HexWiki - User contributions [en]

Sixth row template problem

2009-01-13T19:40:17Z

Na wspak: /* Generalisation */

As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|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 [[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 ==

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
1:HHHHHVHHHHH
2:_____V_____
</hex>

== 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
<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 ==
===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 ... ===
==== ... 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 ==
* [[Theory]]
* [[User:Wccanard|Wccanard]]

== External link ==

* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.

[[category:theory]]
[[category:templates]]

Sixth row template problem

2009-01-12T17:55:36Z

Na wspak: /* 6th row template */

As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|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 [[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 ==

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
1:HHHHHVHHHHH
2:_____V_____
</hex>

== 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:
<hex>
R7 C19 Q0
1: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
1: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
1:BBBBBBBBBRBBBBBBBBB
Rj2
Pg5
Pf6
Pe7
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
</hex>

====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>
</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 ... ===
==== ... 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 ==
* [[Theory]]
* [[User:Wccanard|Wccanard]]

== External link ==

* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.

[[category:theory]]
[[category:templates]]
{{stub}}

Sixth row template problem

2009-01-12T17:54:29Z

Na wspak: /* 6th row template */

As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|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 [[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 ==

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
1:HHHHHVHHHHH
2:_____V_____
</hex>

== 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:
<hex>
R7 C19 Q0
1: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
1: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
1:BBBBBBBBBRBBBBBBBBB
Rj2
Pg5
Pf6
Pe7
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
</hex>

====6th row template====
<hex>
R7 C14 Q0
1:BBBBBBBBBBBRBBB
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>
</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 ... ===
==== ... 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 ==
* [[Theory]]
* [[User:Wccanard|Wccanard]]

== External link ==

* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.

[[category:theory]]
[[category:templates]]
{{stub}}

Edge templates with one stone

2009-01-12T17:48:56Z

Na wspak: /* Sixth row edge template */

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1
Sa2 Sb2 Sc2 Vd2 Si2
Sa3 Sb3
Sa4
</hex>

== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
<hex>
R6 C14
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Rj1 Sl1 Sm1 Sn1
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sm2 Sn2
Sa3 Sb3 Sc3 Sd3 Sn3
Sa4 Sb4
Sa5
</hex>

[[Category: Edge templates]]

Edge templates with one stone

2009-01-11T23:37:33Z

Na wspak: /* Sixth row edge template */

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1
Sa2 Sb2 Sc2 Vd2 Si2
Sa3 Sb3
Sa4
</hex>

== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
<hex>
R6 C18
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Rl1 Sn1 So1 Sp1 Sq1 Sr1
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sq2 Sr2
Sa3 Sb3 Sc3 Sd3 Sr3
Sa4 Sb4
Sa5
</hex>

[[Category: Edge templates]]

Sixth row template problem

2009-01-11T23:31:19Z

Na wspak: /* this note is for temporary use */

As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|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 [[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 ==

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
1:HHHHHVHHHHH
2:_____V_____
</hex>

== 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:
<hex>
R7 C19 Q0
1: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
1: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
1:BBBBBBBBBRBBBBBBBBB
Rj2
Pg5
Pf6
Pe7
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
</hex>

====6th row template====
<hex>
R7 C18 Q0
1:BBBBBBBBBBBRBBBBBBB
Sa2 Sb2 Sc2 Sd2 Se2 Sf2 Sg2 Sh2 Si2 Sj2 Rl2 Sn2 So2 Sp2 Sq2 Sr2
Sa3 Sb3 Sc3 Sd3 Se3 Sf3 Sq3 Sr3
Sa4 Sb4 Sc4 Sd4 Sr4
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 ... ===
==== ... 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 ==
* [[Theory]]
* [[User:Wccanard|Wccanard]]

== External link ==

* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.

[[category:theory]]
[[category:templates]]
{{stub}}

Edge templates with one stone

2009-01-11T21:45:15Z

Na wspak:

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1
Sa2 Sb2 Sc2 Vd2 Si2
Sa3 Sb3
Sa4
</hex>

== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
There are almost certainly no edge templates with one stone on the [[Sixth row|sixth row]], but this is unproven.

[[Category: Edge templates]]

Edge templates with one stone

2009-01-11T21:39:11Z

Na wspak: /* Edge template IV1d = */

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1
Sa2 Sb2 Sc2 Vd2 Si2
Sa3 Sb3
Sa4
</hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1 Si2
Sa2 Sb2 Sc2 Vd2
Sa3 Sb3
Sa4
</hex>
== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
There are almost certainly no edge templates with one stone on the [[Sixth row|sixth row]], but this is unproven.

[[Category: Edge templates]]

Edge templates with one stone

2009-01-11T21:38:47Z

Na wspak: /* Edge template IV1c */

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

=== [[Edge template IV1d]] ====
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1
Sa2 Sb2 Sc2 Vd2 Si2
Sa3 Sb3
Sa4
</hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1 Si2
Sa2 Sb2 Sc2 Vd2
Sa3 Sb3
Sa4
</hex>
== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
There are almost certainly no edge templates with one stone on the [[Sixth row|sixth row]], but this is unproven.

[[Category: Edge templates]]

Edge templates with one stone

2009-01-11T21:21:54Z

Na wspak: /* Comments to above template */

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

==== Comments to above template ====
(what if blue moves here ??)
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3 Hf3
Sa4
Sc5 </hex>

(i belive it meant to be like this. correct me if i'm wrong. na_wspak)
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1
Sa2 Sb2 Sc2 Vd2 Si2
Sa3 Sb3
Sa4
</hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1 Si2
Sa2 Sb2 Sc2 Vd2
Sa3 Sb3
Sa4
</hex>
== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
There are almost certainly no edge templates with one stone on the [[Sixth row|sixth row]], but this is unproven.

