Difference between revisions of "Sixth row template problem"

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As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|open]]:
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In January 2009, it was an open problem, initially stated by javerberg and wccanard in the LG forum, whether there is a one stone sixth row [[edge template]] that uses no stones higher than the sixth row.
  
Is there a one stone sixth row [[template]] that uses no stones higher than the sixth row?
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The answer is "yes", and [[edge template VI1a]] is such a template.
  
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.
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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 6 but none above.
  
 
== Description ==
 
== Description ==
  
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 ?
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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?
  
 
<hex> R7 C11
 
<hex> R7 C11
Line 16: Line 16:
 
== Generalisation ==
 
== Generalisation ==
  
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.
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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 is also referred to as the ''n''th row template problem.
  
== Possible paths to answer ==
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One of the way to prove if there is such an ''n'' is to prove if there is such ''n''−1 for which an (''n''−1)-row-template with one defender stone originally placed next to attacker stone in the same row. Of course if such template exists ''n''th-row-template is still not proven to exist.
===By "hand"...===
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====...answering "Yes" ====
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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.
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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:
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Here is an example for ''n'' = 7
<hex>
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<hex> R8 C11
R7 C19 Q0
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1:HHHHHVHHHHH
1:BBBBBBBBBRBBBBBBBBB
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2:+++++V+++++
Rj2
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3:____HV_____
Si3 Sj3
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Si4
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Sg5 Sh5 Si5 Sj5
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Sf6 Sg6 Si6 Sj6
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Se7 Sf7 Sg7 Sh7 Si7 Sj7
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</hex>
 
</hex>
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!
 
  
We can dispose of 3 moves on the left (and, using mirror symmetry, the corresponding 3 moves on the right), as follows:
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For now it seems like there is no solution for above example.
  
<hex>
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== Possible paths to answer ==
R7 C19 Q0
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1:BBBBBBBBBRBBBBBBBBB
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Rj2
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Pg5
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Pf6
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Pe7
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N:on Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
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</hex>
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At this point, we can use the [[Parallel ladder]] trick as follows:
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=== If the answer is "yes" ===
 
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<hex>
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R7 C19 Q0
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1:BBBBBBBBBRBBBBBBBBB
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Rj2
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Pg5
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Pf6
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Pe7
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Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
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N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
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</hex>
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Now, let's deal with the remaining intrusions!:
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=====One remaining intrusion on the first row=====
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<hex>
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R7 C19 Q0
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1:BBBBBBBBBRBBBBBBBBB
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Rj2
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Bf7
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</hex>
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=====The other remaining intrusion on the first row=====
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<hex>
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R7 C19 Q0
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1:BBBBBBBBBRBBBBBBBBB
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Rj2
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Bg7
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</hex>
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=====The remaining intrusion on the second row=====
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<hex>
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R7 C19 Q0
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1:BBBBBBBBBRBBBBBBBBB
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Rj2
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Bg6
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</hex>
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=====The remaining intrusion on the third row=====
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<hex>
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R7 C19 Q0
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1:BBBBBBBBBRBBBBBBBBB
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Rj2
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Bh5
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</hex>
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=====The remaining intrusion on the fourth row=====
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<hex>
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R7 C19 Q0
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1:BBBBBBBBBRBBBBBBBBB
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Rj2
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Bi4
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</hex>
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=====The remaining intrusion on the fifth row=====
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<hex>
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R7 C19 Q0
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1:BBBBBBBBBRBBBBBBBBB
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Rj2
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Bi3
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</hex>
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====6th row template====
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<hex>
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R7 C14 Q0
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1:BBBBBBBBBRBBBBB
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Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2
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Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3
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Sa4 Sb4 Sc4 Sd4 Sn4
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Sa5 Sb5
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Sa6
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</hex>
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====...answering "No" ====
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This would involve placing a stone on the ''n''th row of a sufficiently wide board, and showing how to always connect to the bottom, either by hand or by computer. Note this does not necessarily identify the minimal template needed.
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.
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=== Computer Aided demonstration ... ===
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=== If the answer is "no" ===
==== ... answering "Yes" ====
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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.
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==== ... answering "No" ====
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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.
TODO
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== See Also ==
 
== See Also ==
Line 138: Line 45:
 
== External link ==
 
== External link ==
  
* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.
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* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 Little Golem thread] where the names were associated.
  
[[category:theory]]
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[[category: Edge templates]]
[[category:templates]]
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[[category: Open problems]]

Latest revision as of 21:45, 28 December 2020

In January 2009, it was an open problem, initially stated by javerberg and wccanard in the LG forum, whether there is a one stone sixth row edge template that uses no stones higher than the sixth row.

The answer is "yes", and edge template VI1a is such a template.

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 templates have been found for sizes up to 6 but none above.

Description

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?

Generalisation

The general problem of knowing if there is n such that there is no one stone edge template on the nth row is also referred to as the nth row template problem.

One of the way to prove if there is such an n is to prove if there is such n−1 for which an (n−1)-row-template with one defender stone originally placed next to attacker stone in the same row. Of course if such template exists nth-row-template is still not proven to exist.

Here is an example for n = 7

For now it seems like there is no solution for above example.

Possible paths to answer

If the answer is "yes"

This would involve placing a stone on the nth row of a sufficiently wide board, and showing how to always connect to the bottom, either by hand or by computer. Note this does not necessarily identify the minimal template needed.

If the answer is "no"

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.

See Also

External link