|templates]] have been found for sizes up to 5 but none above. Answering with "No" to the former question answers the latter.<br />
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== Description ==<br />
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Is there a number m such that the game on the board of width m designed as follows, with Blue's turn to play, is won by Red ?<br />
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<hex> R7 C11<br />
1:HHHHHVHHHHH<br />
2:_____V_____<br />
</hex><br />
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== Generalisation ==<br />
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The general problem of knowing if there is n such that there is no one stone edge template on the n^th row<math>n^th</math> is also referred to as the n-th row template problem.<br />
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== Possible paths to answer ==<br />
===By "hand"...===<br />
====...answering "Yes" ====<br />
This would involve placing a stone on the 6th row of a sufficiently wide board, and showing how to always connect to the bottom. (Note this does not necessarily identify the minimal template needed.) <br />
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Here is a start. Just from [[edge template IV1a]] and [[edge template IV1b]], Blue's first move must be one of the following:<br />
<hex><br />
R7 C19 Q0<br />
1:BBBBBBBBBRBBBBBBBBB<br />
Rj2<br />
Si3 Sj3<br />
Si4<br />
Sg5 Sh5 Si5 Sj5 <br />
Sf6 Sg6 Si6 Sj6<br />
Se7 Sf7 Sg7 Sh7 Si7 Sj7<br />
</hex><br />
Many of these moves will be easy to dismiss. Others will benefit from the [[Parallel ladder]] trick. Of course, symmetry will cut our work in half!<br />
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We can dispose of 3 moves on the left (and, using mirror symmetry, the corresponding 3 moves on the right), as follows:<br />
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<hex><br />
R7 C19 Q0<br />
1:BBBBBBBBBRBBBBBBBBB<br />
Rj2<br />
Pg5 <br />
Pf6 <br />
Pe7<br />
N:on Ri4 Bi5 Rh5 Bg7 Rh6 Bh7<br />
</hex><br />
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At this point, we can use the [[Parallel ladder]] trick as follows:<br />
<br />
<hex><br />
R7 C19 Q0<br />
1:BBBBBBBBBRBBBBBBBBB<br />
Rj2<br />
Pg5 <br />
Pf6 <br />
Pe7<br />
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7<br />
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3<br />
</hex><br />
<br />
====...answering "No" ====<br />
This would involve showing how to connect (in the diagram above) the Blue stones to the right (plus Blue stones on the far right edge) to Blue stones on the left (plus Blue stones on the far left edge), no matter how wide the board is.<br />
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=== Computer Aided demonstration ... ===<br />
==== ... answering "Yes" ====<br />
Such a proof would use the computer to find the template and it's [[carrier]]. Afterwards it should be easy to manually check that every Blue intrusion does not prevent Red from connecting to bottom.<br />
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==== ... answering "No" ====<br />
TODO<br />
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== See Also ==<br />
* [[Theory]]<br />
* [[User:Wccanard|Wccanard]]<br />
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== External link ==<br />
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* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.<br />
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[[category:theory]]<br />
[[category:templates]]<br />
{{stub}}</div>Dvd Avins