[[Category: Edge templates]]

Edge templates with one stone

2009-01-11T21:21:15Z

Na wspak: /* Comments to above template */

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

==== Comments to above template ====
(what if blue moves here ??)
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3 Hf3
Sa4
Sc5 </hex>

(i belive it meant to be like this. correct me if im wrong. -na_wspak)
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1
Sa2 Sb2 Sc2 Vd2 Si2
Sa3 Sb3
Sa4
</hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1 Si2
Sa2 Sb2 Sc2 Vd2
Sa3 Sb3
Sa4
</hex>
== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
There are almost certainly no edge templates with one stone on the [[Sixth row|sixth row]], but this is unproven.

[[Category: Edge templates]]

Edge templates with one stone

2009-01-11T21:15:11Z

Na wspak: /* Edge template IV1c */

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

==== Comments to above template ====
(what if blue moves here ??)
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3 Hf3
Sa4
Sc5 </hex>

(i belive it meant to be like this. correct me if im wrong. -na_wspak)
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sb5 </hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1 Si2
Sa2 Sb2 Sc2 Vd2
Sa3 Sb3
Sa4
</hex>
== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
There are almost certainly no edge templates with one stone on the [[Sixth row|sixth row]], but this is unproven.

[[Category: Edge templates]]

Edge templates with one stone

2009-01-11T21:09:32Z

Na wspak: /* Edge template IV1c */

Here you can see some known '''[[Edge template|edge templates]]''' with one stone to be connected to the bottom row. Not all of them are useful to know. The [[Fifth row|fifth-row]] [[template]] occurs very seldom in real play.

There is some overlap with the article [[Edge templates everybody should know]].

== First row edge template ==
<hex>R1 C1 Va1</hex>

== Second row edge template ==
<hex>R2 C2 Vb1 Sa1</hex>

== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===
Also known as the '''Ziggurat'''.
[[Image:Ziggurat.png]]

=== [[defending against intrusions in template 1-IIIb|Edge template III1b]] ===
[[Image:Template-1-3b.png]]



== Fourth row edge templates ==
=== [[defending against intrusions in template 1-IVa|Edge template IV1a]] ===
[[Image:Template-1-4a.png]]

=== [[Edge template IV1b]] ===
[[Image:Template-1-4b.png]]

=== [[Edge template IV1c]] ===
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3
Sa4
Sc5 </hex>

(what if blue moves here ??)
<hex> R5 C11
Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sj1 Sk1
Sa2 Sb2 Sc2 Sd2 Vf2 Sk2
Sa3 Sb3 Hf3
Sa4
Sc5 </hex>

=== [[Edge template IV1d]] ===
<hex> R5 C9
Sa1 Sb1 Sc1 Sd1 Sh1 Si1 Si2
Sa2 Sb2 Sc2 Vd2
Sa3 Sb3
Sa4
</hex>
== [[edge template V1|Fifth row edge template]] ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>
== Sixth row edge template ==
There are almost certainly no edge templates with one stone on the [[Sixth row|sixth row]], but this is unproven.

[[Category: Edge templates]]

Sixth row template problem

2009-01-11T20:59:07Z

Na wspak:

As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|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 [[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 ==

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
1:HHHHHVHHHHH
2:_____V_____
</hex>

== 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:
<hex>
R7 C19 Q0
1: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
1: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
1:BBBBBBBBBRBBBBBBBBB
Rj2
Pg5
Pf6
Pe7
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
</hex>

====this note is for temporary use====
( this is IMHO the minimal field required for 6th row template. any proves it is not are very welcome. i didnt make analysis on jhex hence i have no record. i will make jhex file and send to anyone interested. i hope it will be soon:) -na_wspak)
<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" ====
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 ==
* [[Theory]]
* [[User:Wccanard|Wccanard]]

== External link ==

* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.

[[category:theory]]
[[category:templates]]
{{stub}}

Sixth row template problem

2009-01-11T20:55:29Z

Na wspak: /* ...answering "Yes" */

As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|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 [[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 ==

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
1:HHHHHVHHHHH
2:_____V_____
</hex>

== 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:
<hex>
R7 C19 Q0
1: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
1: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
1:BBBBBBBBBRBBBBBBBBB
Rj2
Pg5
Pf6
Pe7
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
</hex>

====this note is for temporary use====
( this is IMHO the minimal field needed for 6th row template. any proves it is not are very welcome. i didnt make analysis on jhex hence i have no record. i will make jhex file and send to anyone interested. i hope it will be soon:) -na_wspak)
<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" ====
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 ==
* [[Theory]]
* [[User:Wccanard|Wccanard]]

== External link ==

* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.

[[category:theory]]
[[category:templates]]
{{stub}}

Sixth row template problem

2009-01-11T20:48:48Z

Na wspak:

As of January 2009 the following problem, initially stated by javerberg and wccanard in the LG forum, is still [[open problems|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 [[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 ==

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
1:HHHHHVHHHHH
2:_____V_____
</hex>

== 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:
<hex>
R7 C19 Q0
1: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
1: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
1:BBBBBBBBBRBBBBBBBBB
Rj2
Pg5
Pf6
Pe7
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
</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" ====
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 ==
* [[Theory]]
* [[User:Wccanard|Wccanard]]

== External link ==

* The [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=339 thread] were the names were associated.

[[category:theory]]
[[category:templates]]
{{stub}}