HexWiki - User contributions [en]

Edge template VI1a

2009-01-14T05:02:24Z

Artduval: /* The remaining intrusion on the fourth row */ -- starting the solution

This template is the first one stone 6th row [[edge template|template]] for which a proof has been handwritten.

<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>

== Elimination of irrelevant Blue moves ==

Red has a couple of direct threats to connect, using smaller templates. Blue must play in the carrier of these threats in order to counter them. To prevent Red from connecting Blue must play in the intersection of Red's threats carriers.

=== [[edge template IV1a]] ===

<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

Pi3 Pj3
Ph4 Ri4
Pf5 Pg5 Ph5 Pi5 Pj5
Pe6 Pf6 Pg6 Ph6 Pi6 Pj6
Pd7 Pe7 Pf7 Pg7 Ph7 Pi7 Pj7
</hex>

<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

Pi3 Pj3
Ri4 Pj4
Pg5 Ph5 Pi5 Pj5 Pk5
Pf6 Pg6 Ph6 Pi6 Pj6 Pk6
Pe7 Pf7 Pg7 Ph7 Pi7 Pj7 Pk7
</hex>

=== [[edge template IV1b]] ===

<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

Pi3 Pj3
Ph4 Ri4 Pj4
Pf5 Pg5 Ph5 Pi5 Pj5 Pk5
Pe6 Pf6 Pg6 Pi6 Pj6 Pk6
Pd7 Pe7 Pf7 Pg7 Ph7 Pi7 Pj7 Pk7
</hex>
=== Using the [[parallel ladder]] trick ===

6 moves can furthermore be discared thanks to 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 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

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 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

Pg5
Pf6
Pe7
Ri4 Bi5 Rh5 Bg7 Rh6 Bh7
N:on Rk5 Bj6 Ri6 Bi7 Rl4 Bj5 Rk3
</hex>

=== [[Overlapping connections|Remaining possibilities]] for Blue ===
Blue's first move must be one of the following:
<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

Pi3 Pj3
Pi4
Ph5 Pi5
Pg6 Pi6
Pf7 Pg7 Ph7 Pi7
</hex>

== Specific defence ==
For the moves that intersect all the carriers, Red has to find specific answers. Let's deal with the remaining intrusions!

===One remaining intrusion on the first row===
<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

Bf7
</hex>

===The other remaining intrusion on the first row===
<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

Bg7
</hex>

===The remaining intrusion on the second row===
<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

Bg6
</hex>

===The remaining intrusion on the third row===
<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

Bh5
</hex>

===The remaining intrusion on the fourth row===
<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

Bi4
</hex>

Red should move here:

<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

Bi4 Rh3
</hex>

This gives Red several immediate threats:
From III1a:
<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

Bi4 Rh3
Rg5
Pg4 Ph4
Ph5
Pf6 Pg6 Ph6
Pe7 Pf7 Pg7 Ph7
</hex>

From III1a again:
<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

Bi4 Rh3
Rg5
Pg4 Ph4
Pf5
Pe6 Pf6 Pg6
Pd7 Pe7 Pf7 Pg7
</hex>

From IV1a:
<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

Bi4 Rh3
Rg4
Pf4
Pd5 Pe5 Pf5 Pg5 Ph5
Pc6 Pd6 Pe6 Pf6 Pg6 Ph6
Pb7 Pc7 Pd7 Pe7 Pf7 Pg7 Ph7
</hex>

From IV1b:
<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

Bi4 Rh3
Rg4
Pf4 Ph4
Pd5 Pe5 Pf5 Pg5 Ph5 Pi5
Pc6 Pd6 Pe6 Pg6 Ph6 Pi6
Pb7 Pc7 Pd7 Pe7 Pf7 Pg7 Ph7 Pi7
</hex>

The intersection of all of these leaves:
<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

Bi4 Rh3
Pg4
Pg5
Pg6
Pe7 Pf7 Pg7
</hex>

So we must deal with each of these responses. (Which will not be too hard!)

