Useless triangle - Revision history

Selinger: Some copy-editing; fixed links; added link to capture

2021-09-12T01:47:25Z

Some copy-editing; fixed links; added link to capture

Selinger: Added categories

2020-12-28T21:53:39Z

Added categories

Selinger: Shaded the triangles in question.

2020-12-25T03:56:12Z

Shaded the triangles in question.

Selinger: Use "dead" terminology.

2020-06-13T17:27:29Z

Use "dead" terminology.

Tommah: added to explanation

2015-08-22T20:58:17Z

added to explanation

Halladba: cat. basic

2008-12-02T23:57:57Z

cat. basic

Halladba: link to dead cell

2008-10-13T22:02:22Z

link to dead cell

Roland Illig: removed dead links

2008-01-26T22:18:17Z

removed dead links

Turing: Copied/adapted page from http://hexwiki.tk

2005-02-07T18:31:14Z

Copied/adapted page from http://hexwiki.tk

New page

When a [[piece]]'s neighboring [[Hex (board element)|hex]]es are 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 [[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.

See also the page [[Equivalent patterns]].

@@ Line 1: / Line 1: @@
-When a [[piece]]'s neighboring [[hexes]] are filled by the opponent in such a way 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'''.
+When a [[piece]]'s neighboring [[hex (board element)|hexes]] are occupied by the opponent's pieces (and/or the opponent's edge) in such a way 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'''.
 <hexboard size="7x7" contents="R c5 d7 f7  B b5 c4 d4 d5 e7 S c5 b6 c6 e7 e6 f6"/>
@@ Line 6: / Line 6: @@
 In the above diagram, the red piece at c5 and the empty hexes 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 [[border row]], the piece is [[dead cell|dead]]. It 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.  Notice that any red path through c5 must pass through b6 and c6.  But b6 and c6 are connected to each other; therefore any red path through c5 would remain connected even if the red piece at c5 were removed.  Similar logic holds for the blue piece on e7.
+The important point is that the piece in a useless triangle is [[dead cell|dead]]. It 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.  Notice that any red path through c5 must pass through b6 and c6.  But b6 and c6 are connected to each other; therefore any red path through c5 would remain connected even if the red piece at c5 were removed.  Similar logic holds for the blue piece on e7.
-== See also ==
+The useless triangle is a special case of the more general concept of [[captured cell|capture]].
-[[Dead cell]]
+== See also ==
-[[Equivalent patterns]]
+* [[Dead cell]]
 [[category: Theory]]
 [[category: Intermediate Strategy]]

← Older revision		Revision as of 21:53, 28 December 2020
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	[[Equivalent patterns]]		[[Equivalent patterns]]
		+
		+	[[category: Theory]]
		+	[[category: Intermediate Strategy]]

@@ Line 2: / Line 2: @@
 When a [[piece]]'s neighboring [[hexes]] are filled by the opponent in such a way 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'''.
-<hexboard size="7x7" contents="R c5 d7 f7  B b5 c4 d4 d5 e7"/>
+<hexboard size="7x7" contents="R c5 d7 f7  B b5 c4 d4 d5 e7 S c5 b6 c6 e7 e6 f6"/>
 In the above diagram, the red piece at c5 and the empty hexes b6 and c6 form a useless triangle. The blue piece at e7 and the empty hexes e6 and f6 also form a useless triangle.

@@ Line 6: / Line 6: @@
 In the above diagram, the red piece at c5 and the empty hexes 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 [[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.  Notice that any red path through c5 must pass through b6 and c6.  But b6 and c6 are connected to each other; therefore any red path through c5 would remain connected even if the red piece at c5 were removed.  Similar logic holds for the blue piece on e7.
+The important point is that unless the piece in a useless triangle is in that player's [[border row]], the piece is [[dead cell|dead]]. It 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.  Notice that any red path through c5 must pass through b6 and c6.  But b6 and c6 are connected to each other; therefore any red path through c5 would remain connected even if the red piece at c5 were removed.  Similar logic holds for the blue piece on e7.
 == See also ==

@@ Line 1: / Line 1: @@
-<hex>R7 C7 Q1 Vc5 Hd4 Hc4 Hb5 Hd5     Vd7 He7 Vf7</hex>
+When a [[piece]]'s neighboring [[hexes]] are filled by the opponent in such a way 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'''.
 In the above diagram, the red piece at c5 and the empty hexes 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 border row, the piece has effectively been removed from the game &mdash; that is, it cannot have any effect on the rest of the game regardless of the rest of the position.
+The important point is that unless the piece in a useless triangle is in that player's [[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.  Notice that any red path through c5 must pass through b6 and c6.  But b6 and c6 are connected to each other; therefore any red path through c5 would remain connected even if the red piece at c5 were removed.  Similar logic holds for the blue piece on e7.
-−
-The useless triangle is a kind of [[dead cell]].
 == See also ==
-[[category:Basic Strategy]]
+[[Dead cell]]

← Older revision		Revision as of 23:57, 2 December 2008
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	== See also ==		== See also ==
	* [[Equivalent patterns]]		* [[Equivalent patterns]]
		+
		+	[[category:Basic Strategy]]