HexWiki - User contributions [en]

User:Reiner Martin

2007-11-23T21:42:31Z

Reiner Martin:

I like [[Go]], [[Hex]], Chess, and other strategy games. You can find me on [[Little Golem]].

I currently live in London.

User:Reiner Martin

2007-11-23T21:41:14Z

Reiner Martin:

I like [[Go]], Hex, chess, and other strategy games. You can find me on [[Little Golem]].

I currently live in London.

Template

2007-11-23T21:38:49Z

Reiner Martin: added link

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

* [[Edge template]]s
* [[Ladder template]]s
* [[Interior template]]s

== See also ==

* [[Naming of templates]]

== Reference ==

* [http://www.drking.plus.com/hexagons/hex/templates.html David King's hex template page]

History of computer Hex

2007-11-23T21:27:25Z

Reiner Martin: /* References */

Hex has been invented by matematicians and the [[rules]] are basic. Therefore it is relatively easy to make a simple [[artificial intelligence]] compared to chess for instance. However the [[branching factor]] is huge and a brutal approach is not as efficient in Hex than in Chess. This fact is another similarity to [[Go]].

== Chronology ==

* 1942 Invention of Hex.
* In 1953 Shannon and Moore build an electric machine able to play Hex. The board is considered as an electric circuit with one player's ([[Blue]] for example) hexes being off switches and the other's being on switches, the empty hexes are resistances. The machine measures the global resistance of the circuit between [[Red]]'s edges, tries every move and then chooses the one that minimizes the resistance value of the circuit, trying to make it easier to join the two sides.

== See also ==

* [[History of Hex]]
* [[Computer Hex]]

== References ==

* Anshelevich, Vadim V. [http://home.earthlink.net/~vanshel/VAnshelevich-01.pdf The Game of Hex: An Automatic Theorem Proving Approach to Game Programming]. See paragraphs 2 and 3 for the first Hex playing machine.
* Gardner, Martin (1988). [http://www.amazon.com/Hexaflexagons-Other-Mathematical-Diversions-Scientific/dp/0226282546 Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games]. University of Chicago Press, ISBN 0-226-28254-6.

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History of computer Hex

2007-11-23T21:27:00Z

Reiner Martin: /* References */

Hex has been invented by matematicians and the [[rules]] are basic. Therefore it is relatively easy to make a simple [[artificial intelligence]] compared to chess for instance. However the [[branching factor]] is huge and a brutal approach is not as efficient in Hex than in Chess. This fact is another similarity to [[Go]].

== Chronology ==

* 1942 Invention of Hex.
* In 1953 Shannon and Moore build an electric machine able to play Hex. The board is considered as an electric circuit with one player's ([[Blue]] for example) hexes being off switches and the other's being on switches, the empty hexes are resistances. The machine measures the global resistance of the circuit between [[Red]]'s edges, tries every move and then chooses the one that minimizes the resistance value of the circuit, trying to make it easier to join the two sides.

== See also ==

* [[History of Hex]]
* [[Computer Hex]]

== References ==

* Anshelevich, Vadim V. [http://home.earthlink.net/~vanshel/VAnshelevich-01.pdf , The Game of Hex: An Automatic Theorem Proving Approach to Game Programming]. See paragraphs 2 and 3 for the first Hex playing machine.
* Gardner, Martin (1988). [http://www.amazon.com/Hexaflexagons-Other-Mathematical-Diversions-Scientific/dp/0226282546 Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games]. University of Chicago Press, ISBN 0-226-28254-6.

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History of computer Hex

2007-11-23T21:26:35Z

Reiner Martin: /* References */

Hex has been invented by matematicians and the [[rules]] are basic. Therefore it is relatively easy to make a simple [[artificial intelligence]] compared to chess for instance. However the [[branching factor]] is huge and a brutal approach is not as efficient in Hex than in Chess. This fact is another similarity to [[Go]].

== Chronology ==

* 1942 Invention of Hex.
* In 1953 Shannon and Moore build an electric machine able to play Hex. The board is considered as an electric circuit with one player's ([[Blue]] for example) hexes being off switches and the other's being on switches, the empty hexes are resistances. The machine measures the global resistance of the circuit between [[Red]]'s edges, tries every move and then chooses the one that minimizes the resistance value of the circuit, trying to make it easier to join the two sides.

== See also ==

* [[History of Hex]]
* [[Computer Hex]]

== References ==

* Anshelevich, Vadim V. [http://home.earthlink.net/~vanshel/VAnshelevich-01.pdf , The Game of Hex: An Automatic Theorem Proving Approach to Game Programming]. See paragraphs 2 and 3 for the first Hex playing machine.
* Gardner, Martin (1988). [http://www.amazon.com/Hexaflexagons-Other-Mathematical-Diversions-Scientific/dp/0226282546 Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games]. University of Chicago Press, ISBN 0-226-28254-6

{{stub}}

History of computer Hex

2007-11-23T21:21:46Z

Reiner Martin:

Hex has been invented by matematicians and the [[rules]] are basic. Therefore it is relatively easy to make a simple [[artificial intelligence]] compared to chess for instance. However the [[branching factor]] is huge and a brutal approach is not as efficient in Hex than in Chess. This fact is another similarity to [[Go]].

== Chronology ==

* 1942 Invention of Hex.
* In 1953 Shannon and Moore build an electric machine able to play Hex. The board is considered as an electric circuit with one player's ([[Blue]] for example) hexes being off switches and the other's being on switches, the empty hexes are resistances. The machine measures the global resistance of the circuit between [[Red]]'s edges, tries every move and then chooses the one that minimizes the resistance value of the circuit, trying to make it easier to join the two sides.

== See also ==

* [[History of Hex]]
* [[Computer Hex]]

== References ==

* [http://home.earthlink.net/~vanshel/VAnshelevich-01.pdf Anshelevich, Vadim V., The Game of Hex: An Automatic Theorem Proving Approach to Game Programming] - See paragraphs 2 and 3 for the first Hex playing machine.
* Gardner, Martin (1988), [http://www.amazon.com/Hexaflexagons-Other-Mathematical-Diversions-Scientific/dp/0226282546 Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games], Chicago: University of Chicago Press, ISBN 0-226-28254-6

{{stub}}

History of computer Hex

2007-11-23T21:19:42Z

Reiner Martin: formatting

Hex has been invented by matematicians and the [[rules]] are basic. Therefore it is relatively easy to make a simple [[artificial intelligence]] compared to chess for instance. However the [[branching factor]] is huge and a brutal approach is not as efficient in Hex than in Chess. This fact is another similarity to [[Go]].

== Chronology ==

* 1942 Invention of Hex.
* In 1953 Shannon and Moore build an electric machine able to play Hex. The board is considered as an electric circuit with one player's ([[Blue]] for example) hexes being off switches and the other's being on switches, the empty hexes are resistances. The machine measures the global resistance of the circuit between [[Red]]'s edges, tries every move and then chooses the one that minimizes the resistance value of the circuit, trying to make it easier to join the two sides.

== See also ==

* [[History of Hex]]
* [[Computer Hex]]

== References ==
[http://home.earthlink.net/~vanshel/VAnshelevich-01.pdf See paragraphs 2 and 3 for the first Hex playing machine.]

Gardner, Martin (1988), [http://www.amazon.com/Hexaflexagons-Other-Mathematical-Diversions-Scientific/dp/0226282546 Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games], Chicago: University of Chicago Press, ISBN 0-226-28254-6

{{stub}}

Hex theory

2007-06-08T00:00:26Z

Reiner Martin:

Unlike many other games, it is possible to say certain things about [[Hex]], with absolute certainty. While, for example, nobody seriously believes that black has a winning strategy in [http://en.wikipedia.org/wiki/Chess chess], nobody has been able to prove that such a strategy doesn't exist. In Hex, on the other hand, a simple argument can show that the [[second player]] certainly does not have a winning strategy from the [[starting position]] (when the [[Swap rule|swap option]] is not used). Whether this makes Hex a better game is of course debatable, but many find this attribute charming.

The most important properties of Hex are the following:

* The game can not end in a [[draw]]. ([http://javhar1.googlepages.com/hexcannotendinadraw Proofs] on Javhar's page)
* The [[first player]] has a [[winning strategy]].
* When playing with the swap option, the second player has a winning strategy.

== See also ==

[[Open problems]]

Edge template

2007-06-07T23:58:46Z

Reiner Martin:

An '''edge template''' is a [[pattern]] which guarantees a [[connection]] the [[edge]].

Here is an example of a [[Third row|third-row]] edge template ([[template IIIa]], also known as the ''Ziggurat''):

[[Image:Ziggurat.png]]



The red stone has a certain connection to the bottom, using only the shaded hexagons, even if blue moves first.



Countless edge templates can be constructed, all of them of varying degrees of usefulness and frequency of occurance.

* [[Edge templates everybody should know]]
* [[Edge templates with one stone]]
* [[Edge templates with two adjacent stones]]
* [[Edge templates with a bridge]]

Intermediate (strategy guide)

2006-10-30T21:29:37Z

Reiner Martin: /* Using edge templates to block your opponent */ - fixed typo

''Adapted with permission from Glenn C. Rhoads strategy guide.''

== Loose connections ==
''(See also the article [[Loose connection]])''

[[Adjacent move]]s provide a guaranteed connection but cover little ground. [[Bridge|Two-bridges]] cover twice the distance and are almost as strong. The next best connection when even more distance is required is called the '''loose connection''' — a [[Hex (board element)|hex]] that is a two-bridge plus an adjacent step away.

<hex>R4 C5 Vb2 Sc2 Sc3 Vd3</hex>

The [[piece]]s of the loose connection [[threat]]en to connect via a two-bridge plus an adjacent step [[Multiple threats|in two different ways]] — by playing at either of the marked hexes. Also, the two marked hexes are the only ones that are in the [[overlap]] of the two [[Template|connection patterns]]. Thus, to break a loose connection, one must play in one of the marked hexes.

Another connection pattern that is useful to be aware of is two pieces in the same row or column that are three hexes apart.

<hex>R5 C6 Sd2 Vb3 Pc3 Pd3 Ve3 Sc4</hex>

The pieces threaten to connect via 2 two-bridge steps in two different ways, namely by playing at piece at one of the hexes marked with a *. There are two hexes that are in the overlap between these two connection threats and a move played in either of them breaks the immediate connection (these two hexes are marked with a +). This connection pattern is not as strong as the loose connection.

== The useless triangle ==
''(See also the article [[Useless triangle]])''

When a piece's neighboring hexes are [[occupied hex|filled]] by the [[opponent]] such that that piece has only two empty neighboring hexes that are also [[adjacent]] to each other, then the piece is said to lie in a "'''useless triangle'''."

<hex>R7 C7 Q1 Vc5 Hd4 Hc4 Hb5 Hd5 Vd7 He7 Vf7</hex>

In the above diagram, the red piece at c5 and the [[empty hex]]es b6 and c6 form a useless triangle. The blue piece at e7 and the empty hexes e6 and f6 also form a useless triangle. The important point is that unless the piece in a useless triangle is in that player's [[First row|border row]], the piece has effectively been removed from the game — that is, it cannot have any effect on the rest of the game regardless of the rest of the position.

== Minimal edge templates ==
(See also the page [[Edge templates]])

An '''edge template''' is a pattern of empty hexes that will allow a piece to be [[Connection|connected]] to the [[edge]] even if the opponent has the next move. Just as the two-bridge is a useful connection pattern to know, so are minimal edge templates — the ones of the smallest size. (The templates are numbered according to row of the [[connecting piece]]).

=== [[Template I]] ===

Trivially, a piece on an edge row is connected to the edge.

<hex>R3 C3 Vb3</hex>

=== [[Template II]] ===

<hex>R3 C3 Vb2 Sa3 Sb3</hex>

If the opponent plays inside the template, [[Red (player)|Red]] plays the other move in the template restoring the connection to the edge.

For templates farther away, the general strategy to apply when the opponent intrudes on the template is to make a move in the template that reduces to a smaller and closer template.

=== [[Template IIIa]] ===
''(Also called [[Ziggurat]])''

<hex>R4 C4 Q1 Vc2 Sb3 Sd3</hex>

If the opponent intrudes on the template, then Red plays at one of the two marked points achieving [[template II]]. Since the b3 template and the two-chain/d3 template combination don't overlap, the opponent cannot stop both. (This template exists in a mirror image form with the red piece at d2).

=== [[Template IIIb]] ===

<hex>R4 C5 Vd2 Sb3 Se3 Pc4</hex>

The [[Hex (board element)|hex]] marked with an '+' can be occupied by the opponent! If the opponent intrudes on the template, then Red two-chains to either of the marked hexes and in either case forms [[template II]]. Since the two-chain/template II combinations don't [[overlap]] with each other, the opponent cannot stop both.

=== [[Template IVa]] ===

<hex>R5 C7 Q1 Ve2 Sd3 Se3 Sf3 PD5</hex>

If the opponent intrudes on the template, Red plays to one of hexes marked with '*' forming [[template IIIa]]. The one exception is if the opponent plays at d5, then Red plays to e3 and connects via [[template IIIb]]. (This template has a mirror image form with the red piece at f2.)

=== [[Template IVb]] ===

<hex>R5 C8 Vf2 Sd3 Sg3 Pe4</hex>

The hex marked with a '+' can be occupied by the opponent! If the opponent intrudes on the template, Red two-chains to one of the marked hexes forming [[template IIIa]].

== Forming ladders ==
''(See also the article [[Ladder]])''

A '''ladder''' occurs when one player tries to force a connection to an edge but is kept a constant distance away by the opponent, resulting in a sequence of moves parallel to the edge. The following is an example with Red to play.

<hex>R9 C9 Q1 Ve3 Vd4 Vd5 Vc6 Vc7 Hb8 Hb9 Hd7 He6 Hf6</hex>

Suppose Red plays c8 which forces c9 in reply. Now Red can play the following forcing sequence. d8, d9, e8, e9, f8, f9, etc. A sequence of such plays parallel to an edge is called a ladder. Note that the [[Blue (player)|Blue]]'s responses are forced. If Red blindly continues the ladder all the way to end, then he will simply lose (Blue will get pieces in row 9 from b9 through i9). There is no good reason to ever force a ladder all the way through to end, it only helps your opponent.

=== Ladder escapes ===
''(See also the article [[Ladder escape]])''

Consider the same position as before but suppose Red has an additional piece at h8.

<hex>R9 C9 Q1 Ve3 Vd4 Vd5 Vc6 Vc7 Hb8 Hb9 Hd7 He6 Hf6 Vh8</hex>

This additional piece forms a '''ladder escape''' which allows Red to jump a move ahead of the ladder and win the game. The piece at h2 is called the "[[escape piece]]." Red should now play along the ladder as before, forcing Blue's response at each step. After c8, c9, d8, d9, e8, e9, f8, f9, g8 Red is connected to the bottom.

<hex>R9 C9 Q1 Ve3 Vd4 Vd5 Vc6 Vc7 Hb8 Hb9 Hd7 He6 Hf6 Vh8 Vc8 Hc9 Vd8 Hd9 Ve8 He9 Vf8 Hf9 Vg8</hex>

In general, for a ladder escape to be successful, it should be safely connected to the edge and not interfere with the ladder's [[Projected ladder path|projected path]].

=== Ladder escape templates ===
''(See also the article [[Ladder escape templates]])

* [[Second row|Row-2]] ladders: All of the [[edge template]]s described earlier are valid.
* [[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]] is valid. Also [[template IVa]] is valid if you can double two-bridge to the [[escape piece]] as follows.

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

Red can jump ahead to the escape template by playing at the marked hex.

=== The ladder escape fork ===
''(See also the article [[Ladder escape fork]])''

If you are forced onto a ladder and no convenient escape is present, then you must create one. The best way is to play one of the valid ladder escape templates that threatens another strong connection. Such a move is called a '''ladder escape fork'''. For an example, see the first example in the upcoming section "forcing moves." The first forcing move is a ladder escape fork played just prior to the formation of the ladder (and a very short ladder at that). A ladder escape fork is frequently a [[killer move]].

=== Foiling ladder escapes ===
''(See also the article [[Foiling ladder escapes]])''

In order to successfully stop a ladder escape, you must either block the [[projected ladder path]] from connecting to the escape piece or intrude on the ladder escape template. To successfully stop a ladder escape fork, you need to do both with a single move and almost always with a move that is [[Adjacent move|adjacent]] to the escape piece. The following is an example of foiling a ladder escape fork.

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Hc6 Hd6 He7 Hf7 Hg7 Hi7 Hg9 Sc7 Sb8 Sc8 Sd8 Sa9 Sb9 Sc9 Sd9</hex>

Red has just played a forking ladder escape at d7. This piece is connected to the edge via template IIIa as shown by the marked hexes. Red is threatening to create an unbeatable chain by playing at E6 and the edge template is a valid ladder escape for the row-2 ladder starting G8, F9, F8, etc. To stop this, Blue needs to play a move that blocks the ladder path from connecting to the escape piece and that also intrudes on the escape template. Blue can achieve both aims by playing at D8 (which is adjacent to the escape piece). Red responds by playing C8 re-establishing the connection to the edge (there is nothing better). Now Blue continues by playing E6 blocking the forking path obtaining a [[win|winning position]].

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Ve6 Hc6 Hd6 Hd8 He6 He7 Hf7 Hg7 Hi7 Hg9 Vc8</hex>

Consider the same initial position but with Blue's piece on e7 removed.

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Hc6 Hd6 Hf7 Hg7 Hi7 Hg9 Sc7 Sb8 Sc8 Sd8 Sa9 Sb9 Sc9 Sd9</hex>

This change may look inconsequential but now Blue cannot foil the forking ladder escape. Suppose the play goes d8, c8, e6 as before.

