Difference between revisions of "Liberty"

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(Updated the article on liberties, which was a stub.)
 
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A '''liberty''' of an [[isolated piece]] or a [[group]] is the number of unoccupied [[Hex (board element)|cells]] [[adjacent]] to it. A piece with few liberties is generally weaker than one with many liberties.
+
A '''liberty''' of an [[isolated piece|isolated stone]] or a [[group]] is an unoccupied [[Hex (board element)|cell]] [[chain|adjacent]] to it. A stone or group with few liberties is generally weaker than one with many liberties.
  
−
''Do not play an isolated piece with three or fewer liberties.'' Such a move is always a bad move — it is always possible to find a better one.
+
== Advice for players ==
 +
 
 +
=== Old advice ===
 +
 
 +
An older version of this page, dated 2005, contained the following advice:
 +
 
 +
: ''Do not play an isolated piece with three or fewer liberties. Such a move is always a bad move — it is always possible to find a better one.''
 +
 
 +
The advice is not accurate. We now know that there are some situations where playing a stone with 3 liberties is the unique winning move ([[Wheel#The intrusion at B|see here for an example]]). Nevertheless, it is still a useful guideline in most situations that arise in real games.
 +
 
 +
=== Updated advice ===
 +
 
 +
Below, we analyze groups with up to 3 liberties. Based on the analysis, we can formulate the following principle:
 +
 
 +
: ''Do not play a stone that creates an interior group with 2 or fewer liberties, or with 3 liberties when two of them are adjacent or bolstered.''
 +
 
 +
== Analysis of groups with up to 3 liberties ==
 +
 
 +
=== Definitions ===
 +
 
 +
By an '''interior group''' we mean a group of [[friendly]] stones that is not adjacent to an edge.
 +
 
 +
A liberty L of a group X is said to be '''bolstered''' if it matches one of the following patterns (up to rotation and symmetry):
 +
<hexboard size="3x3"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  visible="-a1 c3 -a3,b3"
 +
  contents="R X:b1,c1 B a2 c2 E L:b2"
 +
  /><hexboard size="3x3"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  visible="-a1 c3 -a3,b3"
 +
  contents="R X:b1 B a2 c1 c2 E L:b2"
 +
  /><hexboard size="3x3"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  visible="-a1 c3 -a3,c2"
 +
  contents="R X:b1 b3 B a2 c1 E L:b2"
 +
  />
 +
 +
=== Groups with one liberty ===
 +
 
 +
If an interior group has only a single liberty, the group is [[dead cell|dead]]. It can never become part of a shortest winning path for the group's owner.
 +
 
 +
Examples:
 +
<hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B c2--e2--e3--d4--b4 R c3 d3"
 +
  visible="area(b3,c2,e2,e3,d4,b4)"
 +
  /> <hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B b4--b3--c2--e2--e3--d4 R c3 d3"
 +
  visible="area(b3,c2,e2,e3,d4,b4)"
 +
  />
 +
In both cases, the red group has a single liberty and is dead.
 +
 
 +
=== Groups with two adjacent liberties ===
 +
 
 +
If an interior group has only two liberties, and these liberties are adjacent to each other, the group is [[dead cell|dead]].
 +
 
 +
Examples:
 +
<hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B b3--c2--e2--e3--d4 R c3 d3"
 +
  visible="area(b3,c2,e2,e3,d4,b4)"
 +
  /> <hexboard size="5x7"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B b4--b3--c2--f2--f3--e4 R c3 d3 e3"
 +
  visible="area(b3,c2,f2,f3,e4,b4)"
 +
  />
 +
In both cases, the red group has two adjacent liberties and is dead.
 +
 
 +
=== Groups with two bolstered liberties ===
 +
 
 +
If an interior group has two non-adjacent liberties, and both of these liberties are bolstered, the group is [[captured cell|captured]]. If the group's owner plays in one liberty, the opponent can play in the other, [[dead cell|killing]] the group.
 +
 
 +
Examples:
 +
<hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B b4--b3--c2 e2--e3--d4 R c3 d3"
 +
  visible="area(b3,c2,e2,e3,d4,b4)"
 +
  /><hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B b3--c2--e2 d4--c4 R c3 d3 a5 f3"
 +
  visible="area(b3,c2,e2,e3,d4,b4) a5 f3"
 +
  />
 +
 
 +
=== Groups with two general liberties ===
 +
 
 +
If an interior group has two non-adjacent liberties that are not both bolstered, then the group is not in general captured. If one of the liberties is bolstered and the other is not, playing in the non-bolstered liberty [[domination|dominates]] playing in the bolstered one for both players.
 +
 
