Selinger: /* Non-existence of pairing strategies */ typo

2022-02-15T00:52:46Z

‎Non-existence of pairing strategies: typo

Selinger: /* Interior templates with pairing strategies */ Minor wording

2022-02-15T00:51:39Z

‎Interior templates with pairing strategies: Minor wording

Selinger: Added categories

2022-02-14T15:26:20Z

Added categories

Selinger: New article on pairing strategies

2022-02-14T15:20:07Z

New article on pairing strategies

New page

A '''pairing strategy''' for a player is specified by a set of pairs of empty cells. Each cell may belong to at most one pair, and not all empty cells need to be part of a pair. The player then plays as follows: whenever the opponent occupies one cell of a pair, the player occupies the other.

Pairing strategies are second-player strategies; they work even if the other player goes first.

Pairing strategies are useful because they are simple to describe and easy to carry out. If a game reaches a point where one of the players can win by a pairing strategy, the game can be considered to be won by that player (unless the player is so weak that they do not see the winning strategy).

A pairing strategy can be described succinctly by numbering the cells, giving the same number to both cells of a pair.

== Examples ==

In the following examples, the pairing strategy is always for [[Red]].

=== Interior templates with pairing strategies ===

Most interior templates have pairing strategies. In fact, with the exception of the [[hammock]], all of the templates listed on the [[Interior_template|interior template page]] have pairing strategies. The simplest template with a pairing strategy is the [[bridge]]:
<hexboard size="3x2"
edges="none"
coords="none"
visible="-a1 b3"
contents="R b1 a3 E 1:(a2 b2)"
/>
Here, there is only one pair, and if Blue plays in one cell, Red plays in the other.
Here are more examples of interior templates with pairing strategies. Many of the templates admit more than one pairing strategy; we only give one for each.

'''The [[wheel]]'''

<hexboard size="3x3"
coords="hide"
edges="none"
visible="-a1 c3"
contents="R a3 R b1 R c2 E 1:(a2 b2) 2:(c1 b3)"
/>

'''The [[trapezoid]]'''

<hexboard size="3x3"
coords="hide"
edges="none"
visible="-c3"
contents="R a1 a3 b3 c1 E 1:(a2 b1) 2:(b2 c2)"
/>

'''The [[crescent]]'''

<hexboard size="3x3"
coords="hide"
edges="none"
visible="-c3"
contents="R a1 a3 b3 c2 E 1:(a2 b1) 2:(b2 c1)"
/>

'''The span'''

<hexboard size="3x4"
coords="hide"
edges="none"
visible="-a1 d3"
contents="R a2 R a3 R c3 R d2 E 1:(c1 b3) 2:(b1 b2) 3:(c2 d1)"
/>

'''The scooter'''

<hexboard size="4x4"
coords="hide"
edges="none"
visible="area(c1,a3,a4,c4,d3,d1)"
contents="R b2 R a4 R c1 R c4 E 1:(a3 b3) 2:(b4 d2) 3:(c2 d1) 4:(c3 d3)"
/>

'''The long crescent'''

<hexboard size="3x6"
coords="hide"
edges="none"
visible="-f3"
contents="R a1 a3 b3 c3 d3 e3 f2 E 1:(a2 b1) 2:(b2 c1) 3:(c2 d1) 4:(d2 e1) 5:(e2 f1)"
/>

=== Edge templates with pairing strategies ===

Most edge templates do not admit pairing strategies, but some of the smaller ones do. Here are some examples.

'''The [[ziggurat]]'''

<hexboard size="3x4"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="R c1 E 1:(b2 d2) 2:(a3 b3) 3:(c2 d1) 4:(c3 d3)"
/>

'''[[Edge template III1b]]'''

<hexboard size="3x5"
coords="none"
edges="bottom"
visible="area(c1,a3,e3,e1)"
contents="R d1 S c3 E 1:(b2 e2) 2:(a3 b3) 3:(c1 c2) 4:(d2 e1) 5:(d3 e3)"
/>

'''[[Edge template IV1a]]'''

<hexboard size="4x8"
coords="none"
edges="bottom"
visible="area(e1,c2,a4,h4,h2,g1)"
contents="R f1 S e3 E 1:(d2 g2) 2:(b3 d3) 3:(f3 h3) 4:(c2 c3) 5:(e1 e2) 6:(g1 f2) 7:(g3 h2) 8:(a4 b4) 9:(c4 d4) 10:(e4 f4) 11:(g4 h4)"
/>