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Ve6 Hc6 Hd6 Hd8 He6 Hf7 Hg7 Hi7 Hg9 Vc8</hex>

Now Red can ladder up to E8 by the sequence G8, F9, F8, E9, E8 achieving the following position.

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Ve6 Hc6 Hd6 Hd8 He6 Hf7 Hg7 Hi7 Hg9 Vg8 Hf9 Vf8 He9 Ve8 Vc8</hex>

Now if Blue stops the e8 piece from connecting to the [[Bottom edge|bottom]] by playing d9, Red responds by playing e7 connecting to the bottom anyway. This example illustrates that a potential foiling move that leaves vulnerable points is unlikely to succeed against a forking ladder escape. The proper handling of ladders and ladder escapes is a complex matter and it is where many games are won or lost.

=== Pre-ladder formations ===

It's important to recognize situations in which a ladder is about to form or which could be formed. Such recognition allows you to play pieces that also serve as ladder escapes before the ladder occurs. It also allows you to play defensive moves that also block potential ladder paths prior to the existence of the ladder. By far the most common pre-ladder formation is the following "[[Bottleneck]] formation."

<hex>R5 C7 Q1 Hd3 Ve3 Hf3 Hd5</hex>

Red can now form a ladder by playing e4, e5, f4, f5, etc. or by playing d4, c5, c4, b5, etc. Such formations typically occur due to blocking a player from directly connecting to an edge as in the following example.

<hex>R5 C7 Q1 Vg1 Ve2 Hf3</hex>

In order to block Red from connecting to the bottom edge, Blue plays d3 creating a [[bottleneck]]. Red responds with e3 squeezing through and then Blue blocks with d5 completing the formation in the previous diagram.

The other common pre-ladder formation occurs when the defender is blocking the connection to an edge via a classic block as in the following diagram.

<hex>R5 C7 Q1 Ve1 Hd4</hex>

Red can form a ladder by playing d3, c4 and then laddering either to the left or right (c3, b4, b3, a4 or e3, e4, f3, f4, etc.)

== Forcing moves ==
''(See also the article [[Forcing move]])''

'''Forcing moves''' are moves that make a threat that your opponent must reply to on their next turn. Common forcing moves include playing in one of the [[Empty hex|open hexes]] in a two-chain (threatening to break the link), intrusion into an edge template, or threatening an immediate strong connection or win. Consider the following position with the [[vertical player]] to move.

<hex>R9 C9 Q1 Vg3 Vg4 Vf5 Vh5 Hc7 Hd8 Hf7 Hh7</hex>

At first glance, the position looks bad for Red, but Red can win by making a couple of forcing moves. He plays at e8 threatening to play at e7 on his next turn which would create an unbeatable winning chain. Blue has little choice but to stop this threat by playing e7 (there is nothing better). The move e8 is a forcing move.

The forcing nature of the move allows Red to place a piece on the other side of Blue's line without giving Blue any time to do anything constructive. The e8 piece on the other side is connected to the bottom and is of critical importance.

Red continues by playing another forcing move at g7. The only move that stops this piece from immediately connecting to the bottom edge is f9. But after f9, Red completes the win by playing at f8.

<hex>R9 C9 Q1 Vg3 Vg4 Vf5 Vh5 Hc7 Hd8 Hf7 Hh7 Ve8 He7 Vg7 Hf9 Vf8</hex>

The group of red pieces near the bottom are connected to the bottom edge. These pieces are connected via two-chains to the [[group]] g3-g4-f5 which is in turn connected to the top edge via edge [[template IIIa]].

(Note: the two forcing moves could just as easily be played in the reverse order. That is Red plays g7, Blue is forced to respond with f9, and then Red plays e8 which threatens to form an unbeatable chain in two distinct ways.)

In general terms, you have three options when responding to a forcing move in a [[Bridge|two-chain]].

# [[Saving a connection|Save]] the link by playing the other move in the two-chain.
# [[Ignoring a threat|Play elsewhere]] (e.g. playing another move may give another way of meeting the threat thus rendering it harmless)
# [[Counterthreat|Respond]] with a forcing move of your own.

=== Breaking edge templates via forcing moves ===

Forcing moves are also the only way to successfully defeat an edge template. This is done by making a [[template intrusion]] that is also a more threatening forcing move. After the opponent responds to the greater threat, you can play another move within the template and destroy the connection to the edge. For example, consider the following position with the vertical player to move.

<hex>R9 C9 Q1 Hc3 Hd3 He3 Hf2 Hg1 Hg3 Vd7 Ve6 Ve5 Ve4 Vf3 Vh5 Si1 Si2 Si3 Si4 Sh2 Sh3 Sh4 Sg4</hex>

The piece on g3 is connected to the right edge via [[template IIIa]] indicated by the '*'s. Red's best move is to play at h2. This intrudes on the edge template, is connected to the top via edge [[template II]], and threatens to complete an unbeatable chain by playing at g2 next turn. Blue can stop this threat only by playing at g2. Then Red Plays i3 breaking Blue's connection to the right.

<hex>R9 C9 Q1 Hc3 Hd3 He3 Hf2 Hg1 Hg3 Vd7 Ve6 Ve5 Ve4 Vf3 Vh5 Vh2 Hg2 Vi3</hex>

=== Using forcing moves to steal territory ===
''(See also the article [[Stealing territory]])''

I'll define '''territory''' to be the number of empty hexes adjacent to your pieces. By playing a forcing move in one of the empty hexes in a two-chain, a player can steal territory at no cost.

<hex>R5 C5 Hc2 Hb4 Vd2 Vd3 Sb3</hex>

In this position, Blue has two more hexes of territory than Red (9 vs. 7 [[adjacent hex]]es). Suppose Red makes the forcing move at the indicated hex and Blue saves the link.

<hex>R5 C5 Hc2 Hb4 Vd2 Vd3 Vb3 Hc3</hex>

Now Red has two more hexes of territory; i.e. Red has stolen 4 hexes of territory without disturbing either player's connections. Significantly, the additional territory is on the other side of Blue's connection where it may potentially be used for a future threat. The additional territory can't hurt and sometimes it makes a crucial difference.

A forcing move is [[Irrelevant move|harmless]] if it gains no territory for the opponent. You should not be worried at all about leaving harmless forcing moves available for your opponent. When you have more than one way of completing a connection with a two-bridge, e.g. when completing the loose connection described previously, you should consider which forcing move is least valuable for your opponent. Consider the following position with Red to play.

<hex>R5 C6 Q1 Vd2 He3 Hb4 Vd4 Hb5</hex>

Red wants to connect the d4 piece to the d2 piece. There are three distinct moves that accomplish this, d3, c4 (two-chaining to d2), and c3 (two-chaining to d4).

There is not much to be said about d3; it [[Direct connection|directly connects]] without altering anything else. c4 connects but gives a potentially useful forcing move to Blue. Blue can respond with c3 and suppose Red saves the connection with d3. Now Blue has gained a free hex of territory, the hex c2 is now directly adjacent to the c3/b4/b5 group when it wasn't previously. Hence, c4 is worse than d3.

Now consider the last remaining possibility c3. This leaves two forcing moves for Blue but both of them are completely harmless! If after c3, Blue plays one of the forcing moves c4 or d3, then Red can save the link and Blue will not have gained any territory at all — any empty hexes adjacent to the forcing piece were already adjacent to Blue's existing pieces. Hence, c3 is just as safe as d3 but significantly, c3 ''gains'' one hex! — b3 is now adjacent to Red's d2/b3 group when it wasn't before. Thus, c3 is better than d3 and is the best of three choices.

== Using edge templates to block your opponent ==

If your opponent has not completed an [[edge template]] but is threatening to do so in multiple ways, then the only defensive moves that stop the immediate threatened connections are those in the overlap between all threatened template connections. Suppose you are trying to stop the vertical player from connecting to the [[bottom edge]] in the following example.

<hex>R5 C6 Q1 Ve2 Hf3</hex>

The vertical player has not formed an edge template but is threatening to do so in the following four different ways.

{|
| <hex>R5 C6 Q1 Ve2 Hf3 Sd3 Se3 Sd4 Sc5 Sd5</hex> || <hex>R5 C6 Q1 Ve2 Hf3 Sd3 Se3 Sc4 Sd4 Se4 Sb5 Sc5 Sd5 Se5</hex>
|-
| <center>''Two-chain to [[template II]] at d4''</center> || <center>''Adjacent move to [[template IIIa]] at d3 and e3''</center>
|-
| <hex>R5 C6 Q1 Ve2 Hf3 Sc3 Sd3 Sb4 Sc4 Sd4 Sa5 Sb5 Sc5 Sd5</hex> || <hex>R5 C6 Q1 Ve2 Hf3 Sc3 Sd3 Se3 Sb4 Sc4 Sd4 Se4 Sa5 Sb5 Sd5 Se5</hex>
|-
| <center>''Adjacent move to template IIIa at d3'' || <center>''Adjacent move to [[template IIIb]] at d3''</center>
|}

The only three [[Hex (board element)|hexes]] in the overlap among all these edge templates are marked on the following diagram. To stop the immediate connection, the horizontal player must play at one of them.

<hex>R5 C6 Q1 Ve2 Hf3 Sd3 Sd4 Sd5</hex>

== On connectivity ==

=== Overlapping connections ===
''(See also the article [[Overlapping connections]])''

One should be alert to the situations where various connections, edge templates, and potential connections overlap at some hex(es). Consider the following position with Blue to move.

<hex>R11 C11 Vj2 Vi4 Vj5 Vi7 Vi9 Vh9 Vg9 Vf9 Se9 Ve8 Vd10 Hg7 Hf7 He6 Hc7 Hc9 He10 Hf10 Hg10 Hh10 Hi10</hex>

At first glance, it appears that Red has an unbreakable winning path. j2 cannot be stopped from connecting to the [[Top edge|top]], j2 is connected the [[group]] f9-g9-h9-i9 through a series of unbreakable two-chains, this group is connected to e8 via a two-chain, e8 is connected to d10 via another two-chain, and d10 cannot be stopped from connecting to the [[Bottom edge|bottom]].

Appearances are deceiving; it is Blue that has a forced win! The [[Weakest link|flaw]] in Red's formation is that the two-chain from f9 to e8 and the two-chain from e8 to d10 [[overlap]] at the hex marked by a '*' in the diagram (e9). Blue should play at e9. By playing in the overlap, Blue is threatening to break ''both'' two-chains containing this hex. Red cannot save them both.

If Red responds with f8, then Blue plays d9 breaking the two-chain and establishing an unbeatable chain. If Red saves the other link by responding with d9, then Blue breaks through with f8 again establishing an unbeatable chain.

=== Disjoint steps ===

When a piece can be connected to a group of pieces in one move in two non-overlapping ways, then they can be thought of as already connected to the group. Consider the following position.

<hex>R5 C5 Q1 Vc2 Vd2 Vb3 Hc3 Ha5 Hd4</hex>

Red's three pieces are connected to the top. How can Red extend this connection downward? By playing at c4! The piece at c4 is connected to the group of three vertical pieces in two non-overlapping ways; namely, through the hexes b4 and d3. The diagrammed connection pattern is a fairly common occurrence and the connection to the piece at c4 is just as strongly connected as the pieces in a two-chain.

=== Groups ===
''(See also the article [[Group]])''

A '''group''' is a collection of pieces that, considered in isolation from the rest of the position, have an unbreakable connection with each other. As you improve, it is important to learn to think in terms of safely connected groups of pieces. To illustrate why, consider the following two hex puzzles taken from [[Cameron Browne]]'s book "[[Hex Strategy]]."

<hex>R11 C11 Q1 Hj2 Hh3 Hc4 Vd4 Hf4 Vi4 Vj4 Vd5 Vg5 Hh5 Vi5 Vk5 Ve6 Hf6 Hg6 Hh6 Hi6 He7 Vg7 Hi7 Vj7 Vc8 Vi9</hex>

It is Blue's turn; how can he win?. The chain of pieces j2-h3-f4 is connected to the right edge and furthermore, Blue has no other way of connecting to the right edge. So to win, Blue has to extend this chain to the left edge. Looked at in isolation, there doesn't seem to be any way to do this, yet by thinking in terms of connected groups, the solution falls out easily.

Notice that the j2-h3-f4 chain threatens to connect to the i7-i6-h6-h5-g6-f6-e7 group in two non-overlapping ways, through locations h4 and f5. Hence, these two groups can be thought of as a single group of pieces already connected to the right edge (this is another example of "'''disjoint steps'''"). Now notice the key hex d6. This hex threatens to connect to Blue's big group in two distinct non-overlapping ways (through e5 and d7) hence a piece played at d6 would be part of the big group (disjoint steps again!). Furthermore, a blue piece at d6 could not be stopped from connecting to the left because the b4 piece acts as a ladder escape. Therefore, d6 wins.

<hex>R11 C11 Q1 Hj2 Vc3 Hd3 Vg3 Hj3 Hc4 He4 Vc5 Vd5 Hg5 Vi5 Vd6 He6 Vd7 Ve7 Vh7 Hb9</hex>

Again, it is Blue's turn and the task is to [[win]]. The c4 piece cannot be stopped from connecting to the left edge since after the block a5, Red can ladder down row B to the escape piece at b9. Hence, the group c4-d3-e4 is connected to the [[left edge]]. The group j2-j3 is connected to the [[right edge]]. Blue has a third group e6-g5. If Blue can play a single move that connects the e6-g5 group to both other groups, then this would be a winning move. Blue has a unique move which does this, namely play at g4. The g4-g5-e6 group is connected to the left group through f4 and e5. It threatens to connect to the j2-j3 group via h3 and i4. h3 is directly connected to g4 and two-chains to j2. None of the hexes involved, h3, i2, and i3, is involved in the connection threat i4 plus the two chain to g5. I.e. the threats don't overlap and hence the connection cannot be stopped. Therefore, g4 wins.

There is another way of coming up with this move. Red threatens to cut off the e6-g5 group to the left by two-chaining from g3 to f5. Red is also threatening to cut off the e6-g5 from the right by two-chaining from g3 to h4. However, these threats overlap and hence, Blue can stop them both by playing in the unique hex contained in the overlap, namely g4 again.

This illustrates that [[Offence equals defense]] in hex. Playing in regions of overlapping threats in order to stop all the threats is a defensive way of thinking. Trying to establish unbreakable connections between groups of your pieces is an offensive way of thinking. In this example, both offensive and defensive thinking techniques lead you to the unique best move. A lot of times defensive thinking is easier but sometimes offensive thinking is.

== Conclusion ==

The first two strategy guides cover what I consider to be the fundamentals of [[hex strategy]]. This information should be enough to move up into the 1800s or 1900s on [[PlaySite]]. To move up the ranks of the red guys (the topmost group) requires the following. First you need to improve your [[opening play]]. Playing any reasonable looking moves during the initial phase of the game is enough for an orange player, but to compete with the reds, you need to [[Consistency|consistently]] play one of the top two or maybe three moves. Unfortunately, I don't know a good way to describe how to do this; I'm not convinced the necessary information can be verbalized. After that, you need to know the [[Minimax]] principle (described in the [[Advanced (strategy guide)|Advanced strategy guide]]).

Also you need a certain mindset, call it [[willpower]] if you like, to move towards the top ranks. You have to try to hold onto every little [[Hex (board element)|hex]] the way a miser hoards gold pieces and you have use every optimization you can no matter how minor it may seem. The most useful optimizations, tricks, and special situations that I've learned so far are included in the Advanced strategy guide. But surely there are other things out there waiting to be discovered.

Tournaments

2005-06-01T22:11:09Z

Reiner Martin: new page

==International Tournament 2005 in Wrocław==

The first international Hex tournament was held in May 2005 in Wrocław, Poland.

Here is some information:
* a [http://masak.org/carl/wroclaw/ blog with results]
* a page with [http://www.photos-wroclaw.prv.pl/ photos from the event]

Physical Hex sets

2005-06-01T22:04:05Z

Reiner Martin: added historic information

A (physical) Hex set was marketed under that name by Parker Bros. starting in 1952. Currently, there seem to be no commercial Hex sets available.

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].

Physical Hex sets

2005-06-01T22:03:38Z

Reiner Martin: added historic information

<nowiki>Insert non-formatted text here</nowiki>
A (physical) Hex set was marketed under that name by Parker Bros. starting in 1952. Currently, there seem to be no commercial Hex sets available.

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].

Physical Hex sets

2005-06-01T22:01:53Z

Reiner Martin: adding histori information

Current events

2005-04-23T21:46:48Z

Reiner Martin:

There will be an international Hex tournament in Wroclaw, Poland, from the 6th to 8th of May 2005 (this must be one of the first or even the first international Hex tournament).

To register, see [http://masak.org/carl/wroclaw/ Wroclaw Hex event 2005 Registration].

Details of the tournament are discussed in a [http://www.littlegolem.net/jsp/forum/topic.jsp?forum=50&topic=174 forum] on the [[Little Golem]] game site.

Main Page

2005-02-25T08:47:29Z

Reiner Martin: restored redirection

#REDIRECT [[Main page]]

Online playing

2005-02-20T11:44:07Z

Reiner Martin:

Hex can be played online either in ''real-time'', where moves are made within minutes (or even seconds), or ''turn-based'', where a player has days for one move.

The most popular sites are
* [[kurnik]] for real-time play,
* and [[Little Golem]] for turn-based play.

