Dead cell - Revision history

Selinger: Added a dead cell pattern.

2025-01-11T00:53:27Z

Added a dead cell pattern.

Selinger: typo

2023-05-25T16:30:37Z

typo

Selinger: /* Usage */ added a link

2023-05-25T16:28:54Z

‎Usage: added a link

Selinger: typo

2023-05-25T16:22:53Z

typo

Selinger: typo

2021-03-19T01:09:08Z

typo

Demer: /* Interaction between multiple dead cells */ made colors have same number of pieces

2021-01-03T21:41:29Z

‎Interaction between multiple dead cells: made colors have same number of pieces

Selinger: Converted to new hexboard diagrams

2020-12-09T02:34:09Z

Converted to new hexboard diagrams

Selinger: Added definition of "alive". Minor copy-editing.

2020-07-06T17:26:38Z

Added definition of "alive". Minor copy-editing.

Selinger: 2x2 is less confusing than 1x1

2020-06-15T04:28:23Z

2x2 is less confusing than 1x1

Selinger: minor typo.

2020-06-15T04:20:13Z

minor typo.

@@ Line 30: / Line 30: @@
    visible="-a1 b1 b3 c3"
    contents="B a2 a3 R c1 c2 E *:b2"
    />

@@ Line 102: / Line 102: @@
 An extreme example of this is the position where the entire Hex board is filled with red pieces. This is obviously a win for Red. Also, assuming the board size is at least 2x2, every single cell on the board is dead. However, if we were to change all of them to blue, it would clearly change the winner of the game.
-However, if a dead cell is currently empty, they it will remain dead no matter what colour it or other empty dead cells are filled with. In particular, a set of several empty dead cells is captured, as a whole, by both players.
+However, if a dead cell is currently empty, then it will remain dead no matter what colour it or other empty dead cells are filled with. In particular, a set of several empty dead cells is captured, as a whole, by both players.
 == See Also ==

@@ Line 82: / Line 82: @@
 It is never advantageous to move in a dead empty cell. It is also never advantageous to move in a cell that the opponent can kill (i.e., turn into a dead cell) with the next move. Such a cell is called ''vulnerable'' for the player who should not move there.
-This concept can be useful in determining which side of a [[bridge]] it is better to intrude upon. In the following example, b3 is vulnerable for Blue. If Blue plays b3, Red can respond at c3, killing b3. Since b3 is now dead, it can be treated as a red piece. Effectively, Blue has gained nothing, and Red has gained two new pieces at b3 and c3. It follows that if Blue wants to intrude into the bridge, she should do so at c3.
+This concept can be useful in determining which side of a [[bridge]] it is better to intrude upon. In the following example, b3 is vulnerable for Blue. If Blue plays b3, Red can respond at c3, killing b3. Since b3 is now dead, it can be treated as a red piece. Effectively, Blue has gained nothing, and Red has gained two new pieces at b3 and c3. It follows that if Blue wants to intrude into the bridge, she should do so at c3. See also the article on [[bolstered template]]s.
 <hex>R5 C5 Q1 Ha3 Vb4 Vc2</hex>

@@ Line 71: / Line 71: @@
 == Usage ==
-Because the colour does not affect the result of the game, dead cells can be treated as if they contained a blue piece or a red piece, without changing the strategic value of the position. This often simplifies the analysis of Hex positions. In particular, dead cells as [[captured cell|captured]] by both players.
+Because the colour does not affect the result of the game, dead cells can be treated as if they contained a blue piece or a red piece, without changing the strategic value of the position. This often simplifies the analysis of Hex positions. In particular, dead cells are [[captured cell|captured]] by both players.
 For example, in the following hypothetical situation (with Red to move), Red might wonder whether the piece at e2 could somehow be useful as a [[ladder escape]].

← Older revision		Revision as of 01:09, 19 March 2021
Line 1:		Line 1:
−	A dead cell is a cell whose colour does not affect the outcome of the game. More formally, in a given Hex position, a cell is ''dead'' if for every way of filling all empty cells of the board with red and ~~black~~ pieces, the winner remains the same when the colour of the dead cell is changed from red to blue or vice versa.	+	A dead cell is a cell whose colour does not affect the outcome of the game. More formally, in a given Hex position, a cell is ''dead'' if for every way of filling all empty cells of the board with red and blue pieces, the winner remains the same when the colour of the dead cell is changed from red to blue or vice versa.

	A cell that is not dead is ''alive''.		A cell that is not dead is ''alive''.

