Sixth row template problem - Revision history

Selinger: Edited categories

2020-12-28T21:45:24Z

Edited categories

Selinger: Cleaned up the article a bit, which was about an open problem that is no longer open.

2020-12-28T19:02:16Z

Cleaned up the article a bit, which was about an open problem that is no longer open.

Turing: /* ... answering "Yes" */ fixed typo

2009-02-23T17:27:48Z

‎... answering "Yes": fixed typo

Na wspak: /* Generalisation */

2009-01-13T19:40:17Z

‎Generalisation

Halladba: moved some part of the article to another one

2009-01-13T10:24:49Z

moved some part of the article to another one

Ypercube at 09:40, 13 January 2009

2009-01-13T09:40:48Z

Ypercube at 09:39, 13 January 2009

2009-01-13T09:39:15Z

Artduval: /* ...answering "Yes" */ -- setting the outline for dealing with remaining intrusions

2009-01-13T04:56:19Z

‎...answering "Yes": -- setting the outline for dealing with remaining intrusions

Halladba: " " was written twice, removed it once

2009-01-12T21:33:40Z

"</hex> " was written twice, removed it once

Na wspak: /* 6th row template */

2009-01-12T17:55:36Z

‎6th row template

@@ Line 47: / Line 47: @@
 * The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 Little Golem thread] where the names were associated.
-[[category:theory]]
+[[category: Edge templates]]
-[[category:edge templates]]
+[[category: Open problems]]

@@ Line 1: / Line 1: @@
-As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|open]]:
+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?
+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.
+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 ==
-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
@@ Line 16: / Line 16: @@
 == 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.
+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 <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 ''n'' = 7
-−
-Here is an example for <i>n</i> = 7
 <hex> R8 C11
 :HHHHHVHHHHH
@@ Line 27: / Line 26: @@
 :____HV_____
 </hex>
 For now it seems like there is no solution for above example.
 == Possible paths to answer ==
-See [[defending against intrusions in template VI1]] for complete proof.
+=== If the answer is "yes" ===
-====6th row template====
-<hex>
-R7 C14 Q0
-: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 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.
-=== Computer Aided demonstration ... ===
+=== If the answer is "no" ===
-==== ... 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" ====
+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
 == See Also ==
@@ Line 64: / Line 45: @@
 == External link ==
-* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.
+* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 Little Golem thread] where the names were associated.
 [[category:theory]]
-[[category:templates]]
+[[category:edge templates]]

@@ Line 53: / Line 53: @@
 === 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.
+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" ====

← Older revision		Revision as of 19:40, 13 January 2009
Line 17:		Line 17:

	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.		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 ==

@@ Line 23: / Line 23: @@
 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:
+See [[defending against intrusions in template VI1]] for complete proof.
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Si3 Sj3
-Si4
-Sg5 Sh5 Si5 Sj5
-Sf6 Sg6 Si6 Sj6
-Se7 Sf7 Sg7 Sh7 Si7 Sj7
-</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:
-−
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Pg5
-Pf6
-Pe7
-N:on Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
-</hex>
-−
-At this point, we can use the [[Parallel ladder]] trick as follows:
-−
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Pg5
-Pf6
-Pe7
-Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
-N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
-</hex>
-−
-Now, let's deal with the remaining intrusions!:
-−
-=====One remaining intrusion on the first row=====
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Bf7
-</hex>
-−
-=====The other remaining intrusion on the first row=====
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Bg7
-</hex>
-−
-=====The remaining intrusion on the second row=====
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Bg6
-</hex>
-−
-=====The remaining intrusion on the third row=====
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Bh5
-</hex>
-−
-=====The remaining intrusion on the fourth row=====
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Bi4
-</hex>
-−
-=====The remaining intrusion on the fifth row=====
-<hex>
-R7 C19 Q0
-:BBBBBBBBBRBBBBBBBBB
-Rj2
-Bi3
-</hex>
-−
 ====6th row template====
 <hex>

← Older revision		Revision as of 09:40, 13 January 2009
Line 142:		Line 142:
	[[category:theory]]		[[category:theory]]
	[[category:templates]]		[[category:templates]]
−	~~{{expand}}~~

@@ Line 142: / Line 142: @@
 [[category:theory]]
 [[category:templates]]
-{{stub}}
+{{expand}}

@@ Line 70: / Line 70: @@
 Sa5 Sb5
 Sa6
 </hex>