Swap rule

2005-02-20T11:17:45Z

Reiner Martin:

Because the [[First player|first player]] has a considerable advantage if he is allowed to make his first move without restrictions, the '''swap rule''' was devised. It states that one player first makes a move, and then the [[Second player|other player]] decides who plays with which colour. The swap rule can be implemented in two ways, as follows. (Assume that the colours are red and blue, with red moving first.)

# The first player places a red [[Piece|piece]] in any [[hexagon (Board element)|hex]] on the [[Board|board]]. Then the other player can either make a move with the blue pieces, in which case he becomes blue, or he can state that he wants to be red. After this the game continues without any more swapping.
# The first player makes a move with the red pieces. The other player either makes a move with the blue pieces, or makes a [[Piece swap|piece swap]]. This means to remove the red piece from the board, and place a blue piece at the hexagon which is the mirror image of the hexagon in which the red piece was placed, with respect to the board's long diagonal.

In [[IRL play|face-to-face play]] the first option is most practical, since it is easier to change colours of the players that removing and adding pieces on the board. It is also less error-prone. On [[Game sites|game sites]] on the Internet the second version is more common, presumably because the colours are determined at the start of the game, and it is easier to change the board position than the colour designation.

The swap rule is also called the "Pie rule", since it resembles the ''You cut, I choose'' rule when sharing a pie between two children.

When playing with the swap rule, the second player has a winning strategy. However, the second player's [[Advantage|advantage]] is much smaller than the advantage of the first player is when playing without swap.

== The generalized swap rule ==
Instead of placing just one piece, the first player can place any number of red and blue pieces, and state which colour has the next move. The second player then decides who has which colour. This version has the drawback that the first player can prepare in advance a position which he has studied extensively, which the second player doesn't see until the game starts, thereby giving the first player an advantage.

Edge templates everybody should know

2005-02-16T22:05:27Z

Reiner Martin:

In all the diagrams Red is the vertical player and wants to connect his stone or stones to the bottom. Empty cells in the diagrams cannot be occupied by the opponent for the template to be valid; cells with a star may be empty or be occupied by either player.

== Third row edge templates with one stone ==
<hex>R3 C4 Vc1 Sa1 Sa2 Sb1</hex>
<hex>R3 C5 Vd1 Sa1 Sa2 Sb1 Sc3</hex>

== Fourth row edge templates with one stone ==
<hex>R4 C7 Ve1 Sa1 Sa2 Sa3 Sb1 Sb2 Sc1 Sd1 Sg1</hex>
<hex>R4 C8 Vf1 Sa1 Sa2 Sa3 Sb1 Sb2 Sc1 Sd1 Sh1 Se3</hex>

== Third row edge templates with two stones ==
<hex>R3 C3 Vb1 Vc1 Sa1</hex>

== Fourth row edge templates with two stones ==
<hex>R4 C4 Vc1 Vd1 Sa1 Sa2 Sa3 Sb1</hex>

== Fifth row edge templates with two stones ==
<hex>R5 C7 Ve1 Vf1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sg1 Sg2</hex>

Edge templates with one stone

2005-02-16T22:02:42Z

Reiner Martin:

Here you can see all 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 ==
<hex>R3 C4 Vc1 Sa1 Sa2 Sb1</hex>
<hex>R3 C5 Vd1 Sa1 Sa2 Sb1 Sc3</hex>

== Fourth row edge templates ==
<hex>R4 C7 Ve1 Sa1 Sa2 Sa3 Sb1 Sb2 Sc1 Sd1 Sg1</hex>
<hex>R4 C8 Vf1 Sa1 Sa2 Sa3 Sb1 Sb2 Sc1 Sd1 Sh1 Se3</hex>

== Fifth row edge template ==
<hex>R5 C10 Vg1 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1 Sd2 Se1 Si1 Sj1 Sj2</hex>

There are almost certainly no edge templates with one stone on the [[Sixth row|sixth row]]. This hasn´t been proved, though.

Help:Contents

2005-02-16T17:00:00Z

Reiner Martin:

[[HexWiki]] is a [http://en.wikipedia.org/wiki/Wiki wiki] for the game of [[Hex]], i.e., it is a website written collaboratively by people from around the world.

The [http://en.wikipedia.org/wiki/Help:Contents help page for Wikipedia] contains guidance and information about participating which is mostly applicable to HexWiki as well (since both the Wikipedia and HexWiki are based on the MediaWiki software).

Help:Contents

2005-02-16T16:59:33Z

Reiner Martin:

Miai

2005-02-16T11:08:26Z

Reiner Martin: added ext link

'''Miai''' is a Japanese term taken from [[Go]], which is also very useful in [[Hex]]. It means that a player has two different options for achieving a specific goal, usually connecting two groups of pieces.

The most basic example is the [[bridge]] as seen below, where the starred cells are said to be miai; if [[Blue (player)|Blue]] plays at one of them, [[Red (player)|Red]] secures his connection by playing at the other.

<hex>R4 C4 Vb2 Vc3 Sb3 Sc2</hex>

===External link===

* [http://senseis.xmp.net/?Miai Article in Sensei's Library on Miai in Go]

Draw

2005-02-16T11:02:19Z

Reiner Martin:

One of the beautiful properties of Hex is that the game can never end in a '''draw''', i.e., there is always a winner.

There are various ways of proving this, for example:

* A [http://www.cs.ualberta.ca/~javhar/hex/hex-galeproof.html proof by David Gale] that used the fact that exactly three hexes meet at every vertex.
* A [http://www.cs.ualberta.ca/~javhar/hex/hex-yproof.html elegant proof] using the [[Y|game of Y]].

In fact, the no-draw property is equivalent to Brouwer's fixed point theorem (a non-trivial theorem from topology saying that any continuous map from the unit square onto itself must contain a fixed point).

Computer Hex

2005-02-16T10:51:27Z

Reiner Martin: more info

There are several computer programs available which play Hex.

* The currently strongest Hex program is [http://six.retes.hu/ Six] by Gábor Melis, running on Linux or Unix.
* The second strongest program available is [http://home.earthlink.net/~vanshel/ Hexy] by Vadim Anshelevich, running on Microsoft Windows. This site also has a interesting paper on aspects of the algorithm used.
* [http://www.cs.ualberta.ca/~queenbee/ Queenbee] by Jack van Rijswijck
* Mongoose by Yngvi Björnsson, Ryan Hayward, Mike Johanson, Morgan Kan, and Nathan Po

The [http://www.cs.unimaas.nl/icga/ International Computer Games Association] also has some [http://www.cs.unimaas.nl/icga/games/hex/ information on Hex]. They organize an annual computer olympiad, which also covers Hex. For example, Six by won the gold medal of the 8th Computer Olympiad in Graz, while Hexy won the gold medal of the 5th Computer Olympiad in London, which appears to be the first computer Hex tournament ever hold.

Hex

2005-02-16T10:34:42Z

Reiner Martin:

'''Hex''' is a game played by two [[player]]s on a [http://en.wikipedia.org/wiki/Rhombus rhombic] [[board]] consisting of [[Hex (board element)|hexagons]]. Each player owns two of the [[Edge|sides]] of the board, and a player [[win]]s if he creates an unbroken [[chain]] between his two sides. Hex is classified as a [[Connection game]].

For a description of how to play the game, see the article [[Rules]].

== More playing ==

If you already have read the [[Rules]] article and you want more thorough information on how to play Hex, proceed to the article about [[Strategy]].

== History of Hex ==

The [[History]] article contains an account of the events surrounding the game so far.

Computer Hex

2005-02-16T10:33:31Z

Reiner Martin:

There are several computer programs available which play Hex.

* The currently strongest Hex program is [http://six.retes.hu/ Six] by Gábor Melis, running on Linux or Unix.
* The second strongest program available is [http://home.earthlink.net/~vanshel/ Hexy] by Vadim Anshelevich, running on Microsoft Windows. This site also has a interesting paper on aspects of the algorithm used.
* [http://www.cs.ualberta.ca/~queenbee/ Queenbee]

The [http://www.cs.unimaas.nl/icga/ International Computer Games Association] also has some [http://www.cs.unimaas.nl/icga/games/hex/ information on Hex]. They organize an annual computer olympiad, which also covers Hex. For example, Six by won the gold medal of the 8th Computer Olympiad in Graz, while Hexy won the gold medal of the 5th Computer Olympiad in London, which appears to be the first computer Hex tournament ever hold.

Computer Hex

2005-02-16T10:32:52Z

Reiner Martin:

There are several computer programs available which play Hex.

* The currently strongest Hex program is [http://six.retes.hu/ Six] by Gábor Melis, running on Linux or Unix).
* The second strongest program available is [http://home.earthlink.net/~vanshel/ Hexy] by Vadim Anshelevich, running on Microsoft Windows. This site also has a interesting paper on aspects of the algorithm used.
* [http://www.cs.ualberta.ca/~queenbee/ Queenbee]

The [http://www.cs.unimaas.nl/icga/ International Computer Games Association] also has some [http://www.cs.unimaas.nl/icga/games/hex/ information on Hex]. They organize an annual computer olympiad, which also covers Hex. For example, Six by won the gold medal of the 8th Computer Olympiad in Graz, while Hexy won the gold medal of the 5th Computer Olympiad in London, which appears to be the first computer Hex tournament ever hold.

Computer Hex

2005-02-16T10:31:24Z

Reiner Martin: new page

There are several computer programs available which play Hex.

* [http://six.retes.hu/ Six] The currently strongest Hex program, by Gábor Melis, running on Linux or Unix).
* [http://home.earthlink.net/~vanshel/ Hexy] The second strongest program available, by Vadim Anshelevich, running on Microsoft Windows. This site also has a interesting paper on aspects of the algorithm used.
* [http://www.cs.ualberta.ca/~queenbee/ Queenbee]

The [http://www.cs.unimaas.nl/icga/ International Computer Games Association] also has some [http://www.cs.unimaas.nl/icga/games/hex/ information on Hex]. They organize an annual computer olympiad, which also covers Hex. For example, Six by won the gold medal of the 8th Computer Olympiad in Graz, while Hexy won the gold medal of the 5th Computer Olympiad in London, which appears to be the first computer Hex tournament ever hold.

Common mistakes

2005-02-16T10:18:58Z

Reiner Martin: removed capitalisation for consistency

== Ladder escaping too early ==

<hex>R6, C10, Q1, Vc6, Ve5, Vg4, Vg3, Vg2, Hd4, Hf3, Hg1, Hh1, Hh3, Hi2, Hb2</hex>

Red to move. In this situation Red has a win with perfect play. He only has to find a good [[Ladder escape|ladder escape]] from the [[Ladder|ladder]] starting at h1, g2. c2 is such an escape. However, playing c2 in this stage of the game is a losing move because Blue can play any of f1, f2, e2. Another good escape from the [[Second row|second line]] would be c3, but once again it fails it this situation. So what should Red do? The only way to win is to play out the ladder to e2 and ''then'' jump to c2:

<hex>R6, C10, Q1, Vc6, Ve5, Vg4, Vg3, Vg2, Hd4, Hf3, Hg1, Hh1, Hh3, Hi2, Hb2, He1, Hf1, Vf2, Ve2, Vc2</hex>

Now Blue cannot block Red's connection. A further development game could be: d2, c3, d3, b5 and Red makes a double [[Bridge|bridge]], connecting c3 and c6.

== Bad bridge intrusion ==

<hex>R6, C10, Q1, Vc6, Ve5, Vg4, Vg3, Vg2, Hd4, Hf3, Hg1, Hh1, Hh3, Hi2, Hb2</hex>

The same situation can be lost if Red intrudes into the d4-f3 bridge at e3. Of course if Blue doesn't see the trap he will fill up the bridge at e4 and then Red's e3 is a ladder escape. But Blue can play f2 and win because the ladder moved from second to the [[Third row|third line]], and Red has no ladder escape from the third line.
''Remember that such a bridge intrusion removes the ladder one line further from the edge, which is often a disadvantage to the attacking player.''

Edge templates with a bridge

2005-02-16T10:17:19Z

Reiner Martin: removed capitalisation for consistency

In the [[edge templates]] below, both stones are guaranteed a connection to the edge.

== Fourth row template ==
<hex>R4 C4 Vc1 Vd2 Sa1 Sa2 Sa3 Sb1</hex>

== Fifth row template ==
<hex>R5 C6 Ve1 Vf2 Sa1 Sa2 Sa3 Sa4 Sb1 Sb2 Sb3 Sc1 Sc2 Sd1</hex>

Piet Hein

2005-02-16T10:16:25Z

Reiner Martin: added links

'''Piet Hein''' (1905-1996) was a Danish poet and mathematician, and the first inventor of Hex.

In Denmark, and in the rest of Scandinavia, he is most famous for his collections of short poems, which he called [http://en.wikipedia.org/wiki/Grook grooks]. Most of them are written in Danish, but some he himself translated into English. The following is an example of these poems, which ought to be taken to heart by all [[Hex]] players.

The road to wisdom?
Well, it's plain
and simple to express:
Err
and err
and err again
but less
and less
and less

==External links==

* [http://www.piethein.com Piet Hein Homepage]
* [http://www.ctaz.com/~dmn1/hein.htm Notes on Piet Hein]
* [http://chat.carleton.ca/~tcstewar/grooks/grooks.html Grooks by Piet Hein]

Strategy

2005-02-16T10:13:15Z

Reiner Martin: removed capitalisation for consistency

== General advice ==
* One important advice for novice players to remember: "In [[Hex]], the best attacking move is often to defend." Instead of trying to connect your [[Edge|sides]], do your best to [[Blocking|impede]] your [[opponent]].
* If you play with a strong player and you lose, try to repeat the opponent's moves and observe the responses. This can be a fast way to learn new bit of strategy.
* Another good way to learn strategy is to analyse games — preferably in your head and not with a program. Talking to senior players is also a good idea.

== General strategy guide ==

These three pages originate from Glenn C. Rhoads' excellent strategy guide. They have mostly been changed to accomodate the transfer to the Wiki format.

* [[Basic]]
* [[Intermediate]]
* [[Advanced]]

== Specific topics ==
* [[Openings]]
* [[Connections]]
* [[Bottlenecks]]
* [[Ladders]]
* [[Ladder handling]]
* [[Templates]]
* [[Equivalent patterns]]
* [[Common mistakes]]
* [[Tips and tricks]]
* [[Territory]]
* [[Local play]], [[Global play]]
* [[Walls]]
* [[Glossary]]
* [[Puzzles]]
* [[Tactics]]

Forcing move

2005-02-16T10:11:46Z

Reiner Martin: removed capitalisation for consistency

A '''forcing move''' is a move that makes a threat that your opponent must reply to on their next [[turn]]. Common forcing moves include playing in one of the [[Empty hex|open hex]]es in a two-chain (threatening to break the link), intrusion into an [[edge template]], or [[threat]]ening an immediate [[strong connection]] or [[win]]. Consider the following position with the [[vertical player]] to move.

<hex>R9 C9 Q1 Vg3 Vg4 Vf5 Vh5 Hc7 Hd8 Hf7 Hh7</hex>

At first glance, the position looks bad for [[Red (player)|Red]], but Red can win by making a couple of forcing moves. He plays at e8 threatening to play at e7 on his next turn which would create an unbeatable [[winning chain]]. [[Blue (player)|Blue]] has little choice but to stop this threat by playing e7 (there is nothing better). The move e8 is a forcing move.

The forcing nature of the move allows Red to place a piece on the other side of Blue's line without giving Blue any time to do anything constructive. The e8 piece on the other side is connected to the [[Bottom edge|bottom]] and is of critical importance.

Red continues by playing another [[forcing move]] at g7. The only move that stops this piece from immediately [[connection|connecting]] to the bottom edge is f9. But after f9, Red completes the win by playing at f8.

<hex>R9 C9 Q1 Vg3 Vg4 Vf5 Vh5 Hc7 Hd8 Hf7 Hh7 Ve8 He7 Vg7 Hf9 Vf8</hex>

The group of red pieces near the bottom are connected to the bottom edge. These pieces are connected via [[chain|two-chains]] to the group g3-g4-f5 which is in turn connected to the [[top edge]] via [[edge template IIIa]].

''(Note: the two forcing moves could just as easily be played in the reverse order. That is Red plays g7, Blue is forced to respond with f9, and then Red plays e8 which threatens to form an unbeatable chain [[Double threat|in two distinct ways]].)''

In general terms, you have three options when responding to a forcing move in a two-chain.

# [[Saving a connection|Save]] the link by playing the other move in the two-chain.
# [[Ignoring a threat|Play elsewhere]] (e.g. playing another move may give another way of meeting the threat thus rendering it harmless)
# [[Counterthreat|Respond]] with a forcing move of your own.

== Breaking edge templates via forcing moves ==

Forcing moves are also the only way to successfully defeat an [[edge template]]. This is done by making a [[template intrusion]] that is also a more threatening forcing move. After the opponent responds to the [[greater threat]], you can play another move within the template and destroy the connection to the edge. For example, consider the following position with the [[vertical player]] to move.

<hex>R9 C9 Q1 Hc3 Hd3 He3 Hf2 Hg1 Hg3 Vd7 Ve6 Ve5 Ve4 Vf3 Vh5 Si1 Si2 Si3 Si4 Sh2 Sh3 Sh4 Sg4</hex>

The piece on g3 is connected to the [[right edge]] via [[Edge template IIIa|template IIIa]] indicated by the '*'s. Red's best move is to play at h2. This intrudes on the edge template, is connected to the top via [[edge template II]], and threatens to complete an unbeatable chain by playing at g2 next turn. Blue can stop this threat only by playing at g2. Then Red Plays i3 breaking Blue's connection to the [[Right edge|right]].