To be continued....

===The remaining intrusion on the fifth row===
<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

Bi3
</hex>

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

Sixth row template problem

2009-01-13T04:56:19Z

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

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>

Now, let's deal with the remaining intrusions!:

=====One remaining intrusion on the first row=====
<hex>
R7 C19 Q0
1:BBBBBBBBBRBBBBBBBBB
Rj2
Bf7
</hex>

=====The other remaining intrusion on the first row=====
<hex>
R7 C19 Q0
1:BBBBBBBBBRBBBBBBBBB
Rj2
Bg7
</hex>

=====The remaining intrusion on the second row=====
<hex>
R7 C19 Q0
1:BBBBBBBBBRBBBBBBBBB
Rj2
Bg6
</hex>

=====The remaining intrusion on the third row=====
<hex>
R7 C19 Q0
1:BBBBBBBBBRBBBBBBBBB
Rj2
Bh5
</hex>

=====The remaining intrusion on the fourth row=====
<hex>
R7 C19 Q0
1:BBBBBBBBBRBBBBBBBBB
Rj2
Bi4
</hex>

=====The remaining intrusion on the fifth row=====
<hex>
R7 C19 Q0
1:BBBBBBBBBRBBBBBBBBB
Rj2
Bi3
</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>

====...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

2009-01-10T18:17:32Z

Artduval: /* ...answering "Yes" */ Taking care of 3 moves with parallel ladders

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

Is there any one stone sixth row [[template]] ?

More generally, it is still unknown whether one stone edge templates can be found for every 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>

====...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}}

Open problems

2009-01-10T15:58:20Z

Artduval: renaming Javerberg-wccanard problem as Sixth row template problem; fixing link to Jory's problem

* Are there cells other than a1 and b1 which are theoretically losing first moves?

* Is it true that for every cell (defined in terms of direction and distance from an [[acute corner]]) there is an ''n'' such that for any [[Board]] of size at least ''n'' that cell is a losing [[opening move]]?

* Conversely, is it true that, for example, c3 is a winning first move on every Hex board of size at least 5?

* [[Sixth row template problem]]: Does there exist an [[edge template]] which guarantees a secure [[connection]] for a [[piece]] on the sixth row?

* Is the [[center hex]] on every Hex board of [[Odd size board|odd size]] a winning opening move?

* Two further open problems are posed by [[Jory]] in the [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=167 Little Golem forum].

[[category:Theory]]

Sixth row template problem

2009-01-10T15:50:24Z

Artduval: changed wording in introduction to match new title

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

Is there any one stone sixth row [[template]] ?

More generally, it is still unknown whether one stone edge templates can be found for every 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!

====...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

2009-01-10T15:46:08Z

Artduval: Javerberg-wccanard problem moved to Sixth row template problem: More descriptive title

As of January 2009 the following problem is still [[open problems|open]]. '''Javerberg-Wccanard Problem''' is simply put as follow.

Is there any one stone sixth row [[template]] ?

More generally, it is still unknown whether one stone edge templates can be found for every 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 '''Javerberg-Wccanard 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!

====...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}}

Javerberg-wccanard problem

2009-01-10T15:46:08Z

Artduval: Javerberg-wccanard problem moved to Sixth row template problem: More descriptive title

#REDIRECT [[Sixth row template problem]]

Sixth row template problem

2009-01-10T15:44:11Z

Artduval: width m (not n), minor wording in intro

Sixth row template problem

2009-01-10T04:24:38Z

Artduval: /* ...answering "Yes" */ Fixing the number of rows (silly mistake)

As of January 2009 the following problem is still [[open problems|open]]. '''Javerberg-Wccanard Problem''' is simply put as follow.

Is there any one stone sixth row [[template]] ?

It is still unknown whether one stone edge templates can be found for every 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 width n such that the game on the board of width n designed as follow with turn to Blue 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 '''Javerberg-Wccanard 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 [[Edge template IV1a]], 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 Sh6 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!

