Difference between revisions of "Sixth row template problem"
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| − | As of January 2009 the following problem is still [[open problems|open]] | + | As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|open]]: |
| − | Is there | + | Is there a one stone sixth row [[template]] that uses no stones higher than the sixth row? |
| − | + | 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. | |
== Description == | == Description == | ||
| − | Is there a | + | 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 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. |
| + | |||
| + | |||
| + | One of the way to prove if there is such an <i>n</i> is to prove if there is such <i>n-1</i> for which an <i>(n-1)</i>-row-template with one defender stone originaly placed next to attacker stone in the same row. Of course if such template exists <i>n</i>-row-template is still not proven to exist. | ||
| + | |||
| + | Here is an example for <i>n</i> = 7 | ||
| + | <hex> R8 C11 | ||
| + | 1:HHHHHVHHHHH | ||
| + | 2:+++++V+++++ | ||
| + | 3:____HV_____ | ||
| + | </hex> | ||
| + | |||
| + | |||
| + | For now it seems like there is no solution for above example. | ||
== Possible paths to answer == | == Possible paths to answer == | ||
| + | ===By "hand"...=== | ||
| + | ====...answering "Yes" ==== | ||
| + | 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.) | ||
| + | |||
| + | See [[defending against intrusions in template VI1]] for complete proof. | ||
| + | ====6th row template==== | ||
| + | <hex> | ||
| + | R7 C14 Q0 | ||
| + | 1:BBBBBBBBBRBBBBB | ||
| + | Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Rj2 Sl2 Sm2 Sn2 | ||
| + | Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sm3 Sn3 | ||
| + | Sa4 Sb4 Sc4 Sd4 Sn4 | ||
| + | Sa5 Sb5 | ||
| + | Sa6 | ||
| + | </hex> | ||
| + | |||
| + | ====...answering "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. | ||
| + | |||
=== Computer Aided demonstration ... === | === Computer Aided demonstration ... === | ||
==== ... answering "Yes" ==== | ==== ... answering "Yes" ==== | ||
| − | Such a proof would use the computer to find the template and | + | Such a proof would use the computer to find the template and its [[carrier]]. Afterwards it should be easy to manually check that every Blue intrusion does not prevent Red from connecting to bottom. |
==== ... answering "No" ==== | ==== ... answering "No" ==== | ||
| Line 36: | Line 68: | ||
[[category:theory]] | [[category:theory]] | ||
[[category:templates]] | [[category:templates]] | ||
| − | |||
Revision as of 17:27, 23 February 2009
As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still open:
Is there a one stone sixth row template that uses no stones higher than the sixth row?
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 5 but none above. Answering with "No" to the former question answers the latter.
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 n^th row<math>n^th</math> 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 originaly placed next to attacker stone in the same row. Of course if such template exists n-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
By "hand"...
...answering "Yes"
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.)
See defending against intrusions in template VI1 for complete proof.
6th row template
...answering "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.
Computer Aided demonstration ...
... answering "Yes"
Such a proof would use the computer to find the template and its carrier. Afterwards it should be easy to manually check that every Blue intrusion does not prevent Red from connecting to bottom.
... answering "No"
TODO
See Also
External link
- The thread were the names were associated.