<hex>R9 C9 Q1 Hc3 Hd3 He3 Hf2 Hg1 Hg3 Vd7 Ve6 Ve5 Ve4 Vf3 Vh5 Vh2 Hg2 Vi3</hex>

== Using forcing moves to steal territory ==

I'll define [[territory]] to be the number of [[empty hex]]es adjacent to your pieces. By playing a forcing move in one of the empty hexes in a two-chain, a player can [[Stealing territory|steal territory]] at no cost.

<hex>R5 C5 Hc2 Hb4 Vd2 Vd3 Sb3</hex>

In this position, Blue has two more hexes of territory than Red (9 vs. 7 adjacent hexes). Suppose Red makes the forcing move at the indicated hex and Blue [[Saving a connection|saves the link]].

<hex>R5 C5 Hc2 Hb4 Vd2 Vd3 Vb3 Hc3</hex>

Now Red has two more hexes of territory; i.e. Red has stolen 4 hexes of territory without disturbing either player's connections. Significantly, the additional territory is on the other side of Blue's connection where it may potentially be used for a future threat. The additional territory can't hurt and sometimes it makes a crucial difference.

A forcing move is [[Irrelevant move|harmless]] if it gains no territory for the opponent. You should not be worried at all about leaving harmless forcing moves available for your opponent. When you have more than one way of completing a connection with a two-bridge, e.g. when completing the [[loose connection]] described previously, you should consider which forcing move is least valuable for your opponent. Consider the following position with Red to play.

<hex>R5 C6 Q1 Vd2 He3 Hb4 Vd4 Hb5</hex>

Red wants to connect the d4 piece to the d2 piece. There are three distinct moves that accomplish this, d3, c4 (two-chaining to d2), and c3 (two-chaining to d4).

There is not much to be said about d3; it [[Direct connection|directly connects]] without altering anything else. c4 connects but gives a potentially useful forcing move to Blue. Blue can respond with c3 and suppose Red saves the connection with d3. Now Blue has gained a free hex of territory, the hex c2 is now directly adjacent to the c3/b4/b5 group when it wasn't previously. Hence, c4 is worse than d3.

Now consider the last remaining possibility, c3. This leaves two forcing moves for Blue but both of them are completely harmless! If after c3, Blue plays one of the forcing moves c4 or d3, then Red can save the link and Blue will not have gained any territory at all — any empty hexes adjacent to the forcing piece were already adjacent to Blue's existing pieces. Hence, c3 is just as safe as d3 but significantly, c3 ''gains'' one hex! — b3 is now adjacent to Red's d2/b3 group when it wasn't before. Thus, c3 is better than d3 and is the best of three choices.

Main Page

2005-02-16T10:10:19Z

Reiner Martin: Main Page moved to Main page

#REDIRECT [[Main page]]

Advanced (strategy guide)

2005-02-16T10:09:41Z

Reiner Martin: removed capitalisation for consistency

== Advanced edge templates ==
=== Template IVc ===

<hex>R4 C4 Q1 Vc1 Vd1 Pc2 Sb3 Sc3 Pb4</hex>

This is a two-piece template and is useful for squeezing edge connections and ladder escapes into relatively small regions. Also, many players are unaware of it. If the opponent intrudes on the template with any move other than those marked by '+', Red two-chains to template II by playing at one of the hexes marked '*'.

==== Solution to intrusion at b4 ====

<hex>R4 C4 Q1 Vc1 Vd1 Hb4 Vd3 Hd2</hex>

<hex>R4 C4 Q1 Vc1 Vd1 Hb4 Vd3 Hd2 Vb3 Sc3 Sa4</hex>

If Blue (the horizontal player) intrudes at b4, then Red responds with d3 — d3 is connected to the edge via template II and threatens a direct connection via d2. So d2 by Blue is forced. Red then two-chains to b3 threatening to connect either via a4 or c3.

==== Solution to intrusion at c2 ====

<hex>R4 C4 Q1 Vc1 Vd1 Hc2 Vb3 Hb2</hex>

<hex>R4 C4 Q1 Vc1 Vd1 Hc2 Vb3 Hb2 Vd2 Hc4 Vc3</hex>

If Blue intrudes at c2, then Red responds with b3; b3 is connected to the edge via template II and threatens a direct connectione via b2. So b2 by Blue is forced. Then Red plays at d2. Red threatens to extend d2 to template II at c3 and d3, and threatens to two-chain from d2 to the edge at c4. The only hex that is in the overlap of all these threats is c4 thus, Blue is forced to play at c4. Then Red plays at c3 completing the connection.

=== Template Va ===

<hex>R6 C10 Q1 Vg2 Pf4 Pd6 Pf6 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

The hexes marked '*' are not relevant to the template.

If Blue intrudes in the template at any hex besides the three marked '+', Red makes a move that reduces the situation to a closer template.

==== Solution to the intrusion at f4 ====

The key move is the response d5. Yielding

<hex>R6 C10 Q1 Vg2 Pe3 Hf4 Vd5 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

Red is threatening to play at the marked hex which would complete the connection of the g2 piece to the bottom. Blue must block by a playing at some hex between Red's two pieces. Red then plays h3 forcing Blue to block at h4 yielding

<hex>R6 C10 Q1 Vg2 He3 Hf4 Vd5 Vh3 Hh4 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

Red has forced the most common pre-ladder formation. Red can get a second row ladder by squeezing through at g4 (Blue blocks at f6). Red's initial key d5 piece acts as a ladder escape which completes the connection. The final position is

<hex>R6 C10 Q1 Vg2 He3 Hf4 Vd5 Vh3 Hh4 Vg4 Hf6 Vf5 He6 Ve5 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

==== Solution to the intrusion at d6 ====

Red's best response is to two-chain to h3. To stop the threatened immediate connection, Blue must block at h4 or play the forcing move g3 in the two-chain. The first play is defeated by the forcing sequence f5, g3, f3, f4, e4 yielding

<hex>R6 C10 Q1 Vg2 Hd6 Vh3 Hh4 Vf5 Hg3 Vf3 Hf4 Ve4 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

Red is now threatening to connect to the bottom in two non-overlapping ways, by playing e5 or by two-chaining to c5. Blue cannot stop both threats with a single move. The other play, g3 (after Red's h3) is defeated by the forcing sequence h2, h4, f5, g4, f3, f4, e4 yielding

<hex>R6 C10 Q1 Vg2 Hd6 Vh3 Hg3 Vh2 Hh4 Vf5 Hg4 Vf3 Hf4 Ve4 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

Red is again threatening to connect to the bottom in the same two non-overlapping ways: by playing at e5 or two-chaining to c5. Blue cannot stop both threats with a single move.

==== Solution to the intrusion at f6 ====

Red's best response is play e5 which is connected to the bottom and forms a loose connection with the g2 piece. To stop the immediate connection, Blue must play in the middle of the loose connection at one of the hexes marked "+" in the following diagram.

<hex>R6 C10 Q1 Vg2 Pf3 Pf4 Ve5 Hf6 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

The move f3 is defeated by the forcing sequence g3, f4, g4 yielding

<hex>R6 C10 Q1 Vg2 Hf3 Ve5 Hf6 Vg3 Hf4 Vg4 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

Red is threatening to connect to the bottom in two non-overlapping ways: by playing at f5 or two-chaining to h5. The alternative response, f4, is defeated by the forcing sequence e3, d6, e6, e4, d4 yielding

<hex>R6 C10 Q1 Vg2 Hf4 Ve5 Hf6 Ve3 Hd6 Ve6 He4 Vd4 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sj3 Sa4 Sb4 Sa5</hex>

Red is threatening to connect to the bottom in two non-overlapping ways, by playing at d5 or c5.

Note that template Va occurs in a mirror-image form (in the mirror image form, the three hexes on the 5th row (from the bottom) are shifted over 1 hex to the G, H, and I columns). It may seem that this template is very strong because it reaches 5 rows into the board but it rarely occurs because of the huge size of the template; the template requires 31 empty hexes and 10 hexes along an edge — the entire edge on the 10x10 board!

Furthermore, the large perimeter makes it more vulnerable to encroaching adjacent plays and forcing moves. Additionally, template area surrounds the 5th row piece on both "shoulders" so that non-overlapping plays from the 5th row piece can occur in only two directions.

=== Template Vb ===

<hex>R6 C10 Q1 Vf2 Vg2 Pf3 Pe5 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Sh2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sh3 Si3 Sj3 Sa4 Sb4 Sc4 Si4 Sj4 Sa5 Sb5 Si5 Sj5 Sa6 Si6 Sj6</hex>

The hexes marked '*' are not relevant to the template.

If the horizontal player Blue intrudes in the template at any hex besides the three marked '+', Red makes a move that reduces the situation to a closer template.

==== Solution to the intrusion at f3 ====

There are several solutions but the simplest is to respond with g3. Blue's only play to stop the immediate connection is f5. Then Red plays e4 yielding

<hex>R6 C10 Q1 Vf2 Vg2 Hf3 Vg3 Hf5 Ve4 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Sh2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sh3 Si3 Sj3 Sa4 Sb4 Sc4 Si4 Sj4 Sa5 Sb5 Si5 Sj5 Sa6 Si6 Sj6</hex>

The e4 piece is connected to the bottom via a 3rd row template and e4 is connected to the other group of red pieces through e3 and f4. Thus, the connection is complete.

==== Solution to the intrusion at e5 ====

Red's best response is g4. This piece is connected to the bottom via a 3rd row template and hence Blue must block at g3. Red then plays e4 yielding

<hex>R6 C10 Q1 Vf2 Vg2 He5 Vg4 Hg3 Ve4 Sa1 Sb1 Sc1 Sd1 Se1 Sf1 Sg1 Sh1 Si1 Sj1 Sa2 Sb2 Sc2 Sd2 Se2 Sh2 Si2 Sj2 Sa3 Sb3 Sc3 Sd3 Sh3 Si3 Sj3 Sa4 Sb4 Sc4 Si4 Sj4 Sa5 Sb5 Si5 Sj5 Sa6 Si6 Sj6</hex>

The e4 piece threatens to connect to the bottom in two non-overlapping ways, to d5 and to g4 (through f4). Hence the connection is unstoppable.

Unlike template Va, this template is not a rare occurrence and many hex players are not familiar with it.

== Advanced templates as ladder escapes ==

Templates IVc and Vb are valid escapes for row 2, row 3, and row 4 ladders. Template Va is not a valid ladder escape.

Exception: Template Vb is not valid for 3rd and 4th row ladders coming from the right side in the above diagram if the Horizontal player has a piece at h3. For the horizontal player to defeat the 3rd row ladder in this case, connecting to h3 must provide a strong threat that the vertical player needs to respond to.

Note: The unique way to win with template Vb and a 2nd row ladder is as follows. As soon as your head ladder piece intrudes on the template, your very next move must be to two-chain up to the 3rd row (this is true no matter which side of the template you are entering from). Then you break off the ladder (this piece will be connected to the edge via a smaller edge template).

== The minimax principle ==
(See also the page [[Minimax]])

Suppose you have multiple ways of establishing/maintaining a connection to an edge. A move that maintains as strong a connection as possible is not preferable to other connection moves because you only need to get some connection; you don't win extra points by connecting more strongly.

In fact it is generally better to play a move that maintains as ''weak'' a connection as possible; the reason being that such a piece may help you extend the connection towards the opposite edge. This principle is sometimes called "mini-maxing."

The idea behind the term is that you are playing a move that maintains a minimal connectivity in one direction while building up (i.e. maximizing) your strength in the other direction. I'll illustrate this with a couple of positions from my games. (Note that this principle applies equally well when establishing/maintaining a connection to ''a group of pieces''.)

<hex>R10 C10 Q1 Va3 Vc6 Hf5</hex>

My opponent, Blue played the minimax move f4. This move maintains a minimal strength connection to the left while building up strength to the right; in fact the f4-f5 group is almost connected to the right edge via template Vb. I responded with my own minimax move d5 (d6 is the other minimax option) yielding the following position.

<hex>R10 C10 Q1 Va3 Vc6 Hf5 Hf4 Vd5</hex>

d5 maintains a minimal strength connection to the bottom while maximizing my strength to the top. (d6 would have maintained a minimal strength connection to the top while maximizing my strength to the bottom.) A move that is even stronger towards the top, such as d4, would be a mistake. My opponent could then block at the bottom with c7, which is connected to the left edge via a 3rd row template and which threatens to link up with the central group. If I try to stop the connection to the central group with e6, my opponent responds with d5 yielding the following position.

<hex>R10 C10 Q1 Va3 Vc6 Hf5 Hf4 Vd4 Hc7 Ve6 Hd5</hex>

d5 is connected to the central group via a 2-chain and the combined threats c5 and d6 guarantee a connection to the left edge (a7 is defeated by c5, b5, b6, a6, b7, a8, b9). I would be in dire straits as the central pair f4-f5 is almost connected to the right edge.

Now back to the game; after my minimax move d5, I can safely meet c7 with e6. Yielding

<hex>R10 C10 Q1 Va3 Vc6 Hf5 Hf4 Vd5 Hc7 Ve6</hex>

In fact, the c7, e6 sequence occurred in the actual game. I eventually won after a close hard fought battle.

<hex>R10 C10 Q1 Va3 Vc6 Vb8 Vc8 Vd8 Ve8 Vg6 Vg7 Vg8 Hf5 Hf7 Hf8 He9 Hd9 Hc9 Hb9 Hd7</hex>

In this position, I was the vertical player and was expecting f6 to which h5 would give me an excellent position (with best play, this position would in fact be winning though this is not obvious). Instead my opponent played the excellent minimax move f4. This move fights in both directions and is in fact a killer move. I can't block the f4-f5 pair from the right due to the forking ladder escape at h9. Thus, I must meekly submit to the forcing sequence f6, e7, e6, d5 yielding

<hex>R10 C10 Q1 Va3 Vc6 Vb8 Vc8 Vd8 Ve8 Vg6 Vg7 Vg8 Hf5 Hf7 Hf8 He9 Hd9 Hc9 Hb9 Hd7 Hf4 Vf6 He7 Ve6 Hd5</hex>

The game is over. The f4-f5 pair is connected to d5 which in turn threatens to connect to left in two non-overlapping ways, c5 (a 3rd row template) and d6, hence the pair is connected to the left. If I try to block at the right, the best I can do is yield a ladder (e.g. h4, h3, j2, i3 and H has a second row ladder) and then the forking ladder escape at h9 wins the game.

In the next example, I am the horizontal player and it is my move.

<hex>C10 R10 Q1 Va3 Vb5 Vb8 Vc7 Vd6 Vd5 Vf4 Hc6 Hc5 Hd4 He4 He5 Hf5</hex>

Most hex players would probably connect to the left side with a7 (or b6 or b7). Despite its apparent necessity, this move actually loses (against best play). Instead I played the winning minimax move d3! By adding a second non-overlapping connection threat to the left, my group of pieces maintains a connection to the left. And despite its modest appearance, d3 also helps out on the right and in fact guarantees a winning connection from f5 to the right by defeating one of the main potential blocking plays.

<hex>C10 R10 Q1 Va3 Vb5 Vb8 Vc7 Vd6 Vd5 Vf4 Hc6 Hc5 Hd4 He4 He5 Hf5 Hd3</hex>

E.g. suppose V tries to block the f5 piece from the right as follows. g5, g4, i3 (or h4), i2 and now I have a forced winning ladder down row 2 — h3, h2, g3, g2, f3, f2, e3, e2 completing the win.

<hex>C10 R10 Q1 Va3 Vb5 Vb8 Vc7 Vd6 Vd5 Vf4 Hc6 Hc5 Hd4 He4 He5 Hf5 Hd3 Vg5 Hg4 Vi3 Hi2 Vh3 Hh2 Vg3 Hg2 Vf3 Hf2 Ve3 He2</hex>

This line clearly shows the usefullness of d3. If I hadn't played d3 (playing a7 instead, for instance), the vertical player could continue d3, d2, a4! and eventually winning with best play (considerable deep analysis is needed to show this).

<hex>C10 R10 Q1 Va3 Vb5 Vb8 Vc7 Vd6 Vd5 Vf4 Hc6 Hc5 Hd4 He4 He5 Hf5 Ha7 Vg5 Hg4 Vi3 Hi2 Vh3 Hh2 Vg3 Hg2 Vf3 Hf2 Ve3 He2 Vd3 Hd2 Va4</hex>

Minimax moves are not always "parallel" moves. The principle of maintaining a minimal amount of connectivity in one direction while maximizing your strength in the opposite direction is more general than that. The final example from a game of mine illustrates non-parallel mini-max moves. I was the vertical player and opened with 1. a3 and my opponent responded with 1... e4 yielding the following position.

<hex>C10 R10 Q1 Va3 He4</hex>

I played the minimax move 2. f5 yielding

<hex>C10 R10 Q1 Va3 He4 Vf5</hex>

By connecting as far away as possible from the top, I increase my strength towards the bottom. (i.e. I am maintaining a minimal strength connection to the top while maximizing my strength towards the bottom). Before playing such a move, I have to verify that my opponent can't stop me from reaching the top. I could meet the attempted block with 2...g4 or 2...h2 by getting a third row ladder (2...g4 3.f4 g2 4.f3, etc. or 2...h2 3.g3 g2 4.f3, etc.), laddering down to e3, and then playing b4 (how to play a third row to a3 is described in a later section). I would be happy with such a line. My opponent however played the excellent e3. This move takes away the ladder, hence forcing me to reconnect to the top, while at the same time increasing his strength to the left.