====...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

2009-01-10T03:59:55Z

Artduval: /* Possible paths to answer */ Starting the solution of the 6th row template

As of January 2009 the following problem is still [[open problems|open]]. '''Javerberg-Wccanard Problem''' is simply put as follow.

Is there any one stone sixth row [[template]] ?

It is still unknown whether one stone edge templates can be found for every 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 width n such that the game on the board of width n designed as follow with turn to Blue 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 '''Javerberg-Wccanard 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 [[Edge template IV1a]], Blue's first move must be one of the following:
<hex>
R6 C19 Q0
1:BBBBBBBBBRBBBBBBBBB
Si2 Sj2
Si3
Sg4 Sh4 Si4 Sj4
Sf5 Sg5 Sh5 Si5 Sj5
Se6 Sf6 Sg6 Sh6 Si6 Sj6
</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!

====...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}}

Template

2008-05-21T03:27:02Z

Artduval: link "interior template" to "unnamed interior template" for now

A '''template''' is a [[pattern]] which guarantees some kind of [[connection]]. There are several different (and sometimes overlapping) types:

* [[:Category:Edge templates |Edge templates]]
* [[Ladder template]]s
* [[Unnamed interior templates|Interior template]]s

== See also ==

* [[Naming of templates]]

== Reference ==

* [http://www.drking.plus.com/hexagons/hex/templates.html David King's Hex template page]
* [http://www.f.kth.se/~rydh/Hex/templates.html Jonatan Rydh's Hex edge templates page]

[[category:templates]]
[[category:connection types]]

Unnamed interior templates

2008-05-21T03:21:28Z

Artduval: gm

These are [[template]]s for which no name has been invented yet.
<hex> R3 C4
Sa1
Va2 Vd2
Va3 Vc3 Sd3</hex>

<hex> R4 C4
Sa1
Va2 Vd2
Va3 Vd3
Sd4</hex>

<hex> R3 C3
Va1 Vc1
Vc2
Vb3 Sc3</hex>
[[category:interior templates]]

Physical Hex sets

2008-05-03T16:25:10Z

Artduval: including printable boards

A (physical) Hex set was marketed under that name by Parker Bros. starting in 1952. Today, hand-made Hex sets can be bought at [http://www.mattesmedjan.se/produkter/index.html Mattesmedjan] in Sweden.

Here are some ideas on building a set:
* the page [http://www.nada.kth.se/~rydh/Hex/hexPics.html Hex Boards] has photos of wooden boards as used in the International Tournament 2005 in Wroclaw, Poland;
* Miguel Garcia has build a nice [http://members.fortunecity.es/zeycus/hexboard/hexboard.html set using steel nuts and ball-bearings];
* or you can use an [http://gregconquest.com/hex.html erase board and magnets].
* Łukasz Rygało submited [http://www.boardgamegeek.com/image/167362 this board] to [http://www.boardgamegeek.com BoardGameGeek].
* In the city of Alicante we have made [http://www.flickr.com/photos/liopic/1688139952 this board] with steel nuts and color-glass balls. We are looking for red and blue glass balls, though.

You can also print out the [[Printable_boards]], in sizes up to 14x14.

Talk:Printable boards

2008-05-03T02:29:59Z

Artduval: horizontal/vertical boards and .gif's

I've just added boards with horizontal/vertical orientation, using the hex commands. But that doesn't make nice .gif files like the diamond shaped ones. If someone can make .gif's of horizontal/vertical boards, that would be great. [[User:Artduval|Artduval]] 04:29, 3 May 2008 (CEST)

Printable boards

2008-05-03T02:23:51Z

Artduval: /* Horizontal/vertical orientation */

Here are some printable Hex boards.

