Difference between revisions of "Sixth row template problem"
(include no higher stone in intro, clarify language) |
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N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3 | N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3 | ||
</hex> | </hex> | ||
| + | |||
| + | (for temporary use: this is IMHO the minimal field needed for 6th row template. any proves its not are very welcome.) | ||
| + | <hex> | ||
| + | R7 C19 Q0 | ||
| + | 1:BBBBBBBBBRBBBBBBBBB | ||
| + | Si2 Rj2 Sk2 | ||
| + | Sg3 Sh3 Si3 Sj3 Sk3 Sl3 | ||
| + | Se4 Sf4 Sg4 Sh4 Si4 Sj4 Sk4 Sl4 Sm4 | ||
| + | Sc5 Sd5 Se5 Sf5 Sg5 Sh5 Si5 Sj5 Sk5 Sl5 Sm5 Sn5 | ||
| + | Sb6 Sc6 Sd6 Se6 Sf6 Sg6 Sh6 Si6 Sj6 Sk6 Sl6 Sm6 Sn6 | ||
| + | Sa7 Sb7 Sc7 Sd7 Se7 Sf7 Sg7 Sh7 Si7 Sj7 Sk7 Sl7 Sm7 Sn7 | ||
| + | </hex> | ||
====...answering "No" ==== | ====...answering "No" ==== | ||
Revision as of 20:48, 11 January 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.
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.)
Here is a start. Just from edge template IV1a and edge template IV1b, Blue's first move must be one of the following:
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:
At this point, we can use the Parallel ladder trick as follows:
(for temporary use: this is IMHO the minimal field needed for 6th row template. any proves its not are very welcome.)
...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 it's 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.