<hex>C10 R10 Q1 Va3 He4 Vf5 He3</hex>

Here I played the minimax move g4. g4 has the potential to help block my opponent from going across the bottom of the board (e.g. Blue e7, Red f7, Blue f6, Red h5 and now g4 is helping out) or equivalently helps me to connect downwards on the right. I.e. g4 maintains a minimal strength connection towards the top while maximizing my strength towards the bottom. Note that a stronger move towards the top such as g3 does not have the same potential to help out towards the bottom. This potential may seem remote but in fact I would not have won the game without it! The rest of the game does not illustrate minimaxing but it is instructive nevertheless. The most important variation is as follows (there were two mistakes in the actual game which took the game out of the path it should have followed into a shorter less instructive branch).

{| cellspacing="0" style="width: 500px"
| 3. || g4 || h3 || Again forcing me to reconnect to the top and hence, getting a free hex that could potentially help him connect to the right (h3 provides an escape for ladders up row I).
|-
| 4. || g3 || f8 || As good of a block towards the bottom that there is.
|-
| 5. || e8 ||   || Necessary.
|-
| 5. || ... || e9 || Essential. Note that this move stops any ladders on row 9 coming from the right and using e8 as an escape.
|-
| 6. || g7! ||   || Excellent. Reconnecting e8 to the bottom with the d8 or d9 is defeated by e7. I can't stop e7 from connecting to the left because e4-e3 provides enough help (e.g. 6.d8 e7 7.c7 d6 8.b6 c6 9.c7 b5 10.c5 c4) nor could I stop e7 from connecting to the right — e7 is aided by f8, e9, and h3 (e.g. 6.d8 e7 7.f7 f6 8.h5 g6 9.h6 g7 10.h7 h8 11.g8 f10 12.g9 g10 13.i9 h9 14.j7 i7 and the h3 piece provides the ladder escape). 6.g7! may look strange and unconventional but it maintains a very slim advantage.
|}

<hex>C10 R10 Q1 Va3 He4 Vf5 He3 Vg4 Hh3 Vg3 Hf8 Ve8 He9 Vg7</hex>

{| cellspacing="0" style="width: 500px"
|-
| 6. || ... || f7 || The toughest move to meet. Note that 6.... g8 loses quickly to 7.d8 e7 8.f7 f6 9.h5
|-
| 7. || e7 ||   || I can't stop f7-f8-e9 from connecting to the right, so I block to the left in the only satisfactory way. I can't allow my opponent to connect from his f7 towards e3-e4.
|-
| 7. || ... || d9 || An essential block.
|-
| 8. || c8 ||   || c9 is no good. I can't allow my opponent to come up row D towards e3-e4. I need to force him as far away from e3-e4 as possible.
|-
| 8. || ... || b10 ||  
|-
| 9. || a10 || b9 || I've forced my opponent into a second row ladder which is not sufficient because my a3 piece is just barely inside the Vb template.
|-
| 10. || a9 || b8 ||  
|-
| 11. || a8 || f2! || A nice idea. e4-e3-f2 is just strong enough for the second row ladder to work and at the same time, it threatens to cut off my main group of pieces from the top. This looks like it wins but there is a way out!
|}

<hex>C10 R10 Q1 Va3 He4 Vf5 He3 Vg4 Hh3 Vg3 Hf8 Ve8 He9 Vg7 Hf7 Ve7 Hd9 Vc8 Hb10 Va10 Hb9 Va9 Hb8 Va8 Hf2</hex>

{| cellspacing="0" style="width: 500px"
|-
| 12. || i2! ||   || A subtle improvement over simply reconnecting with g2 or h2 (both of which are losing). This maintains the connection and i2 interferes with the usefulness of h3. This may seem insignificant but it makes the difference between winning and losing! (In hex, the difference between winning and losing is often very slight)
|-
| 13. || ... || c6 || Continuing with his plan to connect to the left which now works due to f2.
|-
| 14. || h8! ||   || The unique move that stops my opponent from connecting on the right. Note that 14.g8 is not satisfactory. 14.g8 g6 15.i5 h9. h9 is a forking ladder escape that guarantees a connection to the right.
|-
| 14. || ... || g6 ||  
|-
| 15. || i5 || h7 || Note that h7 is threatens to connect to the horizontal player's main group in two distinct ways, through g8 and through h6, and thus is connected to his main group.
|-
| 16. || g8 ||   || Not necessary but it doesn't hurt as the reply is forced.
|-
| 16. || ... || h6 ||  
|-
| 17. || j6 || i9 ||  
|-
| 18. || i8 || g10 ||  
|-
| 19. || f10 || h4 ||  
|-
| 20. || j3 ||   || I now have an unbreakable chain on the right side.
|}

<hex>C10 R10 Q1 Va3 He4 Vf5 He3 Vg4 Hh3 Vg3 Hf8 Ve8 He9 Vg7 Hf7 Ve7 Hd9 Vc8 Hb10 Va10 Hb9 Va9 Hb8 Va8 Hf2 Vi2 Hc6 Vh8 Hg6 Vi5 Hh7 Vg8 Hh6 Vj6 Hi9 Vi8 Hg10 Vf10 Hh4 Vj3</hex>

== Special situations, tricks, etc. ==

=== Reconnecting edge template IIIa after an intrusion ===

<hex>R4 C10 Q1 Ve2 Hd3 Pf2 Se3</hex>

In this diagram, suppose you are Red and Blue has just played d3 intruding upon the third row template connecting your e2 to the bottom. Most hex players would reconnect with e3 without giving it much if any thought, yet there three distinct ways to reconnect and there is often a reason for preferring one over the other.

A second way for Red to reconnect is to play f2 — the hex f2 and the empty hexes g2,e3,f3,g3,d4,e4,f4, and g4 form edge template IIIa; hence f2 has an unbreakable connection to the bottom and f2 is connected to e2.

The potential advantage of reconnecting with f2 over e3 is that it is easier to connect other pieces to the the group e2-f2 than to the group e2-e3 (e.g. h1 is a two-chain away from f2 but is not a two-chain away from either e2 nor e3). The extra connection possibilities can make a critical difference. For example, consider the following position with Red to move.

<hex>R9 C9 Q1 Vg2 Vf3 Ve4 Vd5 Vd6 Vh3 Vh4 Vf7 Hi4 Hc6 Hb8 Hc8 He6 Hf6 Hg6</hex>

Red can win by laddering 1. d7 d8 2. e7. Suppose instead Red plays 1.h5 intruding on the g6 edge template. If Blue reconnects with h6, then Red would have nothing else to do except play the winning line. So Blue reconnects with g7 making the win tougher. (Red could still win by d7, d8, e7, e9, f8, f9, h8! — a forking ladder escape which decides the issue).

Now suppose that Red again intrudes on the edge template with 2. h6. Now the game continues 2...g8 (again reconnecting by playing parallel to the edge) 3. h7 (persistent) h8, 4. d7 d8, 5. e7 e9! and now Blue has an unbreakable winning chain at the bottom. By reconnecting with the parallel moves instead of the direct reconnection, Blue's group had a new way to connect to the left and this extra possibility turned a defeat into a win.

So is it always better to reconnect with the parallel move? No!! Sometimes the parallel reconnection can lose the game while the simple direct connection wins! The potential weakness of the parallel reconnection is that your opponent might then be able to use a double threat to defeat the edge connection. For example, consider the following position with Red to move.

<hex>R9 C9 Q1 Vh3 Vg2 Vf3 Ve4 Vd5 Vd6 Hc6 Hb8 Hc8 He6 Hg5 Hi3</hex>

With best play Blue wins, so Red tries 1. h4. If Blue responds with the direct reconnection h5, then the win is assured and Red may as well resign. Suppose instead that Blue reconnects with 1... g6. Then Red can respond with 2.h7! — this forking ladder escape is a killer. Red now has two disjoint winning threats, laddering from d7 to h7 and play i5 (This double two-chain cutoff threat occurs in situations besides cutting off third row edge templates. It is well worth being familiar with this idea.). Blue cannot stop them both so Red wins.

But this doesn't exhaust the reconnection possibilities. There is a third way to reconnect; a way that most players don't seem to discover.

<hex>R4 C10 Q1 Ve2 Hd3</hex>

Again starting at the initial position in this section, Red's e2 piece is connected to the bottom via edge template IIIa and Blue intrudes upon it with d3. In addition to e3 and f2, Red can reconnect with the surprising f1!

<hex>R4 C10 Q1 Ve2 Hd3 Vf1</hex>

Red is threatening to connect e2-f1 to the bottom with e3. If Blue tries to block this with e3, then Red can reconnect by playing g2. g2 is connected to the bottom via template IIIa (Blue's e9 piece is just outside of this template) and h3 connects to f1 via a two-chain.

But what if Blue blocks with e4 instead of e3? (note the e4 is within the g2 piece's edge template). Then Red can still reconnect by playing as follows. 1. e3 d4 (forced) 2. g3 f3 (forced again) 3.g2 ending up with the following position.

<hex>R4 C10 Q1 Ve2 Hd3 Vf1 He4 Ve3 Hd4 Vg3 Hf3 Vg2</hex>

How does this method compare to the previous two? Compared to the parallel reconnection, it is quite a bit more susceptible to forking plays and plays that encroach upon the increased area that is needed to reconnect, but by playing away from the edge, you have even more potential to connect the edge group towards the opposite edge. Sometimes the extra connection possibilities generated by playing away from the edge is exactly what is needed.

For example consider the beautiful solution to the following position (I wish I could take the credit for its discovery but the original over the board play was found by Tom239 on _Playsite_ (he was at the orange level at the time!). The position below is a slight modification of one constructed by Kevin O'Gorman, the maintainer of the Ohex data base).

<hex>R10 C10 Q1 Va2 Va3 Va4 Va5 Va6 Va7 Va8 Va9 Ha10 Hb2 Hb3 Hb4 Hb5 Hb6 Hb7 Hb8 Vc6 Hc7 Vd5 Hd6 Vd7 Ve4 He5 Vf3 Hf4 Vg2 Hg3 Vh1 Hh2 Hi2</hex>

It is Red's move. To win, Red must connect his a9 piece to bottom. To do this, Red must make some ladder escape that additionally must somehow use the d7 piece to threaten another way to connect to the ladder. This looks impossible but yet there is a way. Red can win by starting with 1.b9 b10 2.c9 c10 3.f8!!

<hex>R10 C10 Q1 Va2 Va3 Va4 Va5 Va6 Va7 Va8 Va9 Ha10 Hb2 Hb3 Hb4 Hb5 Hb6 Hb7 Hb8 Vc6 Hc7 Vd5 Hd6 Vd7 Ve4 He5 Vf3 Hf4 Vg2 Hg3 Vh1 Hh2 Hi2 Vb9 Hb10 Vc9 Hc10 Vf8</hex>

This brilliant move is the only way to win. 3.g7 is defeated only by 3...d9 and 3.d9 d10 4.g7 is defeated only by 4...f8 (it takes a ''lot'' of analysis to demonstrate these claims). Blue's only good attempt is to intrude on the edge template with 3... e9. But Red can defeat this by reconnecting with 4.g7! (this is what Red had in mind when playing 3.f8!!)

<hex>R10 C10 Q1 Va2 Va3 Va4 Va5 Va6 Va7 Va8 Va9 Ha10 Hb2 Hb3 Hb4 Hb5 Hb6 Hb7 Hb8 Vc6 Hc7 Vd5 Hd6 Vd7 Ve4 He5 Vf3 Hf4 Vg2 Hg3 Vh1 Hh2 Hi2 Vb9 Hb10 Vc9 Hc10 Vf8 He9 Vg7</hex>

Now f8-g7 has an unbreakable connection to the bottom and Red threatens two distinct ways of connecting this group back to the group containing c9; Red threatens f6, double two-chaining between d7 and g7, and Red threatens e8 two-chaining to c9. Blue's only possible defense is the forcing move 4...d8. This interferes with the immediate connection threat between c9 and f8, and it prepares to meet the f6 threat with c8 cutting off d7 from c9. But this move is still not sufficient because after 4...d8, Red can win with 5.d9 d10 (forced) 6.e8.

In practice, you can think of the parallel reconnection as your "standard" response (more often than not, it is the correct choice). But if the potential threat to cut off the parallel play from the edge is serious, then go with the direct reconnection. The "away" reconnection entails a substantially increased risk of being cut off from the edge but if you can see that it will be safe or if you need the stronger connection possibilities towards the opposite edge, then go with the "away" connection.

=== Third row ladder to a3 and its symmetric analogues ===
(See also the page [[a3 escape trick]])

The following position is from one of my games.

<hex>R10 C10 Q1 Hg2 Vi2 He3 Hg3 Hb4 Hc4 He4 Vf4 Vg4 Vb5 Vc5 Ve6 Vf6 Hh10</hex>

I am Blue and it is my move. Red's e6-f6(-f4-g4) group is connected to bottom via template Vb. Red's i2 piece is connected to the top via edge template II. In order to stop these two groups from connecting to each and completing a win, I must start laddering down column H. So I ladder down to h6 forcing Red to follow down column I to i6 yielding the following position.

<hex>R10 C10 Q1 Hg2 Vi2 He3 Hg3 Hb4 Hc4 He4 Vf4 Vg4 Vb5 Vc5 Ve6 Vf6 Hh10 Hh3 Vi3 Hh4 Vi4 Hh5 Vi5 Hh6 Vi6</hex>

My h10 piece is ''not'' a valid ladder escape. If I ladder all the way down to h10, then Red follows down to i8 and his response to h9 is not i9 but j9!

<hex>R10 C10 Q1 Hg2 Vi2 He3 Hg3 Hb4 Hc4 He4 Vf4 Vg4 Vb5 Vc5 Ve6 Vf6 Hh10 Hh3 Vi3 Hh4 Vi4 Hh5 Vi5 Hh6 Vi6 Hh7 Vi7 Hh8 Vi8 Hh9 Vj9</hex>

Red has a winning chain on the right side. You might think I could win by instead laddering down one more hex, and then double two-chain to the h10 piece yielding the following position.

<hex>R10 C10 Q1 Hg2 Vi2 He3 Hg3 Hb4 Hc4 He4 Vf4 Vg4 Vb5 Vc5 Ve6 Vf6 Hh10 Hh3 Vi3 Hh4 Vi4 Hh5 Vi5 Hh6 Vi6 Hh7 Vi7 Hg9</hex>

This may appear to settle the matter in my favor but in actuality, Red has a winning position! Red can win by 1. h8 (h9 also works but h8 is slightly more deceptive). If I respond by saving the link, i.e. by 1...g8, then Red wins by playing 2.h9 g10 (forced) 3. j9.

<hex>R10 C10 Q1 Hg2 Vi2 He3 Hg3 Hb4 Hc4 He4 Vf4 Vg4 Vb5 Vc5 Ve6 Vf6 Hh10 Hh3 Vi3 Hh4 Vi4 Hh5 Vi5 Hh6 Vi6 Hh7 Vi7 Hg9 Vh8 Hg8 Vh9 Hg10 Vj9</hex>

Red has an unbreakable winning chain down the right. Instead it is better for me to respond to Red's 1.h8 with 1...h9. My g9-h9-h10 group is now solidly connected to the right but Red can continue 2.g8 and I cannot stop g8 from connecting to the bottom because of the help provided by Red's e6-f6 pieces (work it out!)

In the initial position I cannot afford to ladder down any farther than g6. If I ladder down one more hex, I lose against best play no matter what. If there are no other pieces in the area, as is the case here, then the strongest way to play it is to ladder down one hex short of the hex that could double two-chain to the "almost-escape" piece, and then two chain up from the almost-escape piece which in our present case yields the following position.

<hex>R10 C10 Q1 Hg2 Vi2 He3 Hg3 Hb4 Hc4 He4 Vf4 Vg4 Vb5 Vc5 Ve6 Vf6 Hh10 Hh3 Vi3 Hh4 Vi4 Hh5 Vi5 Hh6 Vi6 Hg9</hex>

Red has three tries to stop the connection between the h6 and g9 pieces.

* g8 is defeated by continuing the ladder down (try it!).
* h7 and h8 are best met by f8 (double two-chaining in the same direction).
* Meeting the play h8 with g8 (connecting up to h6) doesn't work for the same reason that laddering down to h7 and double two-chaining to h10 doesn't work (work it out and you should see what I mean).

Also, note that Red's attempt h9 is of no consequence. Against h9 you should save the link with g10 and then again meet either h7 or h8 with f8.

In the actual game my opponent played h7 and I responded with f8. f8 threatens to connect with with h6 through g7. So my opponent played g7 to which I responded with f7. Again this threatens a winning connection from f7 to h6 through g6. So my opponent played g6 and I responded with c9 with a winning position.

<hex>R10 C10 Q1 Hg2 Vi2 He3 Hg3 Hb4 Hc4 He4 Vf4 Vg4 Vb5 Vc5 Ve6 Vf6 Hh10 Hh3 Vi3 Hh4 Vi4 Hh5 Vi5 Hh6 Vi6 Hg9 Vh7 Hf8 Vg7 Hf7 Vg6 Hc9</hex>

Further play no longer concerns the topic under discussion but the remaining moves were d9, e7, d7, d8, b9, c8, a8, b8, a9, b7, a7, d6, resigns. My opponent doesn't need to see g8, f9, h9, g10, j9, i8

The key play of two-chaining up from the escape piece is also useful in another common type of third row ladder position. For example, consider the following position with the vertical player to move.