==Diamond orientation==

=== Size 4===
[[Image:Hex04.gif]]
[[Image:Hex04e.gif]]

===Size 5===

[[Image:Hex05.gif]]
[[Image:Hex05e.gif]]

===Size 6===
[[Image:Hex06.gif]]

[[Image:Hex06e.gif]]
===Size 7===
[[Image:Hex07.gif]]

[[Image:Hex07e.gif]]
===Size 8===
[[Image:Hex08.gif]]

[[Image:Hex08e.gif]]
===Size 9===
[[Image:Hex09.gif]]

[[Image:Hex09e.gif]]
=== Size 10 ===
[[Image:Hex10.gif]]

[[Image:Hex10e.gif]]
===Size 11===
[[Image:Hex11.gif]]

[[Image:Hex11e.gif]]
===Size 12===
[[Image:Hex12.gif]]

[[Image:Hex12e.gif]]
===Size 13===
[[Image:Hex13.gif]]

[[Image:Hex13e.gif]]
===Size 14===
[[Image:Hex14.gif]]

[[Image:Hex14e.gif]]
== Bent Hex ==
[[Image:hex_bent4_lines.gif]]

==Horizontal/vertical orientation==

To print these, you'll need to take a picture, and you'll probably want to enlarge the picture.

===Size 4===
<hex>
R4 C4
</hex>

===Size 5===
<hex>
R5 C5
</hex>

===Size 6===
<hex>
R6 C6
</hex>

===Size 7===
<hex>
R7 C7
</hex>

===Size 8===
<hex>
R8 C8
</hex>

===Size 9===
<hex>
R9 C9
</hex>

===Size 10===
<hex>
R10 C10
</hex>

===Size 11===
<hex>
R11 C11
</hex>

===Size 12===
<hex>
R12 C12
</hex>

===Size 13===
<hex>
R13 C13
</hex>

===Size 14===
<hex>
R14 C14
</hex>

== See also ==
[[Printable Y boards]]

Printable boards

2008-05-03T02:15:24Z

Artduval: adding boards with horizontal/vertical orientation

Here are some printable Hex boards.

==Diamond orientation==

=== Size 4===
[[Image:Hex04.gif]]
[[Image:Hex04e.gif]]

===Size 5===

[[Image:Hex05.gif]]
[[Image:Hex05e.gif]]

===Size 6===
[[Image:Hex06.gif]]

[[Image:Hex06e.gif]]
===Size 7===
[[Image:Hex07.gif]]

[[Image:Hex07e.gif]]
===Size 8===
[[Image:Hex08.gif]]

[[Image:Hex08e.gif]]
===Size 9===
[[Image:Hex09.gif]]

[[Image:Hex09e.gif]]
=== Size 10 ===
[[Image:Hex10.gif]]

[[Image:Hex10e.gif]]
===Size 11===
[[Image:Hex11.gif]]

[[Image:Hex11e.gif]]
===Size 12===
[[Image:Hex12.gif]]

[[Image:Hex12e.gif]]
===Size 13===
[[Image:Hex13.gif]]

[[Image:Hex13e.gif]]
===Size 14===
[[Image:Hex14.gif]]

[[Image:Hex14e.gif]]
== Bent Hex ==
[[Image:hex_bent4_lines.gif]]

==Horizontal/vertical orientation==

===Size 4===
<hex>
R4 C4
</hex>

===Size 5===
<hex>
R5 C5
</hex>

===Size 6===
<hex>
R6 C6
</hex>

===Size 7===
<hex>
R7 C7
</hex>

===Size 8===
<hex>
R8 C8
</hex>

===Size 9===
<hex>
R9 C9
</hex>

===Size 10===
<hex>
R10 C10
</hex>

===Size 11===
<hex>
R11 C11
</hex>

===Size 12===
<hex>
R12 C12
</hex>

===Size 13===
<hex>
R13 C13
</hex>

===Size 14===
<hex>
R14 C14
</hex>

== See also ==
[[Printable Y boards]]

Equivalent patterns

2008-05-02T03:45:41Z

Artduval: /* Example 1 */ gm

We say that two hex [[pattern]]s (subsets of a board) are '''equivalent patterns''' if, when one of them occurs embedded in ''any'' hex board, it could be replaced by the other and the side who has winning strategy does not change.

