Difference between revisions of "Sixth row template problem"
m (→... answering "Yes": fixed typo) |
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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. | |
| − | + | 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 | + | 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 ? | + | 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 | ||
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== Generalisation == | == Generalisation == | ||
| − | The general problem of knowing if there is n such that there is no one stone edge template on the n | + | 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. |
| + | 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. | ||
| − | + | Here is an example for ''n'' = 7 | |
| − | + | ||
| − | Here is an example for | + | |
<hex> R8 C11 | <hex> R8 C11 | ||
1:HHHHHVHHHHH | 1:HHHHHVHHHHH | ||
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3:____HV_____ | 3:____HV_____ | ||
</hex> | </hex> | ||
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For now it seems like there is no solution for above example. | For now it seems like there is no solution for above example. | ||
== Possible paths to answer == | == Possible paths to answer == | ||
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| − | + | === If the answer is "yes" === | |
| − | ==== | + | |
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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 | + | |
| − | === | + | === If the answer is "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. | |
| − | + | ||
== See Also == | == See Also == | ||
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== External link == | == External link == | ||
| − | * The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] | + | * The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 Little Golem thread] where the names were associated. |
[[category:theory]] | [[category:theory]] | ||
| − | [[category:templates]] | + | [[category:edge templates]] |
Revision as of 19:02, 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.
Contents
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
- The Little Golem thread where the names were associated.