<hex>R10 C10 Q1 He2 Vi2 Va3 Hf3 Hc4 Vd4 He4 Hf4 Vh4 Vi4 Vc5 Hd5 Hf5 Hh5 Vi5 Vc6 Hf6 Hg6 Hi6 Vc7 Hd7 He7 Vf7 Hg7 Vh7 Vc8 Vd8 Ve8 Hg8 Hh8 Hb9 Vc9</hex>

Red has a chain running from the bottom at c9 up to d4. The only way Red can win is to connect this group to the top. Red can ladder d3, c3, b3 but as we saw earlier, the a3 piece is not a valid ladder escape. But Red can still win by two-chaining from a3 to b4.

<hex>R10 C10 Q1 He2 Vi2 Va3 Hf3 Hc4 Vd4 He4 Hf4 Vh4 Vi4 Vc5 Hd5 Hf5 Hh5 Vi5 Vc6 Hf6 Hg6 Hi6 Vc7 Hd7 He7 Vf7 Hg7 Vh7 Vc8 Vd8 Ve8 Hg8 Hh8 Hb9 Vc9 Vb4</hex>

This threatens a winning connection to c5 through b5. If Blue blocks this with b5, then Red plays the ladder because now the pair a3-b4 are a valid ladder escape. If instead Blue blocks off the ladder with say c3, then Red wins with the line b5, b3, a4 (forced), b1, d2!

<hex>R10 C10 Q1 He2 Vi2 Va3 Hf3 Hc4 Vd4 He4 Hf4 Vh4 Vi4 Vc5 Hd5 Hf5 Hh5 Vi5 Vc6 Hf6 Hg6 Hi6 Vc7 Hd7 He7 Vf7 Hg7 Vh7 Vc8 Vd8 Ve8 Hg8 Hh8 Hb9 Vc9 Vb4 Hc3 Vb5 Hb3 Va4 Hb1 Vd2</hex>

d2 is a forking ladder escape; it threatens d3 and it provides an escape to the 2nd row ladder starting with b2. Blue cannot stop both winning threats with a single move, thus Red wins.

a3/j8 is a common opening move. If you frequently play it or play against somebody who does, then you will run into these 3rd row ladder situations and hence, it will be beneficial to learn how to play them.

=== The parallel ladder trick ===
(See also the page [[Parallel ladder]])

Consider the following position with Red to play.

<hex>R10 C10 Q1 Hc1 Vd2 Vd3 He3 Vf3 Vd4 Ve4 Hf4 Hg4 Ve5 Vc6 Vd6 He6 Hi6 Hc7 Vd7 Ha8 Hb8 Vc8 Hd8 Hb10</hex>

All of Red's pieces form a connected group. This group is connected to the top. At the bottom, Red has a second row ladder with no possible escape on the left. The potential escapes on the right are inadequate. For example, suppose Red ladders to f9. Then tries to escape with 5.h9 g9 6.h8 g8 7.h7 f7.

<hex>R10 C10 Q1 Hc1 Vd2 Vd3 He3 Vf3 Vd4 Ve4 Hf4 Hg4 Ve5 Vc6 Vd6 He6 Hi6 Hc7 Vd7 Ha8 Hb8 Vc8 Hd8 Hb10 Vc9 Hc10 Vd9 Hd10 Ve9 He10 Vf9 Hf10 Vh9 Hg9 Vh8 Hg8 Vh7</hex>

Now Red's only reasonable try is 8.g7 f8. Now 9.g6 loses to 9...f5 and 9.h5 loses to the forcing sequence 9...g6 10.h6 h4 11.g5 f5. All the other escape attempts likewise fail. Is Red done for?

No! Red can create a sufficient escape by making use of a parallel ladder. In the original position Red plays 1.e7. How can Blue stop Red from connecting to the bottom? d9 lets Red two-chain from e7 to f8 connecting to the bottom; e9 and e10 allow d9 which is connected to the bottom and threatens to connect to Red's big group through c9 and e8; d10 loses to e8, f9 (forced), c10; hence, Blue is forced to play the parallel ladder move 1...e8. It is simplest for Red to repeat this and ladder to f7 forcing the 2...f8 response.

<hex>R10 C10 Q1 Hc1 Vd2 Vd3 He3 Vf3 Vd4 Ve4 Hf4 Hg4 Ve5 Vc6 Vd6 He6 Hi6 Hc7 Vd7 Ha8 Hb8 Vc8 Hd8 Hb10 Ve7 He8 Vf7 Hf8</hex>

Now Red now goes back to the second row ladder and tries to escape. What have we gained by preceding this with the parallel ladder moves? When trying to escape, the threat to connect to d7-e7-f7 is stronger than the previous weak threat to connect to d7. This extra threat will let us push our escape chain farther up the board and in this case, just far enough to win the game.

Red's winning sequence is long but rather simple because every one of Blue's replies is forced. As before, Red ladders to f9 and escapes with 7. h9. Play continues 7...g9 8.h8 g8 9.h7 g7 10.h6 g6 11.h5. Red is threatening to play g5 with the double winning threats f5 and f6. But if Blue blocks this, say with 11...g5, then Red continues 12.i3 i2 13.h3 and 14.g3 completes the win.

I have managed to pull this trick off from one row farther back; i.e. with ladders on row 3 and 5 but this occurs far less frequently and you have to examine some additional defensive possibilities. Consider the following position.

<hex>R10 C10 Q1 Vd5 He5 Vd6 Ve6 Hb7 Vc7 Hd7 Hb9</hex>

Red has just played e6 trying the parallel ladder trick. With the closer ladder on row 2, we saw that Blue was forced to respond with the parallel ladder play e7. But here Blue has two additional possibilities e8 and c9 (the only other possibility where Red doesn't have a way to force his group to connect to the bottom is c10. But Red can respond with f8 and now Blue has nothing better than e7, g6).

e8 yields a second row ladder after d8, e7, c8, c10, d9. The play c9 also leads to a second row ladder after the likely f7, f8, e8 (d9 is met by e7) d10. In the latter case, Red could again try the parallel ladder trick by playing g7. Of course, the existence of other pieces in the area can change the possibilities.

Intermediate (strategy guide)

2005-02-16T10:07:45Z

Reiner Martin: removed capitalisation for consistency

''Adapted with permission from Glenn C. Rhoads strategy guide.''

== Loose connections ==
''(See also the article [[Loose connection]])''

[[Adjacent move]]s provide a guaranteed connection but cover little ground. [[Bridge|Two-bridges]] cover twice the distance and are almost as strong. The next best connection when even more distance is required is called the '''loose connection''' — a [[Hex (board element)|hex]] that is a two-bridge plus an adjacent step away.

<hex>R4 C5 Vb2 Sc2 Sc3 Vd3</hex>

The [[piece]]s of the loose connection [[threat]]en to connect via a two-bridge plus an adjacent step [[Multiple threats|in two different ways]] — by playing at either of the marked hexes. Also, the two marked hexes are the only ones that are in the [[overlap]] of the two [[Template|connection patterns]]. Thus, to break a loose connection, one must play in one of the marked hexes.

Another connection pattern that is useful to be aware of is two pieces in the same row or column that are three hexes apart.

<hex>R5 C6 Sd2 Vb3 Pc3 Pd3 Ve3 Sc4</hex>

The pieces threaten to connect via 2 two-bridge steps in two different ways, namely by playing at piece at one of the hexes marked with a *. There are two hexes that are in the overlap between these two connection threats and a move played in either of them breaks the immediate connection (these two hexes are marked with a +). This connection pattern is not as strong as the loose connection.

== The useless triangle ==
''(See also the article [[Useless triangle]])''

When a piece's neighboring hexes are [[occupied hex|filled]] by the [[opponent]] such that that piece has only two empty neighboring hexes that are also [[adjacent]] to each other, then the piece is said to lie in a "'''useless triangle'''."

<hex>R7 C7 Q1 Vc5 Hd4 Hc4 Hb5 Hd5 Vd7 He7 Vf7</hex>

In the above diagram, the red piece at c5 and the [[empty hex]]es b6 and c6 form a useless triangle. The blue piece at e7 and the empty hexes e6 and f6 also form a useless triangle. The important point is that unless the piece in a useless triangle is in that player's [[First row|border row]], the piece has effectively been removed from the game — that is, it cannot have any effect on the rest of the game regardless of the rest of the position.

== Minimal edge templates ==
(See also the page [[Edge templates]])

An '''edge template''' is a pattern of empty hexes that will allow a piece to be [[Connection|connected]] to the [[edge]] even if the opponent has the next move. Just as the two-bridge is a useful connection pattern to know, so are minimal edge templates — the ones of the smallest size. (The templates are numbered according to row of the [[connecting piece]]).

=== [[Template I]] ===

Trivially, a piece on an edge row is connected to the edge.

<hex>R3 C3 Vb3</hex>

=== [[Template II]] ===

<hex>R3 C3 Vb2 Sa3 Sb3</hex>

If the opponent plays inside the template, [[Red (player)|Red]] plays the other move in the template restoring the connection to the edge.

For templates farther away, the general strategy to apply when the opponent intrudes on the template is to make a move in the template that reduces to a smaller and closer template.

=== [[Template IIIa]] ===
''(Also called [[Ziggurat]])''

<hex>R4 C4 Q1 Vc2 Sb3 Sd3</hex>

If the opponent intrudes on the template, then Red plays at one of the two marked points achieving [[template II]]. Since the b3 template and the two-chain/d3 template combination don't overlap, the opponent cannot stop both. (This template exists in a mirror image form with the red piece at d2).

=== [[Template IIIb]] ===

<hex>R4 C5 Vd2 Sb3 Se3 Pc4</hex>

The [[Hex (board element)|hex]] marked with an '+' can be occupied by the opponent! If the opponent intrudes on the template, then Red two-chains to either of the marked hexes and in either case forms [[template II]]. Since the two-chain/template II combinations don't [[overlap]] with each other, the opponent cannot stop both.

=== [[Template IVa]] ===

<hex>R5 C7 Q1 Ve2 Sd3 Se3 Sf3 PD5</hex>

If the opponent intrudes on the template, Red plays to one of hexes marked with '*' forming [[template IIIa]]. The one exception is if the opponent plays at d5, then Red plays to e3 and connects via [[template IIIb]]. (This template has a mirror image form with the red piece at f2.)

=== [[Template IVb]] ===

<hex>R5 C8 Vf2 Sd3 Sg3 Pe4</hex>

The hex marked with a '+' can be occupied by the opponent! If the opponent intrudes on the template, Red two-chains to one of the marked hexes forming [[template IIIa]].

== Forming ladders ==
''(See also the article [[Ladder]])''

A '''ladder''' occurs when one player tries to force a connection to an edge but is kept a constant distance away by the opponent, resulting in a sequence of moves parallel to the edge. The following is an example with Red to play.

<hex>R9 C9 Q1 Ve3 Vd4 Vd5 Vc6 Vc7 Hb8 Hb9 Hd7 He6 Hf6</hex>

Suppose Red plays c8 which forces c9 in reply. Now Red can play the following forcing sequence. d8, d9, e8, e9, f8, f9, etc. A sequence of such plays parallel to an edge is called a ladder. Note that the [[Blue (player)|Blue]]'s responses are forced. If Red blindly continues the ladder all the way to end, then he will simply lose (Blue will get pieces in row 9 from b9 through i9). There is no good reason to ever force a ladder all the way through to end, it only helps your opponent.

=== Ladder escapes ===
''(See also the article [[Ladder escape]])''

Consider the same position as before but suppose Red has an additional piece at h8.

<hex>R9 C9 Q1 Ve3 Vd4 Vd5 Vc6 Vc7 Hb8 Hb9 Hd7 He6 Hf6 Vh8</hex>

This additional piece forms a '''ladder escape''' which allows Red to jump a move ahead of the ladder and win the game. The piece at h2 is called the "[[escape piece]]." Red should now play along the ladder as before, forcing Blue's response at each step. After c8, c9, d8, d9, e8, e9, f8, f9, g8 Red is connected to the bottom.

<hex>R9 C9 Q1 Ve3 Vd4 Vd5 Vc6 Vc7 Hb8 Hb9 Hd7 He6 Hf6 Vh8 Vc8 Hc9 Vd8 Hd9 Ve8 He9 Vf8 Hf9 Vg8</hex>

In general, for a ladder escape to be successful, it should be safely connected to the edge and not interfere with the ladder's [[Projected ladder path|projected path]].

=== Ladder escape templates ===
''(See also the article [[Ladder escape templates]])

* [[Second row|Row-2]] ladders: All of the [[edge template]]s described earlier are valid.
* [[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]] is valid. Also [[template IVa]] is valid if you can double two-bridge to the [[escape piece]] as follows.

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

Red can jump ahead to the escape template by playing at the marked hex.

=== The ladder escape fork ===
''(See also the article [[Ladder escape fork]])''

If you are forced onto a ladder and no convenient escape is present, then you must create one. The best way is to play one of the valid ladder escape templates that threatens another strong connection. Such a move is called a '''ladder escape fork'''. For an example, see the first example in the upcoming section "forcing moves." The first forcing move is a ladder escape fork played just prior to the formation of the ladder (and a very short ladder at that). A ladder escape fork is frequently a [[killer move]].

=== Foiling ladder escapes ===
''(See also the article [[Foiling ladder escapes]])''

In order to successfully stop a ladder escape, you must either block the [[projected ladder path]] from connecting to the escape piece or intrude on the ladder escape template. To successfully stop a ladder escape fork, you need to do both with a single move and almost always with a move that is [[Adjacent move|adjacent]] to the escape piece. The following is an example of foiling a ladder escape fork.

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Hc6 Hd6 He7 Hf7 Hg7 Hi7 Hg9 Sc7 Sb8 Sc8 Sd8 Sa9 Sb9 Sc9 Sd9</hex>

Red has just played a forking ladder escape at d7. This piece is connected to the edge via template IIIa as shown by the marked hexes. Red is threatening to create an unbeatable chain by playing at E6 and the edge template is a valid ladder escape for the row-2 ladder starting G8, F9, F8, etc. To stop this, Blue needs to play a move that blocks the ladder path from connecting to the escape piece and that also intrudes on the escape template. Blue can achieve both aims by playing at D8 (which is adjacent to the escape piece). Red responds by playing C8 re-establishing the connection to the edge (there is nothing better). Now Blue continues by playing E6 blocking the forking path obtaining a [[win|winning position]].

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Ve6 Hc6 Hd6 Hd8 He6 He7 Hf7 Hg7 Hi7 Hg9 Vc8</hex>

Consider the same initial position but with Blue's piece on e7 removed.

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Hc6 Hd6 Hf7 Hg7 Hi7 Hg9 Sc7 Sb8 Sc8 Sd8 Sa9 Sb9 Sc9 Sd9</hex>

This change may look inconsequential but now Blue cannot foil the forking ladder escape. Suppose the play goes d8, c8, e6 as before.

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Ve6 Hc6 Hd6 Hd8 He6 Hf7 Hg7 Hi7 Hg9 Vc8</hex>

Now Red can ladder up to E8 by the sequence G8, F9, F8, E9, E8 achieving the following position.

<hex>R9 C9 Q1 Vg3 Vf4 Vf5 Vg5 Vh6 Vh7 Vd7 Ve6 Hc6 Hd6 Hd8 He6 Hf7 Hg7 Hi7 Hg9 Vg8 Hf9 Vf8 He9 Ve8 Vc8</hex>

Now if Blue stops the e8 piece from connecting to the [[Bottom edge|bottom]] by playing d9, Red responds by playing e7 connecting to the bottom anyway. This example illustrates that a potential foiling move that leaves vulnerable points is unlikely to succeed against a forking ladder escape. The proper handling of ladders and ladder escapes is a complex matter and it is where many games are won or lost.

=== Pre-ladder formations ===

It's important to recognize situations in which a ladder is about to form or which could be formed. Such recognition allows you to play pieces that also serve as ladder escapes before the ladder occurs. It also allows you to play defensive moves that also block potential ladder paths prior to the existence of the ladder. By far the most common pre-ladder formation is the following "[[Bottleneck]] formation."

<hex>R5 C7 Q1 Hd3 Ve3 Hf3 Hd5</hex>

Red can now form a ladder by playing e4, e5, f4, f5, etc. or by playing d4, c5, c4, b5, etc. Such formations typically occur due to blocking a player from directly connecting to an edge as in the following example.

<hex>R5 C7 Q1 Vg1 Ve2 Hf3</hex>

In order to block Red from connecting to the bottom edge, Blue plays d3 creating a [[bottleneck]]. Red responds with e3 squeezing through and then Blue blocks with d5 completing the formation in the previous diagram.

The other common pre-ladder formation occurs when the defender is blocking the connection to an edge via a classic block as in the following diagram.

<hex>R5 C7 Q1 Ve1 Hd4</hex>

Red can form a ladder by playing d3, c4 and then laddering either to the left or right (c3, b4, b3, a4 or e3, e4, f3, f4, etc.)

== Forcing moves ==
''(See also the article [[Forcing move]])''

'''Forcing moves''' are moves that make a threat that your opponent must reply to on their next turn. Common forcing moves include playing in one of the [[Empty hex|open hexes]] in a two-chain (threatening to break the link), intrusion into an edge template, or threatening an immediate strong connection or win. Consider the following position with the [[vertical player]] to move.

<hex>R9 C9 Q1 Vg3 Vg4 Vf5 Vh5 Hc7 Hd8 Hf7 Hh7</hex>

At first glance, the position looks bad for Red, but Red can win by making a couple of forcing moves. He plays at e8 threatening to play at e7 on his next turn which would create an unbeatable winning chain. Blue has little choice but to stop this threat by playing e7 (there is nothing better). The move e8 is a forcing move.

The forcing nature of the move allows Red to place a piece on the other side of Blue's line without giving Blue any time to do anything constructive. The e8 piece on the other side is connected to the bottom and is of critical importance.