 +
Examples: Both players prefer "a" to "b".
 +
<hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B c4--b4--b3--c2 e2--e3 R c3 d3 E a:d4 b:d2"
 +
  visible="area(b3,c2,e2,e3,d4,b4)"
 +
  /><hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B b3--c2--e2 d4--c4 R c3 d3 f3 E a:b4 b:e3"
 +
  visible="area(b3,c2,e2,e3,d4,b4) f3"
 +
  />
 +
 
 +
For all groups with two general liberties, if it is the opponent's turn, they can [[dead cell|kill]] the group by taking away one of the liberties. Therefore, no player should play a move that creates an interior group with only 2 liberties.
 +
 
 +
=== Groups with three liberties ===
 +
 
 +
If an interior group has 3 liberties, it is not in general captured.
 +
 
 +
Examples: Three non-adjacent liberties, zero or one of which are bolstered.
 +
<hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B all R b2 c2 E b1 a3 c3"
 +
  visible="area(a2,b1,d1,d2,c3,a3)"
 +
  /><hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B all R b2 c2 E c1 a3 c3"
 +
  visible="area(a2,b1,d1,d2,c3,a3)"
 +
  />
 +
 
 +
However, if two of the liberties are adjacent or bolstered, then if it is the opponent's turn, the opponent can [[dead cell|kill]] the group by playing in the remaining liberty. Therefore, no player should play a move that creates an interior group with only 3 liberties of which two are adjacent or bolstered.
 +
 
 +
Examples: Two of three liberties are adjacent or bolstered. Blue can kill the red group by playing at "a".
 +
<hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B c4--b4--b3--c2 e2--e3 R c3 d3 E a:d4 c2"
 +
  visible="area(b3,c2,e2,e3,d4,b4)"
 +
  /><hexboard size="5x6"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B b3--c2--e2 d4--c4 R c3 d3 f3 E a:b4 d2"
 +
  visible="area(b3,c2,e2,e3,d4,b4) f3"
 +
  />
 +
 
 +
== See also ==
 +
 
 +
* [[Bolstered template]]
 +
* [[Dead cell]]
 +
* [[Captured cell]]
 +
* [[Domination]]
 +
 
 +
[[Category:Definition]]
 +
[[Category:Theory]]

Latest revision as of 14:56, 25 October 2025

A liberty of an isolated stone or a group is an unoccupied cell adjacent to it. A stone or group with few liberties is generally weaker than one with many liberties.

Advice for players

Old advice

An older version of this page, dated 2005, contained the following advice:

Do not play an isolated piece with three or fewer liberties. Such a move is always a bad move — it is always possible to find a better one.

The advice is not accurate. We now know that there are some situations where playing a stone with 3 liberties is the unique winning move (see here for an example). Nevertheless, it is still a useful guideline in most situations that arise in real games.

Updated advice

Below, we analyze groups with up to 3 liberties. Based on the analysis, we can formulate the following principle:

Do not play a stone that creates an interior group with 2 or fewer liberties, or with 3 liberties when two of them are adjacent or bolstered.

Analysis of groups with up to 3 liberties

Definitions

By an interior group we mean a group of friendly stones that is not adjacent to an edge.

A liberty L of a group X is said to be bolstered if it matches one of the following patterns (up to rotation and symmetry):

XXL
XL
XL

Groups with one liberty

If an interior group has only a single liberty, the group is dead. It can never become part of a shortest winning path for the group's owner.

Examples:

In both cases, the red group has a single liberty and is dead.

Groups with two adjacent liberties

If an interior group has only two liberties, and these liberties are adjacent to each other, the group is dead.

Examples:

In both cases, the red group has two adjacent liberties and is dead.

Groups with two bolstered liberties

If an interior group has two non-adjacent liberties, and both of these liberties are bolstered, the group is captured. If the group's owner plays in one liberty, the opponent can play in the other, killing the group.

Examples:

Groups with two general liberties

If an interior group has two non-adjacent liberties that are not both bolstered, then the group is not in general captured. If one of the liberties is bolstered and the other is not, playing in the non-bolstered liberty dominates playing in the bolstered one for both players.

Examples: Both players prefer "a" to "b".

ba
ba

For all groups with two general liberties, if it is the opponent's turn, they can kill the group by taking away one of the liberties. Therefore, no player should play a move that creates an interior group with only 2 liberties.

Groups with three liberties

If an interior group has 3 liberties, it is not in general captured.

Examples: Three non-adjacent liberties, zero or one of which are bolstered.

However, if two of the liberties are adjacent or bolstered, then if it is the opponent's turn, the opponent can kill the group by playing in the remaining liberty. Therefore, no player should play a move that creates an interior group with only 3 liberties of which two are adjacent or bolstered.

Examples: Two of three liberties are adjacent or bolstered. Blue can kill the red group by playing at "a".

a
a

See also

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