Wheel - Revision history

Selinger: Updated witness for intrusion B.

2025-10-25T12:58:43Z

Updated witness for intrusion B.

Selinger: Added Hexanna's example from the talk page.

2023-04-18T17:33:24Z

Added Hexanna's example from the talk page.

Selinger: Fixed an error. See talk page.

2023-04-17T18:57:09Z

Fixed an error. See talk page.

Selinger: Added a link.

2022-11-27T16:07:51Z

Added a link.

Selinger: Converted to new hexboard diagrams

2020-12-07T04:51:06Z

Converted to new hexboard diagrams

Selinger: Copy-editing; clarified that empty cells are part of template.

2020-07-06T21:28:36Z

Copy-editing; clarified that empty cells are part of template.

Halladba: added to category interior templates + created a title for u-turn

2007-12-09T20:33:00Z

added to category interior templates + created a title for u-turn

Taral at 13:27, 28 March 2005

2005-03-28T13:27:01Z

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

2005-02-10T07:16:35Z

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

New page

The '''wheel''' consists of three [[Friendly piece|same-coloured pieces]] around a central point.

<hex>R5 C5 Vc2 Vd3 Vb4 Sd2 Sb3 Sc4</hex>

The red pieces are strongly [[Connection|connected]] to each other. [[Blue (player)|Blue]] cannot break the connection between them, and his available forcing moves are not very good. It also has a big circumference, so that it is easy to connect to it from other parts of the board. The drawback is that it requires three pieces and is rather compact, so that it is often not efficient enough.

Blue has three possible [[forcing move]]s, which are marked by stars. If Blue plays in one of them, [[Red (player)|Red]] can restore the connection by playing in any of the other two starred cells. (He can also play at the midpoint, but it's usually not so good.) By doing this, Red strengthens his connection, and the influence his pieces has on the rest of the board. It is therefore usually better for Blue not to [[Intrusion|invade]].

Blue should never attempt to invade by playing at the midpoint. Red can then [[Restoring|restore]] the connection between his pieces by playing in any starred cell. Red's position improves, while Blue's piece is useless.

Note that the position below, where Blue has already intruded into one of the starred cells, is a very weak one. In this case, Blue should play at the midpoint, which is now the point where two [[bridge]]s overlap.

<hex>R5 C5 Vc2 Vd3 Vb4 Hc4</hex>

@@ Line 27: / Line 27: @@
 ==The intrusion at B==
-While in most cases, it is not a good idea to [[intrusion|intrude]] into the wheel at "B", there are some positions where a move at "B" is winning and all of the moves at "A" are losing. Such positions are quite rare. Here is an example, with Blue to move.
+While in most cases, it is not a good idea to [[intrusion|intrude]] into the wheel at "B", there are some positions where a move at "B" is winning and all other moves are losing. Such positions are quite rare. Here is an example, with Blue to move.
-<hexboard size="5x5"
+<hexboard size="7x7"
-  coords="show"
+   contents="B a1--e1 g1 a2 b2 a3 f3 e4 b5 c5 b6 c6 d6 g6 b7 c7 g7--e7 f5 R f1 d2 g2--g5 e3 a4--a7 d7 c4 d5
-  edges="show"
+            E A:(c3 e2 d4) B:d3 S area(c3,c4,d4,e3,e2,d2)"
-   contents="R d1 e2 c3 e5 B c1 b3 b5 E B:d2 *:d4 A:(e1 c2 d3) S area(d1,c2,c3,d3,e2,e1)"
    />
-Note that "B" is the only winning move inside the wheel. However, there is also another winning move for Blue outside the wheel, at "*".
+Note that "B" is the only winning move, both inside and outside the wheel. This position comes from the paper "On 3-terminal positions in Hex".
 [[category:interior templates]]

@@ Line 24: / Line 24: @@
    />
 If only two of the cells marked "A" are empty, and not all three of them, it is not a wheel and it is not a template. The U-turn is a very weak position. The U-turn sometimes arises when a player tries to connect via [[bridge]]s and takes a sharp turn. The problem is that  Blue can play at B, leaving Red unable to defend both bridges.
 [[category:interior templates]]

@@ Line 11: / Line 11: @@
 Blue has three possible [[forcing move]]s, labelled A. If Blue plays in one of them, [[Red (player)|Red]] can restore the connection by playing in any of the other cells labelled A. (She can also play at B, but it's usually not as good.) By doing this, Red strengthens her connection, and the influence her pieces have on the rest of the board. It is therefore usually better for Blue not to [[Intrusion|intrude]].
-Blue should never attempt to intrude by playing at B. Red can then restore the connection between her pieces by playing at any A. Red's position improves, while Blue's piece is not very useful. Although Blue still has to opportunity to intrude in one of the remaining cells marked A, it would have been better for Blue to do this in the first place.
+Blue should not usually attempt to intrude by playing at B. Red can then restore the connection between her pieces by playing at any A. Red's position improves, while Blue's piece is not very useful. Although Blue still has to opportunity to intrude in one of the remaining cells marked A, it would have been better for Blue to do this in the first place.
 ==U-turn==