Red continues by playing another forcing move at g7. The only move that stops this piece from immediately connecting to the bottom edge is f9. But after f9, Red completes the win by playing at f8.

<hex>R9 C9 Q1 Vg3 Vg4 Vf5 Vh5 Hc7 Hd8 Hf7 Hh7 Ve8 He7 Vg7 Hf9 Vf8</hex>

The group of red pieces near the bottom are connected to the bottom edge. These pieces are connected via two-chains to the [[group]] g3-g4-f5 which is in turn connected to the top edge via edge [[template IIIa]].

(Note: the two forcing moves could just as easily be played in the reverse order. That is Red plays g7, Blue is forced to respond with f9, and then Red plays e8 which threatens to form an unbeatable chain in two distinct ways.)

In general terms, you have three options when responding to a forcing move in a [[Bridge|two-chain]].

# [[Saving a connection|Save]] the link by playing the other move in the two-chain.
# [[Ignoring a threat|Play elsewhere]] (e.g. playing another move may give another way of meeting the threat thus rendering it harmless)
# [[Counterthreat|Respond]] with a forcing move of your own.

=== Breaking edge templates via forcing moves ===

Forcing moves are also the only way to successfully defeat an edge template. This is done by making a [[template intrusion]] that is also a more threatening forcing move. After the opponent responds to the greater threat, you can play another move within the template and destroy the connection to the edge. For example, consider the following position with the vertical player to move.

<hex>R9 C9 Q1 Hc3 Hd3 He3 Hf2 Hg1 Hg3 Vd7 Ve6 Ve5 Ve4 Vf3 Vh5 Si1 Si2 Si3 Si4 Sh2 Sh3 Sh4 Sg4</hex>

The piece on g3 is connected to the right edge via [[template IIIa]] indicated by the '*'s. Red's best move is to play at h2. This intrudes on the edge template, is connected to the top via edge [[template II]], and threatens to complete an unbeatable chain by playing at g2 next turn. Blue can stop this threat only by playing at g2. Then Red Plays i3 breaking Blue's connection to the right.

<hex>R9 C9 Q1 Hc3 Hd3 He3 Hf2 Hg1 Hg3 Vd7 Ve6 Ve5 Ve4 Vf3 Vh5 Vh2 Hg2 Vi3</hex>

=== Using forcing moves to steal territory ===
''(See also the article [[Stealing territory]])''

I'll define '''territory''' to be the number of empty hexes adjacent to your pieces. By playing a forcing move in one of the empty hexes in a two-chain, a player can steal territory at no cost.

<hex>R5 C5 Hc2 Hb4 Vd2 Vd3 Sb3</hex>

In this position, Blue has two more hexes of territory than Red (9 vs. 7 [[adjacent hex]]es). Suppose Red makes the forcing move at the indicated hex and Blue saves the link.

<hex>R5 C5 Hc2 Hb4 Vd2 Vd3 Vb3 Hc3</hex>

Now Red has two more hexes of territory; i.e. Red has stolen 4 hexes of territory without disturbing either player's connections. Significantly, the additional territory is on the other side of Blue's connection where it may potentially be used for a future threat. The additional territory can't hurt and sometimes it makes a crucial difference.

A forcing move is [[Irrelevant move|harmless]] if it gains no territory for the opponent. You should not be worried at all about leaving harmless forcing moves available for your opponent. When you have more than one way of completing a connection with a two-bridge, e.g. when completing the loose connection described previously, you should consider which forcing move is least valuable for your opponent. Consider the following position with Red to play.

<hex>R5 C6 Q1 Vd2 He3 Hb4 Vd4 Hb5</hex>

Red wants to connect the d4 piece to the d2 piece. There are three distinct moves that accomplish this, d3, c4 (two-chaining to d2), and c3 (two-chaining to d4).

There is not much to be said about d3; it [[Direct connection|directly connects]] without altering anything else. c4 connects but gives a potentially useful forcing move to Blue. Blue can respond with c3 and suppose Red saves the connection with d3. Now Blue has gained a free hex of territory, the hex c2 is now directly adjacent to the c3/b4/b5 group when it wasn't previously. Hence, c4 is worse than d3.

Now consider the last remaining possibility c3. This leaves two forcing moves for Blue but both of them are completely harmless! If after c3, Blue plays one of the forcing moves c4 or d3, then Red can save the link and Blue will not have gained any territory at all — any empty hexes adjacent to the forcing piece were already adjacent to Blue's existing pieces. Hence, c3 is just as safe as d3 but significantly, c3 ''gains'' one hex! — b3 is now adjacent to Red's d2/b3 group when it wasn't before. Thus, c3 is better than d3 and is the best of three choices.

== Using edge templates to block your opponent ==

If your opponent has not completed an [[edge template]] but is threatening to do so in multiple ways, then the only defensive moves that stop the immediate threatened connections are those in the overlap between all threatened template connections. Suppose you are trying to stop the vertical player from connecting to the [[bottom edge]] in the following example.

<hex>R5 C6 Q1 Ve2 Hf3</hex>

The vertical player has not formed an edge template but is threatening to do so in the following four different ways.

{|
| <hex>R5 C6 Q1 Ve2 Hf3 Sd3 Se3 Sd4 Sc5 Sd5</hex> || <hex>R5 C6 Q1 Ve2 Hf3 Sd3 Se3 Sc4 Sd4 Se4 Sb5 Sc5 Sd5 Se5</hex>
|-
| <center>''Two-chain to [[template II]] at d4''</center> || <center>''Adjacent move to [[template IIIa]] at d3 and e9''</center>
|-
| <hex>R5 C6 Q1 Ve2 Hf3 Sc3 Sd3 Sb4 Sc4 Sd4 Sa5 Sb5 Sc5 Sd5</hex> || <hex>R5 C6 Q1 Ve2 Hf3 Sc3 Sd3 Se3 Sb4 Sc4 Sd4 Se4 Sa5 Sb5 Sd5 Se5</hex>
|-
| <center>''Adjacent move to template IIIa at d3'' || <center>''Adjacent move to [[template IIIb]] at d3''</center>
|}

The only three [[Hex (board element)|hexes]] in the overlap among all these edge templates are marked on the following diagram. To stop the immediate connection, the horizontal player must play at one of them.

<hex>R5 C6 Q1 Ve2 Hf3 Sd3 Sd4 Sd5</hex>

== On connectivity ==

=== Overlapping connections ===
''(See also the article [[Overlapping connections]])''

One should be alert to the situations where various connections, edge templates, and potential connections overlap at some hex(es). Consider the following position with Blue to move.

<hex>R11 C11 Vj2 Vi4 Vj5 Vi7 Vi9 Vh9 Vg9 Vf9 Se9 Ve8 Vd10 Hg7 Hf7 He6 Hc7 Hc9 He10 Hf10 Hg10 Hh10 Hi10</hex>

At first glance, it appears that Red has an unbreakable winning path. j2 cannot be stopped from connecting to the [[Top edge|top]], j2 is connected the [[group]] f9-g9-h9-i9 through a series of unbreakable two-chains, this group is connected to e8 via a two-chain, e8 is connected to d10 via another two-chain, and d10 cannot be stopped from connecting to the [[Bottom edge|bottom]].

Appearances are deceiving; it is Blue that has a forced win! The [[Weakest link|flaw]] in Red's formation is that the two-chain from f9 to e8 and the two-chain from e8 to d10 [[overlap]] at the hex marked by a '*' in the diagram (e9). Blue should play at e9. By playing in the overlap, Blue is threatening to break ''both'' two-chains containing this hex. Red cannot save them both.

If Red responds with f8, then Blue plays d9 breaking the two-chain and establishing an unbeatable chain. If Red saves the other link by responding with d9, then Blue breaks through with f8 again establishing an unbeatable chain.

=== Disjoint steps ===

When a piece can be connected to a group of pieces in one move in two non-overlapping ways, then they can be thought of as already connected to the group. Consider the following position.

<hex>R5 C5 Q1 Vc2 Vd2 Vb3 Hc3 Ha5 Hd4</hex>

Red's three pieces are connected to the top. How can Red extend this connection downward? By playing at c4! The piece at c4 is connected to the group of three vertical pieces in two non-overlapping ways; namely, through the hexes b4 and d3. The diagrammed connection pattern is a fairly common occurrence and the connection to the piece at c4 is just as strongly connected as the pieces in a two-chain.

=== Groups ===
''(See also the article [[Group]])''

A '''group''' is a collection of pieces that, considered in isolation from the rest of the position, have an unbreakable connection with each other. As you improve, it is important to learn to think in terms of safely connected groups of pieces. To illustrate why, consider the following two hex puzzles taken from [[Cameron Browne]]'s book "[[Hex Strategy]]."

<hex>R11 C11 Q1 Hj2 Hh3 Hc4 Vd4 Hf4 Vi4 Vj4 Vd5 Vg5 Hh5 Vi5 Vk5 Ve6 Hf6 Hg6 Hh6 Hi6 He7 Vg7 Hi7 Vj7 Vc8 Vi9</hex>

It is Blue's turn; how can he win?. The chain of pieces j2-h3-f4 is connected to the right edge and furthermore, Blue has no other way of connecting to the right edge. So to win, Blue has to extend this chain to the left edge. Looked at in isolation, there doesn't seem to be any way to do this, yet by thinking in terms of connected groups, the solution falls out easily.

Notice that the j2-h3-f4 chain threatens to connect to the i7-i6-h6-h5-g6-f6-e7 group in two non-overlapping ways, through locations h4 and f5. Hence, these two groups can be thought of as a single group of pieces already connected to the right edge (this is another example of "'''disjoint steps'''"). Now notice the key hex d6. This hex threatens to connect to Blue's big group in two distinct non-overlapping ways (through e5 and d7) hence a piece played at d6 would be part of the big group (disjoint steps again!). Furthermore, a blue piece at d6 could not be stopped from connecting to the left because the b4 piece acts as a ladder escape. Therefore, d6 wins.

<hex>R11 C11 Q1 Hj2 Vc3 Hd3 Vg3 Hj3 Hc4 He4 Vc5 Vd5 Hg5 Vi5 Vd6 He6 Vd7 Ve7 Vh7 Hb9</hex>

Again, it is Blue's turn and the task is to [[win]]. The c4 piece cannot be stopped from connecting to the left edge since after the block a5, Red can ladder down row B to the escape piece at b9. Hence, the group c4-d3-e4 is connected to the [[left edge]]. The group j2-j3 is connected to the [[right edge]]. Blue has a third group e6-g5. If Blue can play a single move that connects the e6-g5 group to both other groups, then this would be a winning move. Blue has a unique move which does this, namely play at g4. The g4-g5-e6 group is connected to the left group through f4 and e5. It threatens to connect to the j2-j3 group via h3 and i4. h3 is directly connected to g4 and two-chains to j2. None of the hexes involved, h3, i2, and i3, is involved in the connection threat i4 plus the two chain to g5. I.e. the threats don't overlap and hence the connection cannot be stopped. Therefore, g4 wins.

There is another way of coming up with this move. Red threatens to cut off the e6-g5 group to the left by two-chaining from g3 to f5. Red is also threatening to cut off the e6-g5 from the right by two-chaining from g3 to h4. However, these threats overlap and hence, Blue can stop them both by playing in the unique hex contained in the overlap, namely g4 again.

This illustrates that [[Offence equals defense]] in hex. Playing in regions of overlapping threats in order to stop all the threats is a defensive way of thinking. Trying to establish unbreakable connections between groups of your pieces is an offensive way of thinking. In this example, both offensive and defensive thinking techniques lead you to the unique best move. A lot of times defensive thinking is easier but sometimes offensive thinking is.

== Conclusion ==

The first two strategy guides cover what I consider to be the fundamentals of [[hex strategy]]. This information should be enough to move up into the 1800s or 1900s on [[PlaySite]]. To move up the ranks of the red guys (the topmost group) requires the following. First you need to improve your [[opening play]]. Playing any reasonable looking moves during the initial phase of the game is enough for an orange player, but to compete with the reds, you need to [[Consistency|consistently]] play one of the top two or maybe three moves. Unfortunately, I don't know a good way to describe how to do this; I'm not convinced the necessary information can be verbalized. After that, you need to know the [[Minimax]] principle (described in the [[Advanced (strategy guide)|Advanced strategy guide]]).

Also you need a certain mindset, call it [[willpower]] if you like, to move towards the top ranks. You have to try to hold onto every little [[Hex (board element)|hex]] the way a miser hoards gold pieces and you have use every optimization you can no matter how minor it may seem. The most useful optimizations, tricks, and special situations that I've learned so far are included in the Advanced strategy guide. But surely there are other things out there waiting to be discovered.

Unlur

2005-02-16T10:05:08Z

Reiner Martin: removed capitalisation for consistency

Unlur is a [[connection game]] invented by [[Jorge Gómez Arrausi]] in [http://en.wikipedia.org/wiki/2001 2001]. It won the 2002 Unequal Forces Game Design Competition which was sponsored by the [[Abstract Games Magazine]], [[About Board Games]] and the [[Strategy Gaming Society]].

It is played by two [[player (general)|players]], black and white, on a [[hexagonal grid of hexagons]], as in the figure. The two players have different objectives, and must therefore use different strategies to achieve their goals.



A [[Y (chain)|Y]] is a [[chain (Y)|chain]] [[connection|connecting]] three non-adjacent sides. A [[line (Y)|line]] is a chain connecting two opposite sides.

Black wins if he creates a black Y, while White wins if he creates a white line. However, a player loses if he achieves his opponent's objective. That is, if Black creates a line without at the same time creating a Y, White wins. Similarly, if White creates a Y without at the same time creating a line, Black wins. If a player creates a line and a Y in the same move, he wins.

== The contract ==

White clearly has an easier objective than Black, since a line requires fewer pieces than a Y. Therefore a generalizations of the [[Swap rule|pie rule]] is used to balance the game.

In the first phase of the game both players play with the black pieces. A move consits of placing a black piece anywhere on the board, except at the border. Instead of making a move a player may [[Pass (Y)|pass]]. In that case, the player passing becomes Black for the rest of the game, while the other player becomes White, and the players then play with their respective pieces. White makes the first move.

The first phase is called the [[contract phase]].

Basic (strategy guide)

2005-02-16T10:04:48Z

Reiner Martin: removed capitalisation for consistency

''Adapted with permission from Glenn C. Rhoads strategy guide.''

== Rules of Hex ==
''(See also the article [[Rules]])''

[[Hex]] is a two player [http://en.wikipedia.org/wiki/Board_game board game] played on an ''n × n'' grid of [[hexagon]]s.

A [[Move|turn]] in Hex consists of placing a [[piece]] of your [[color]] on a [[Hex (board element)|hexagon]]. The [[first player]]'s goal is to form an unbroken [[chain]] of hexes of his color that [[connection|connects]] the [[top edge|top]] to the [[bottom edge|bottom]] while the [[second player]] tries to form an unbroken chain connecting the [[left edge|left]] and [[right edge|right]] sides.

[[Swap rule]]: After the initial play only, the second player has the option of either responding with his turn or swapping sides taking the initial play as his first turn.

Without the swap rule, the first player has a strong [[advantage]]. The swap rule equalizes this advantage by forcing the first player to make a move that leads to a roughly equal game. If the first player makes a very strong opening move, the second player will swap sides and start with an advantage. If the first player makes a very weak opening move, the second player won't swap and again will start with an advantage.

== Basic strategy ==

Notation: the [[row]]s of the board are [[coordinates|indexed]] by numbers and the [[column]]s are indexed by letters. Individual hexes are referred to by listing the column index followed by the row index; e.g. hex c2 is the one in column c row 2. Here at [[HexWiki:About|HexWiki]], red pieces belong to the [[vertical player|"vertical" player]], and blue pieces belong to the [[horizontal player|"horizontal" player]]. An empty 4 × 4 board looks like follows.

<hex>R4 C4 Q1</hex>

=== The two-bridge ===
''(See also the article [[Bridge]])''

The [[pattern|formation]] consisting of two pieces that are [[non-adjacent]] but have two empty [[adjacent|neighboring]] hexes in common is referred to as a [[bridge|two-bridge]]; e.g. the pieces on b2 and c3, and the [[empty hex]]es b3 and c2 in the following diagram form a two-bridge.

<hex>R4 C4 Q1 Vb2 Vc3</hex>

The two pieces are almost as strongly connected as a solid chain from b2 to c3. The opponent can attempt to break this connection only by playing a piece at either b3 or c2, and no matter which one the opponent plays, you can play the other and restore the link. For most purposes you can think of the two-bridge pieces as already being connected. By connecting pieces via two-bridges, you can spread across the board twice as fast as by playing adjacent hexes.

{| border="0" cellpadding="2"
| <hex>R7 C7 Q1 Ve1 Ve2 Vd3</hex> || <hex>R7 C7 Q1 Ve2 Vd4 Vc6</hex>
|-
| <center>''Expanding by adjacent moves''</center> || <center>''Expanding by two-chains''</center>
|}

Considered in isolation the pieces in a two-chain are [[strong connection|connected]] but sometimes a two-chain can be broken by playing a piece in the middle of a two-chain that contains some other [[threat]] that must be immediately answered. After the opponent answers the threat, you can then play in the other hex in the two-chain breaking the connection.

Also, playing in the middle of a two-chain can be a good play even when the opponent should and does respond by saving the link. The reason being that the piece played may be useful later.

=== Blocking moves ===

When you have no pieces in the area, it is usually best to start blocking at a distance. If you block too close, then the opponent can simply flow around the attempted block. For example, suppose you are trying to stop the vertical player from connecting to the bottom in the following diagram.