== Example 1 ==
For example (using '*' for holes), the two patterns below are equivalent, and an example of what is known as the [[Useless triangle]].

<hex>R2 C7 Q1 Va2 Vb1 Vc1 Vc2 Hb2 Ve2 Vf1 Vg1 Vg2 Vf2 Sa1 Sd1 Sd2 Se1</hex>

If Vertical was the last to move in the left pattern, he has '''captured''' the opponent stone in b2, an both players should regard it as another stone of Vertical.

The knowledge of equivalent patterns turns out to be very useful to play Hex, because it allows you to disregard some pieces in the board, or prune the analysis tree. In my opinion, some patterns lead to positions that are much more clear than other equivalent patterns. My intention here is to write always in second place such pattens, and the use that I make of equivalent patterns is that in my games, mentally I always replace the first patterns by the equivalent counterparts.

Here are several examples of pairs of equivalent positions, and a short explanation on the way to prove them so.

== Example 2 ==
<hex>R3 C9 Va2 Vb1 Vc1 Vd1 Vd2 Vc3 Hb2 Hc2 Vf2 Vg1 Vh1 Vi1 Vi2 Vh3 Vg2 Vh2 Sa1 Sa3 Sb3 Sd3 Se3 Sf3 Sg3 Si3 Se1 Sf1 Se2</hex>

The equivalence is obtained by applying two times the example 1.

Both examples before are instances of the following rule to produce equivalence pairs. Given a [[chain]] G, let the '''neighborhood''' of G, neigh(G), be the set of cells next to one of those in G but not belonging to it. In a given pattern P1, suppose that G is a chain owned by the player A, and that C is a cell in neigh(G) such that any cell of A next to C belongs to G. Let P2 be the pattern that results when A occupies C (removing an opponent stone, if necessary), therefore P2 contains a chain G' containing G and C. If neigh(G')=neigh(G) then P1 and P2 are equivalent.

This rule justifies the following equivalent pairs:

== Example 3 ==
<hex>R3 C6 Va2 Vb1 Vc1 Hb2 Hb3 Vd2 Ve1 Vf1 Ve2 He3 Sa1 Sa3 Sc2 Sc3 Sd1 Sd3 Sf2 Sf3</hex>

== Example 4 ==
<hex>R3 C6 Ha3 Hb3 Vb1 Vc1 Hb2 Hd3 He3 Ve1 Vf1 Ve2 Sa1 Sa2 Sc2 Sc3 Sd1 Sd2 Sf2 Sf3</hex>

Another rule producing equivalent patterns: If there are two empty cells C1 and C2 in a pattern, such that if the opponent of the player A occupies one of them, A can occupy the other capturing the latter, then an equivalent position is obtained if both C1 and C2 are occupied by A.

Equivalent pairs obtained with such rule:

== Example 5 ==
<hex>R3 C7 Va1 Vb1 Vc1 Va3 Ve1 Vf1 Vg1 Ve3 Ve2 Vf2 Sc2 Sd2 Sb3 Sc3 Sd3 Sf3 Sg2 Sg3 Sd1</hex>

== Example 6 ==
<hex>R3 C5 Va3 Va1 Vb1 Vb3 Ve1 Vd1 Vd2 Vd3 Ve2 Ve3 Sc2 Sc3 Sc1</hex>

== Example 7 ==
<hex>R2 C9 Va2 Vb1 Vc1 Vd1 Vd2 Sa1 Se1 Se2 Sf1 Vf2 Vg1 Vg2 Vh1 Vh2 Vi1 Vi2</hex>

Using both rules together:

== Example 8 ==
<hex>R3 C9 Vc1 Vb3 Ha2 Hd2 Vh1 Vg3 Vg2 Vh2 Hf2 Hi2 Sa1 Sb1 Sd1 Se1 Sf1 Sg1 Si1 Se2 Sa3 Sc3 Sd3 Se3 Sf3 Sh3 Si3</hex>