'''[[Edge template III2a]]'''

<hexboard size="3x3"
coords="none"
edges="bottom"
visible="-a1"
contents="R b1 c1 E 1:(a3 c2) 2:(a2 b2) 3:(b3 c3)"
/>

'''[[Edge template III2d]]'''

<hexboard size="3x3"
coords="none"
edges="bottom"
visible="area(c1,a3,c3)"
contents="R c1 a3 E 1:(b2 c2) 2:(b3 c3)"
/>

'''[[Edge template IV2d]]'''

<hexboard size="4x4"
coords="none"
edges="bottom"
visible="area(d1,a4,d4)"
contents="R d1 b3 E 1:(c2 d2) 2:(a4 b4) 3:(c3 d3) 4:(c4 d4)"
/>

=== Non-square boards ===

As shown in more detail on the page on [[Hex_theory#Winning_strategy_for_non-square_boards|Hex theory]], on boards of size ''n''×''m'' where ''n'' ≠ ''m'', the player with the shorter distance between their edges can win by a simple pairing strategy, even if the other player goes first:
<hexboard size="5x4"
coords="none"
contents="E 1:a1 2:b1 3:c1 4:d1 5:a2 6:b2 7:c2 8:a3 9:b3 10:a4
E 1:d5 2:d4 3:d3 4:d2 5:c5 6:c4 7:c3 8:b5 9:b4 10:a5"
/>

== Non-existence of pairing strategies ==

An example of a interior template that does not admit a pairing strategy is the hammock:
<hexboard size="3x4"
coords="hide"
edges="none"
visible="-d3"
contents="R arrow(10):a1 R c1 R b3 R arrow(4):d2
E a:b1 b:a2 c:b2 d:a3 e:c2 f:d1 g:c3"
/>
To see why, assume there was such a strategy. Then clearly ''a'' must be paired with ''b'', or else Blue could get both ''a'' and ''b'' and disconnect the template's leftmost stone. For similar reasons, ''c'' must be paired with ''d'', or else Blue could get ''a'', ''c'', and ''d''. Now since Blue can get ''b'' and ''c'', ''e'' must be paired with ''f''. On the other hand, since Blue can get ''a'' and ''c'', ''e'' must be paired with ''g''. Thus, we arrived at a contradiction, showing that there is no pairing strategy.

== Pairing strategies in Reverse Hex ==

Pairing strategies can also be used in Reverse Hex to force the opponent to make a connection. See [[Reverse_Hex#Pairing_strategies|Reverse Hex]] for more details.

← Older revision		Revision as of 15:26, 14 February 2022
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	Pairing strategies can also be used in Reverse Hex to force the opponent to make a connection. See [[Reverse_Hex#Pairing_strategies\|Reverse Hex]] for more details.		Pairing strategies can also be used in Reverse Hex to force the opponent to make a connection. See [[Reverse_Hex#Pairing_strategies\|Reverse Hex]] for more details.
		+
		+	[[Category:Theory]]
		+	[[Category:Strategy]]

@@ Line 146: / Line 146: @@
 == Non-existence of pairing strategies ==
-An example of a interior template that does not admit a pairing strategy is the hammock:
+An example of an interior template that does not admit a pairing strategy is the hammock:
 <hexboard size="3x4"
    coords="hide"

@@ Line 20: / Line 20: @@
    contents="R b1 a3 E 1:(a2 b2)"
    />
-Here, there is only one pair, and if Blue plays in one cell, Red plays in the other.
+In the bridge, there is only one pair, and if Blue plays in one cell, Red plays in the other.
-Here are more examples of interior templates with pairing strategies. Many of the templates admit more than one pairing strategy; we only give one for each.
+The following are more examples of interior templates with pairing strategies. Many of the templates admit more than one pairing strategy; we only give one for each.
 '''The [[wheel]]'''

Pairing strategy - Revision history

Selinger: /* Non-existence of pairing strategies */ typo

Selinger: /* Interior templates with pairing strategies */ Minor wording

Selinger: Added categories

Selinger: New article on pairing strategies