<hex>R11 C11 Q1 Vh1 Vh2 Vh3 Vg4 Vg5 Vg6</hex>

If you try to block by playing adjacent to the [[leading piece]], say by playing at g7, then the vertical player can simply step around it at f7 (see diagram below). Then the attempted block at say e8, could similarly be met by playing at f8. Obviously, you are not making any progress here.

<hex>R11 C11 Q1 Vh1 Vh2 Vh3 Vg4 Vg5 Vg6 Hg7 Vf7 He8 Vf8</hex>

Another try from the original position would be to block at a two-chain distance away at f8 (see diagram below). This is better than the [[adjacent block]] but sometimes the opponent can flow around this too by two-chaining at a angle — e.g. by playing h7 in response to f8. (h7 should be met by either h8 or g9.)

<hex>R11 C11 Q1 Vh1 Vh2 Vh3 Vg4 Vg5 Vg6 Hf8 Vh7</hex>

Another possibility is to combine the above two ideas by first doing an adjacent block at g7 and then if the vertical player responds with f7, you block at a two-chain distance away at e9. Then your opponent cannot two-chain towards the right because of the initial g7 piece.

<hex>R11 C11 Q1 Vh1 Vh2 Vh3 Vg4 Vg5 Vg6 Hg7 Vf7 He9</hex>

A good block in the original position is to block at one hex farther back than the two-chain block at either e9 or f9 (sometimes this is referred to as the classic block). For example suppose H blocks at f9 (see diagram below). Two-chaining to f8 is met by e9. Two-chaining to the lower right (h7) is met by h8 and two-chaining towards the lower-left (e7) is met by d8. By blocking at a distance, you have a move or two before the advancing head reaches the blocking pieces. Note that when the board size is smaller than 11 × 11, then the classic block is much less useful due to the lack of space.

<hex>R11 C11 Q1 Vh1 Vh2 Vh3 Vg4 Vg5 Vg6 Hf9 Vf8 He9 Vh7 Hh8 Ve7 Hd8</hex>

== General principles ==

=== A position is only as good as the weakest link ===
''(See also the page [[Weakest link]])''

Thus, with each move you should attempt to either improve your weakest link or make your opponent's weakest link even weaker. A move which does both is a strong move. For example, in the position below the hex f6 is the weakest point in the Red's best connection across the board. It is also the weakest link in the Blue's best connection across the board. Thus, the player whose turn it is to move would be wise to play at f6. In fact, whoever plays next has a forced win after playing f6.

<hex>R11 C11 Q1 Vh2 Vg4 Hb5 Hh5 Hc6 Sf6 Hi6 Ve8 Vd10</hex>

=== Offense equals defense ===
''(See also the page [[Offense equals defense]])''

In Hex, good [[offense]] and good [[defense]] are entirely equivalent. If you complete a connection between your sides, then your opponent is prevented from completing theirs. Conversely, if your opponent is prevented from completing a connection, then you must have completed yours (draws cannot occur in Hex). Furthermore, the only way to complete a connection is to prevent your opponent from making a connection and the only way to prevent your opponent from connecting is to complete your connection. In a very real sense, you don't have to worry about whether you should play offensively or defensively since they are the same. The critical point to remember is that unless you are making a sequence of [[forcing move|forcing plays]], it is generally easier to think in terms of good defense than good offense regardless of whether you are currently winning or losing. This point about thinking defensively should frequently be used with point 3.1 above. Often I will look for the connection that my opponent is going to have the toughest time making (point 3.1 above), e.g suppose that my opponent's most difficult connection to complete looks like the connection to the right edge of the board. Then I'll look for good defensive moves that make it even more difficult for my opponent to connect up to the right edge. This is very sound strategy.

=== Momentum ===
''(See also the page [[Momentum]])''

The player who is dictating the play is said to have the '''momentum'''. Alternatively, the momentum is against the player who is being forced to respond to the opponent. The player with the momentum usually has the advantage and this advantage is often decisive. You should generally not hand over the momentum to the opponent unless you have a very good reason for doing so. In well played close matches, the momentum often swings between the two players with each move.

=== Multiple threats per move ===
''(See also the page [[Multiple threats]])''

Whenever possible, a player should make each move achieve at least two different goals or threats. Moves that contain only a single threat are generally not hard to meet. If a move contains multiple threats, the opponent may not be able to stop all the threats with a single move.

=== The center ===

The central region of the board is strategically the most important area. From the center, connections can spread out in many directions giving you more flexibility and options than starting from an edge. Furthermore, centrally played pieces are more nearly equidistant from both of your edges — this is related to point 3.1 about improving your weakest link. The greater distance apart two pieces are, the harder they are to connect up, i.e. their potential link is weaker.

== The opening ==

Without the [[swap rule]], the initial move would be easy. Playing in the [[center hex]] is the strongest opening move. The weakest opening move is to play in one of the acute corners (a1 and the opposite corner) and is one of only two opening moves that are a proven loss (without the swap option). The other is right next to it at b1. Suppose the [[vertical player]] moves first. Which opening moves should you swap and which should you not swap? The following is my personal rules for the 10 × 10 board.

=== 10 × 10 swap rules ===

# Don't swap any of Vertical's border row moves except for the obtuse corner.
# Don't swap a2, or b2 (nor the symmetrically equivalents i9 and j9).
# Swap all other initial moves.

Note: the possible theoretical exceptions to these rules are the opening moves a2, b2, c2 and a3 (and their symmetric equivalents). The winning/losing margin with these moves is so razor thin that nobody has been able to determine with any confidence whether these moves should theoretically be swapped or not.

=== Good opening moves on the 10 × 10 ===

The best opening moves against an experienced player are the [[border hex]]es (except don't open a1!) and b2 and c2. b2—d2 are probably the only good non-border moves against an experienced player (b2 is essentially equivalent to the move a2 which is a popular opening choice and there is almost no difference between b2 and c2). Against lesser experienced players you can play something stronger such as one out from the obtuse corner (b9/i2) because they might not realize its strength and even if they do swap, they may not be capable of taking advantage of it anyway.

a2/b2 and a3 both lead to a balanced game and seem to be the most popular choices. Except for games between expert players, you can safely play either side of a2/b2 or a3 and have an equal chance of winning (and similarly for other opening plays). Also some variation in opening play is generally good. Varying your opening is the first thing to try against an opponent that seems to have your number. Sometimes you can find a weakness in a player's personal swap rules by trying out different openings.

=== The second and third moves ===

A very common but not the only good response to a border opening is to play in one of the two central hexes e6 or f5. The third move in response to a central reply should be a blocking move on the side of e6/f5 that is farthest from the edge. e6 is one hex closer to the left edge and f5 is one hex closer to the right edge. In accordance with the principle of exploiting your opponent's weakest link, you should therefore block f5 on the left and e6 on the right. Thus, a typical opening sequence would be a2, ''swap'', f5, c6. In my opinion, the strength of the central response is overrated; practically any move that is not in one of the 3 rows closest to your border rows and that is also not too close to the opponent's border, is a near equally good response. If there is any difference in strength, it is for all practical purposes non-existent.

=== Board size ===

Hex can be played on any size [[board]]. If the board is [[Small boards|too small]], the game becomes trivial and uninteresting. The "standard" size at the online site [[PlaySite]] is 10 × 10 but in my opinion, this is just a little too small and the "standard" size should really be 11 × 11 (11 × 11 is the standard size at [[pbmserv|the PBM play by email site]]). Some experienced players prefer a larger board such as 14 × 14 or 17 × 17. As the board size gets larger and larger, the game becomes more subtle and strategic. Hex is actually of comparable complexity and depth to the oriental board game [http://en.wikipedia.org/wiki/Go_%28board_game%29 Go] played on the same size board (many Go players consider Go to be the deepest and most complex perfect information strategy game ever invented).

A typical hex game fills about one-third of the board. We can use this to get a good estimate of the average number of moves for any board size.

* 10 × 10: 16 moves per side
* 11 × 11: 20 moves per side
* 14 × 14: 28 moves per side
* 17 × 17: 48 moves per side
* 19 × 19: 60 moves per side (this is the standard size in Go)

One of the pleasant aspects of Hex is that games generally do not last as long as in other strategy games of comparable complexity (e.g. Go typically lasts around 100 moves per side). The 11 × 11 game is very good and takes only about 20 moves per side. For those wanting a more complex game, the 14 × 14 game provides it without having the length of the game blow up to marathon proportions.

== Reference bibliography ==

''Hex Strategy: Making the Right Connections'', by Cameron Browne, A.K. Peters Ltd., 2001. — The strategy part of this book is generally very sound. The primary exception is that the suggested opening swap rules are not correct at all.

Rules

2005-02-16T10:02:24Z

Reiner Martin: removed capitalisation for consistency

== Shape of the board ==
The [[Hex]] [[Board|board]] is composed of [[hexagon]]s, arranged in an ''n × n'' [http://en.wikipedia.org/wiki/Rhombus rhombus], where ''n'' is an integer greater than one. Thus Hex can be played on boards of different sizes, and which size is considered standard varies. [[Piet Hein]] used a board of size 11, while [[John Nash]] settled on size 14. Nowadays people usually use the sizes that are available on [[Game sites|game sites]] on the Internet offering Hex. [[Playsite]] uses sizes 10, 14 and 18; [[Little Golem]] offers the sizes 13 and 19, while [[kurnik]] has boards of sizes 9 to 19. (See the article [[Board size]] for more.)

It is also possible to play on boards of size ''m × n'' where ''m'' and ''n'' are distinct integers, but in this case a winning strategy is known for the player with shortest distance between his sides.

This is how a typical Hex board looks:

<hex></hex>

== Playing the game ==
Hex is played by two players, who take turns placing pieces on the board. The players have different colours, say red and blue. (The players themselves are sometimes referred to as [[Red (player)|Red]] and [[Blue (player)|Blue]]) The four [[Edge|edges]] of the board are coloured with the same colours, in such a way that parallel edges have the same colour. Red wins if he can build a continuous chain between the two red edges, and blue wins if he can build such a chain between the blue edges.

For more information on how best to utilize the rules, see [[Strategy]].

== The [[swap rule]] ==
The above rules give a strong [[Advantage|advantage]] to the [[First player|first player]]. Because of this, the [[swap rule]] (also called the Pie rule) was introduced. Suppose that Red is to make the first move. Then the first player places a Red [[piece]] on the board, in any [[Hexagon (Board element)|cell]] he likes. After that, the [[Second player|other player]] decides who will play Red and who will play Blue. Then whoever was decided to be blue makes the next move, and the game continues normally (that's how it works at [http://www.gamerz.net/pbmserv pbmserv]). Another possibility to swap (without changing sides of each player) is to make a [[Piece swap|piece swap]], that is, the second player in his first move removes the very first stone which was placed on field with coordinates (x;y) and places his/her own stone on field (y;x) (that's how it works at [[Little Golem]], [[Playsite]] and [[kurnik]]).

Ladder

2005-02-16T10:01:00Z

Reiner Martin: removed capitalisation for consistency

A '''ladder''' is a situation in which both players move perpendicularly to an [[edge]] making a straight unbroken [[chain]]. The [[attacking player]]'s chain is almost always further from the edge. Here is an example of the ladder starting from b4 and c3.

<hex>R4, C10, Vc1, Vc2, Hb2, Hd2, Hb4, Vc3, Hc4, Vd3, Hd4, Ve3, He4, Vf3, Hf4, Vj3</hex>

Red (vertical) to move. In this situation red has a sure win due to his [[Ladder escape|ladder escape]] on j3. Of course he can start another ladder playing a4 but in that case, blue has a ladder escape on b2.

== Pre-ladder formations ==

<hex>R5 C5 Hb2 Vc2 Hd2 Hb4</hex>

<hex>R5 C5 Vb2 Va4 Ha5</hex>

<hex>R5 C5 Ve1 Vd3 He3 Hc4 Hd4</hex>

== Ladder handling ==

There are three ways for the attacking player to "win" in a ladder situation:

# Making use of an existing [[Ladder escape|ladder escape]] piece
# Creating a new ladder escape with a [[Ladder escape fork]]
# [[Cornering]]

== Specific ladder strategies ==

[[Parallel ladder|Parallel ladder trick]]

[[Yielding]]

Handicap

2005-02-16T09:59:18Z

Reiner Martin: /* Start with pieces on the board */

Playing with '''handicap''' means to give one of the [[player]]s (preferably the weaker one) an advantage at the start of the game. The point of this is to make the game more even, so that it will be challenging for both players.

In [[Hex]] there is no standard way of playing with handicap, and because of this it is not very common to do so. This ought to be changed.

There are several ways a handicap could be implemented.

== Non-rhombic board ==
One natural way is to play with an ''m × n'' [[board]] where ''m'' is distinct from ''n'', and let the weaker player have the shortest distance between his sides. Unfortunately, this doesn't work very well, since there exists an easy, explicit winning strategy for the player with shortest distance.

== Start with pieces on the board ==
Alternatively one can use the kind of handicap used in [[Go]]: The weaker player places a certain number of [[pieces]] on the board as his [[first move]]. A 1-piece handicap is the same as playing ordinary Hex without the [[swap option]]. With a handicap of two or more pieces, the first player either places the stones as he likes, or according to some pre-defined rules.

The drawback of this option is that even a 1-piece handicap gives a very big [[advantage]]. At least this is true on [[Small boards|smaller boards]] (such as the 10 × 10 board). On larger boards, such as 19 × 19 this may be a good solution, and weak players may even be allowed to place three or four pieces against the strongest players.

Rules (Hex)

2005-02-16T09:54:23Z

Reiner Martin: Rules (Hex) moved to Rules

#REDIRECT [[Rules]]

Go

2005-02-16T09:52:13Z

Reiner Martin:

'''Go''' is a 3000 years old game, originally from [http://en.wikipedia.org/wiki/China China]. It is played on a rectangular grid of size 19 × 19, and the players play by alternatingly placing pieces of their colour on the intersections on the board. The object of the game is to get the largest territory.

There are many similarities between Go and [[Hex]], and people who like Go often find Hex interesting, and vice versa.

Some basic information on Go can be found in the [http://en.wikipedia.org/wiki/Go_(board_game) Wikipedia article on Go].
Complete rules are available at [http://senseis.xmp.net Sensei's Library], which is a [http://en.wikipedia.org/wiki/Wiki wiki] for Go.

Go

2005-02-16T09:50:53Z

Reiner Martin: ref to wikipedia

'''Go''' is a 3000 years old game, originally from [http://en.wikipedia.org/wiki/China China]. It is played on a rectangular grid of size 19 × 19, and the players play by alternatingly placing pieces of their colour on the intersections on the board. The object of the game is to get the largest territory.

There are many similarities between Go and [[Hex]], and people who like Go often find Hex interesting, and vice versa.

Some basic information can be found in the [http://en.wikipedia.org/wiki/Go_(board_game) Wikipedia article on Go].
Complete rules can be found at [http://senseis.xmp.net Sensei's Library], which is a [http://en.wikipedia.org/wiki/Wiki wiki] for Go.

Go (board game)

2005-02-16T09:48:41Z

Reiner Martin: Go (board game) moved to Go

#REDIRECT [[Go]]

Open problems

2005-02-15T08:33:52Z

Reiner Martin: ref to problems posed on LG

* 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 [[Hex 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?

* 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/topic.jsp?forum=50&topic=167 Little Golem forum].

Little Golem

2005-02-15T08:26:25Z

Reiner Martin: added external links to wikipedia for the other games

'''Little Golem''' is a turn-based server maintained by [[Richard Malaschitz]] from [http://en.wikipedia.org/wiki/Slovakia Slovakia]. It offers Hex on the sizes 13 and 19, and many of the top players are playing there.

In fact, the most prestigous title in the Hex world today, may very well be the [[Champion]] title at Little Golem. It is currently held by [[Tim Shih]].

Other games available on Little Golem are [http://en.wikipedia.org/wiki/Go_(board_game) Go], [http://en.wikipedia.org/wiki/Chess Chess], [http://en.wikipedia.org/wiki/Twixt Twixt], [http://en.wikipedia.org/wiki/Reversi Reversi], [http://en.wikipedia.org/wiki/Dvonn Dvonn], [http://en.wikipedia.org/wiki/Amazons_(game) Amazons], [[Golem's word game]], [[Four in a row]], [http://en.wikipedia.org/wiki/Gomoku Gomoku], [[Street Soccer]] and [http://en.wikipedia.org/wiki/Dots_and_Boxes Dots and Boxes].

The address is [http://www.littlegolem.net http://www.littlegolem.net].

For real-time hex games, [[Kurnik]] is the most popular site.

Kurnik

2005-02-15T08:15:56Z

Reiner Martin: typo

<div style="float:right;margin-left:1em">
[[Image:Kurnik.gif|200px|kurnik logo]]
</div>
'''kurnik''' is a Polish site featuring a number of online games, and is arguably the best real-time server on the net for [[Hex]].

Because most players on kurnik are from [http://en.wikipedia.org/wiki/Poland Poland], it is a good idea to know a few [[Polish phrases]].

For turn-based (non-real-time) hex games, [[Little Golem]] is the most popular site.

== Addresses ==

* http://www.kurnik.org — English version
* http://www.kurnik.pl — Polish version
* http://www.kurnik.org/intl/cs/ — Chech version
* http://www.kurnik.org/intl/et/ — Estonian version
* http://www.kurnik.org/intl/es/ — Spain version
* http://www.kurnik.org/intl/fr/ — French version
* http://www.kurnik.org/intl/it/ — Italian version
* http://www.kurnik.org/intl/hu/ — Hungarian version
* http://www.kurnik.org/intl/nl/ — Dutch version