@@ Line 7: / Line 7: @@
    />
-The red pieces are strongly [[Connection|connected]] to each other. [[Blue (player)|Blue]] cannot break the connection between them, and his available forcing moves are not very good. The wheel also has a large circumference, so that it is easy to connect to it from other parts of the board. The drawback is that it requires three pieces and is rather compact, so that it is often not efficient enough to cover a large area.
+The red pieces are strongly [[Connection|connected]] to each other. [[Blue (player)|Blue]] cannot break the connection between them, and his available forcing moves are not very good. The wheel also has a large circumference, so that it is easy to connect to it from other parts of the board. The drawback is that it requires three pieces and is rather compact, so that it is often not [[efficiency|efficient]] enough to cover a large area.
 Blue has three possible [[forcing move]]s, labelled A. If Blue plays in one of them, [[Red (player)|Red]] can restore the connection by playing in any of the other cells labelled A. (She can also play at B, but it's usually not as good.) By doing this, Red strengthens her connection, and the influence her pieces have on the rest of the board. It is therefore usually better for Blue not to [[Intrusion|intrude]].

@@ Line 1: / Line 1: @@
-The '''wheel''' consists of three same-coloured pieces around a central point, along with the four empty cells marked "A" and "B".
+The '''wheel''' is a 3-stone [[interior template]]. It consists of three same-coloured pieces around a central point, along with the four empty cells marked "A" and "B".
-[[Image:Wheel.png]]
+The red pieces are strongly [[Connection|connected]] to each other. [[Blue (player)|Blue]] cannot break the connection between them, and his available forcing moves are not very good. The wheel also has a large circumference, so that it is easy to connect to it from other parts of the board. The drawback is that it requires three pieces and is rather compact, so that it is often not efficient enough to cover a large area.
-<!--<hex>R5 C5 Vc2 Vd3 Vb4 Sd2 Sb3 Sc4</hex>-->
+Blue has three possible [[forcing move]]s, labelled A. If Blue plays in one of them, [[Red (player)|Red]] can restore the connection by playing in any of the other cells labelled A. (She can also play at B, but it's usually not as good.) By doing this, Red strengthens her connection, and the influence her pieces have on the rest of the board. It is therefore usually better for Blue not to [[Intrusion|intrude]].
-The red pieces are strongly [[Connection|connected]] to each other. [[Blue (player)|Blue]] cannot break the connection between them, and his available forcing moves are not very good. It also has a big circumference, so that it is easy to connect to it from other parts of the board. The drawback is that it requires three pieces and is rather compact, so that it is often not efficient enough.
+Blue should never attempt to intrude by playing at B. Red can then restore the connection between her pieces by playing at any A. Red's position improves, while Blue's piece is not very useful. Although Blue still has to opportunity to intrude in one of the remaining cells marked A, it would have been better for Blue to do this in the first place.
-−
-Blue has three possible [[forcing move]]s, labelled A. If Blue plays in one of them, [[Red (player)|Red]] can restore the connection by playing in any of the other cells labelled A. (He can also play at B, but it's usually not so good.) By doing this, Red strengthens his connection, and the influence his pieces has on the rest of the board. It is therefore usually better for Blue not to [[Intrusion|invade]].
-−
-Blue should never attempt to invade by playing at B. Red can then [[Restoring|restore]] the connection between his pieces by playing at any A. Red's position improves, while Blue's piece is useless.
 ==U-turn==
-Note that the position below, where Blue has already intruded into one of the starred cells, is a very weak one. In this case, Blue should play at the midpoint, which is now the point where two [[bridge]]s overlap.
+The wheel should never be confused with the ''U-turn'', shown here:
-<hex>R5 C5 Vc2 Vd3 Vb4 Hc4</hex>
+<hexboard size="3x3"
 [[category:interior templates]]

← Older revision		Revision as of 21:28, 6 July 2020
Line 1:		Line 1:
−	The '''wheel''' consists of three ~~[[Friendly piece\|~~same-coloured pieces]] around a central point.	+	The '''wheel''' consists of three same-coloured pieces around a central point, along with the four empty cells marked "A" and "B".

	[[Image:Wheel.png]]		[[Image:Wheel.png]]