HexWiki - User contributions [en]

Ladder escape template

2010-05-06T15:39:44Z

MJongo: /* External links */

A minimal [[edge template]] which can be used as a [[ladder escape]].

Below, '''*''' stands for template carrier, '''+''' stands for projected ladder path.

== [[Second row|Row-2]] ladders ==

All of the common edge templates are valid. For instance:

===[[Template II]]===
<hex> R3 C8
Vb2 Vc2 Vd2 Pe2 Pf2 Pg2 Vh2
Ha3 Hb3 Hc3 Hd3 Pe3 Pf3 Pg3 Sh3</hex>

===[[Template IVa]]===
<hex> R4 C11
Vi1 Sj1
Sg2 Sh2 Si2 Sj2 Sk2
Vb3 Vc3 Pd3 Pe3 Pf3 Sg3 Sh3 Si3 Sj3 Sk3
Ha4 Hb4 Hc4 Pd4 Pe4 Pf4 Sg4 Sh4 Si4 Sj4 Sk4</hex>

== [[Third row|Row-3]] ladders==

Templates [[Template II|II]], [[Template IIIa|IIIa]], and [[Template IVa|IVa]] are valid.

== [[Fourth row|Row-4]] ladders==

===[[Template IIIa]]===
Template IIIa is valid.
<hex> R5 C7
Vb2 Vc2 Vd2
Ha3 Hb3 Hc3 Hd3 Vf3 Sg3
Se4 Sf4 Sg4
Sd5 Se5 Sf5 Sg5</hex>

===[[Template IVa]]===
Also template IVa is valid if you can [[double bridge]] to the [[escape piece]] as follows.

<hex>R6 C9 Vb3 Vc3 Vd3 Sf2 Vg3 Ha4 Hb4 Hc4 Hd4</hex>

[[Red]] can [[jumping|jump]] ahead to the escape template by playing at the marked [[cell]].

== External links ==
* [http://www.drking.org.uk/hexagons/hex/templates.html David King's Hex template page]

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

Edge templates with one stone

2010-05-01T00:03:59Z

MJongo: /* Edge template IV1d */ Wrong 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 ==
[[Image:Template-1.png]]


== Second row edge template ==
[[Image:Template-2.png]]


== Third row edge templates ==
=== [[Ziggurat|Edge template III1a]] ===

[[Image:Ziggurat.png]]
Also known as the '''Ziggurat'''.

=== [[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]] ===
[[Image:Template-1-4c.png]]


=== [[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 ==
<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]]

Y

2010-04-28T17:54:46Z

MJongo: /* Extension to Hex */

The game of Y is a [[connection game]] invented by [[Craige Schenstead]] and [[Charles Titus]]. In its original form, it is played on a [[triangular grid of hexagons]]. There are two [[Player (general)|players]], who have one colour each, and a move consists of placing a piece of your colour in one of the hexagons on the board. The winner is the first player to complete a [[chain]] connecting all three sides of the board. Y is a kind of generalisation of [[Hex]], perhaps the one the nearest from it, but there are some strategic peculiarities, such as [[corner template]]s

[[Image:Y-board-straight.gif]]

== No draws ==

Y cannot end in a draw. That is, once the board is complete there must be one and only one winner.

=== Less than two winners ===
There cannot be two winners at the same time. If there were, each player would have a region of the board touching all three sides of the triangle as well as the opponent's region. Considering the three sides as regions themselves, this gives a map of five regions, each of which is adjacent to the other four. However, this is impossible, as the graph K5 is non-planar.

=== At least one winner ===
It can be proved by an algorithm that once a board is complete there is at least one player linking the 3 sides. Let the "state" of a board refer to the answer to the question "Is there at least one winner?" We want to prove that the state of every board is "Yes".

''First step'': if there is a pawn group (red for instance) completely surrounded by the opponent (blue for instance) let's consider the board with this pawn group replaced by opponent's pawns (blue ones). The new board has the same status as the older one as the remote group was not winning and the new big (blue) one is winning iff it was in the former board. Also note that there is still at least one group left.

Repeat this step until there is no completely surrounded group more (of either color). The board obtained has the same state as the original.

''Second step'': if there is a pawn group surrounded by the opponent and a side, removing it does not change the state of the board (for similar reasons as in step 1) and there is still at least one group left.

Repeat this step.

''Third step'': if there is a pawn group surrounded by the opponent and two sides removing it does not change the state of the board (for similar reasons as in step 1) and there is still at least one group left.

Repeat this step.

It is quite clear that after applying this algorithm there is no group connected to more than 1 opponent's groups. No group is connected to zero sides and one opponent's group, no group is connected to one side and one opponent's group, no group is connected to two sides and one opponent's group. No group can be connected to 0 1 or 2 sides without connecting an opponent group. Moreover there is at least one group left. Hence this group left is connected to 3 sides.

So the state of the board is "yes"; as it is the same as the state of the beginning board, there was a winner to begin with.

Note that this algorithm ends because the number of different groups is finite.

=== Extension to [[Hex]] ===

The proof above extends to Hex because a Hex game can be seen as a subset of a Y game.

For instance consider the following Y board of size 8. (The star marked hexes are not a part of the board.)
<hex> C8 R8
1:SSSSSSS_
2:SSSSSS__
3:SSSSS___
4:SSSS____
5:SSS_____
6:SS______
7:S_______</hex>

We can simulate a Hex game of [[size]] 4 on it.
<hex> C8 R8
1:SSSSSSSR
2:SSSSSSRR
3:SSSSSRRR
4:SSSSRRRR
5:SSSB____
6:SSBB____
7:SBBB____
8:BBBB____</hex>

Now the only way to win for Blue is to cross the board horizontally, whereas the only way for Red to do so is to cross the board verticaly.

Each game of Hex on size n Hex board can be played on a Y board of size 2n or 2n-1 with the rules of Y, player just need to play some stones to "construct" the Hex board.

== The first player wins ==
In Y the [[strategy-stealing argument]] can be applied. It proves that the second player has no winning strategy. The argument is that if the second player had a winning strategy, then the first player could chose a random first move and then pretend that she is the second player and apply the stratey. An important point is that an extra pawn is not a disadvantage in Y. Y is a complete and perfect information game in wich no draw can be conceived, so there is a winning strategy for one player. The second player has no winning strategy so the first player has one.

== Swap ==

The [[Swap rule]] can be used in Y too, the corner are bad moves to be played so there may well exist average moves to begin with. Further information on [[where to swap (y)|where to swap]].

== Variations ==
The game is usually played with the [[swap_rule]]. Alternatively, one can play Double-move Y, also known as [[Master Y]]: The first player places one piece on the board, and each subsequent move consists of placing two pieces on the board. This is a pretty challenging variant, even on small boards.

The inventors tried out a number of alternative playing grids, and eventually concluded that the most suitable one is the following. The pieces are placed on the intersections (like in [[Go]]).

[[Image:Y-board-bent.gif]]

== See also ==

You can find more boards here: [[Printable Y boards]]

Help for [[programming the bent Y board]]

Try this [[Y puzzle]].

== On the web ==

* http://www.gamepuzzles.com/gameofy.htm
* http://www.gamepuzzles.com/revugy.htm (Games magazine reviews)
* http://home.flash.net/~markthom/html/the_game_of_y.html
* http://www.iggamecenter.com/ ('''igGameCenter''' - play "Y" online with other opponents from your iGoogle homepage)

[[Category: Y]]
[[Category: Theory]]