== Example 9 ==
''(generalization of the Example 5, for any horizontal length)''

<hex>R3 C11 Va3 Vb3 Vc3 Vd3 Ve3 Vc1 Vd1 Ve1 Hc2 Hd2 Vg3 Vh2 Vh3 Vi1 Vi2 Vi3 Vj1 Vj2 Vj3 Vk1 Vk2 Vk3 Sa1 Sa2 Sb1 Sf1 Sf2 Sf3 Sg1 Sg2 Sh1</hex>

Third rule for equivalent patterns (rather obvious) rule: Any area surrounded by a single chain of each enemy may be randomly filled. This happens because the outcome of the game does not depend on it at all.

== Practical example ==

Let us see a practical example. In game [http://www.littlegolem.net/jsp/game/game.jsp?gid=206040&nmove=55 #206040] at [[Little Golem]], the situation after 55. m2 is shown in the board below.

<hex>R13 C13 Q1 Hc1 Hd6 Hd8 Hd9 Hd10 Hd11 Hd12 He6 He12 Hf5 Hf10 Hf12 Hg5 Hg6 Hg12 Hh6 Hh8 Hh9 Hh12 Hi5 Hi11 Hj4 Hj6 Hj11 Hk3 Hk11 Hl11 Vc8 Vc9 Vc10 Vc11 Vc12 Vd7 Ve7 Ve8 Ve10 Ve11 Vf4 Vf6 Vf8 Vf11 Vg11 Vh5 Vh7 Vh11 Vi6 Vi8 Vi10 Vj5 Vj10 Vk4 Vk10 Vl10 Vm2</hex>

Clearly, m2 is strongly connected to the top, because the stone in f4 is a [[ladder escape]]. On the other hand, it is strongly connected to the bottom exactly if blue cannot connect k3 with the right, using j6 and maybe the [[group]] in h8-h9-f10 as a ladder escape. In fact he cannot do it, and it is much clearer if some patterns are locally replaced by other equivalent ones, rendering:

<hex>R13 C13 Q1 Hc1 Hd6 Vd8 Vd9 Vd10 Vd11 Vd12 He6 Ve12 Hf5 Vf10 Vf12 Hg5 Hg6 Vg12 Hh6 Vh8 Vh9 Vh12 Hi5 Vi11 Hj4 Hj6 Vj11 Hk3 Vk11 Vl11 Vc8 Vc9 Vc10 Vc11 Vc12 Vd7 Ve7 Ve8 Ve10 Ve11 Vf4 Hf6 Vf8 Vf11 Vg11 Hh5 Vh7 Vh11 Vi6 Vi8 Vi10 Vj5 Vj10 Vk4 Vk10 Vl10 Vm2 Ve9 Vf7 Vf9 Vg7 Vg8 Vg9 Vg10 Vh10 Vi7 Vi9 Vb13 Vc13 Vm10 Vm11 Vi12 Vj12 Vk12 Vl12 Vm12 Vd13 Ve13 Vf13 Vg13 Vh13 Vi13 Vj13 Vk13 Vl13 Vm13</hex>

The changes have been:

* f6 and h5 swap color, as in Example 1.

* The horizontal group from d8 to l11 is completely wrapped by a strongly connected group belonging to the opponent, except for a narrow section (typically, 2 cells). This kind of groups typically are captured, exactly in the same reason as in Example 1.

* If blue moves to i7, red moves to i9 and conversely, in both cases enclosing the blue group h8-h9-f10 as before. So, we can use the second rule for detecting equivalent patterns, capturing it.

* The equivalence in Example 5 can be used for the stone in c12.

* The equivalence in Example 4 can be used, adding a stone for vertical at m11.

* The area in the bottom of the board is now surrounded by a red chain and a blue chain. Therefore, it may be filled as we please.

The blue stone in j6 remained, completely alone and too near to the red group to be of any use in such a small region, so it is obvious that vertical has won.

[[Category: Strategy]]
[[Category: Computer Hex]]
[[Category: Theory]]