@@ Line 96: / Line 96: @@
 While the colour of a dead cell does not affect whether a position is winning or losing, it can affect whether other cells are dead. For example, in the following position, both c3 and d3 are dead.
-<hexboard size="4x6" coords="show" contents="B a4 B b4 B d4 B e4 R c2 R d2 R e2 R c3 R d3"/>
+<hexboard size="4x6" coords="show" contents="B a4 b4 d4 e4 f4 R c2 c3 d2 d3 e2"/>
 Therefore we can colour c3 blue without changing the strategic value of the position. Alternatively, we can colour d3 blue without changing its value. However, we cannot colour ''both'' c3 and d3 blue, as this would actually change the outcome of the game. In other words, if we change the colour of c3, then d3 is no longer dead, and vice versa.

@@ Line 7: / Line 7: @@
 [[Useless triangle]]s are examples of dead cells.
-In the following [[pattern]]s, the dead cell is labelled with a star.
+In the following [[pattern]]s, the dead cell is labelled with a star. Some patterns include empty cells; however, these cells have only been included for context; they do not need to be empty. They can be occupied by stones of either color or even be outside the board.
-<hex>R4 C5 Q0
-    Hc2 Hd2
- Hb3 Sc3 Hd3</hex>
-<hex>R5 C5 Q0
+=== Dead cells in the interior ===
-   Hc2 Hd2
-    Sc3 Hd3
- Vb4</hex>
-<hex>R5 C5 Q0
+<hexboard size="2x3"
-        Hd2
+  coords="none"
- Vb3 Sc3 Hd3
+  edges="none"
-   Vb4</hex>
+  visible="-a1"
-<hex>R5 C5 He2 Hd3 Se3</hex>
+<hexboard size="3x3"
-<hex>R5 C5 He2 Se3 He4</hex>
+<hexboard size="3x3"
-<hex>R5 C5 He2 Se3 Vd4</hex>
+=== Dead cells near edges and corners ===
-<hex>R5 C5 Se3 Vd3 Vd4</hex>
+<hexboard size="3x4"
-<hex>R5 C5 He4 Se5</hex>
+<hexboard size="2x5"
 == Usage ==

@@ Line 1: / Line 1: @@
-A dead cell is a cell whose colour does not affect the outcome of the game.
+A dead cell is a cell whose colour does not affect the outcome of the game. More formally, in a given Hex position, a cell is ''dead'' if for every way of filling all empty cells of the board with red and black pieces, the winner remains the same when the colour of the dead cell is changed from red to blue or vice versa.
-More formally, in a given Hex position, a cell is ''dead'' if for every way of filling all empty cells of the board with red and black pieces, the winner remains the same when the colour of the dead cell is changed from red to blue or vice versa.
+A cell that is not dead is ''alive''.
 [[Useless triangle]]s are examples of dead cells.
@@ Line 41: / Line 43: @@
 However, the cell e2 is dead, and therefore the position is strategically equivalent to the one where e2 is blue. It follows that e2 cannot possibly be useful to Red as a ladder escape, or for any other purpose.
-It is never advantageous to move in a dead cell. It is also never advantageous to move in a cell that the opponent can kill (i.e., turn into a dead cell) with the next move. Such a cell is called ''vulnerable'' for the player who should not move there.
+It is never advantageous to move in a dead empty cell. It is also never advantageous to move in a cell that the opponent can kill (i.e., turn into a dead cell) with the next move. Such a cell is called ''vulnerable'' for the player who should not move there.
-This concept can be useful in determining which side of a [[bridge]] it is better to intrude upon. In the following example, b3 is vulnerable for Blue. If Blue plays b3, Red can respond at c3, killing b3. Since b3 is now dead, it can be treated as a red piece. Effectively, Blue has gained nothing, and Red has gained two new pieces at b3 and c3. It follows that if Blue wants to invade the bridge, she should do so at c3.
+This concept can be useful in determining which side of a [[bridge]] it is better to intrude upon. In the following example, b3 is vulnerable for Blue. If Blue plays b3, Red can respond at c3, killing b3. Since b3 is now dead, it can be treated as a red piece. Effectively, Blue has gained nothing, and Red has gained two new pieces at b3 and c3. It follows that if Blue wants to intrude into the bridge, she should do so at c3.
 <hex>R5 C5 Q1 Ha3 Vb4 Vc2</hex>

@@ Line 63: / Line 63: @@
 An extreme example of this is the position where the entire Hex board is filled with red pieces. This is obviously a win for Red. Also, assuming the board size is at least 1x2, every single cell on the board is dead. However, if we were to change all of them to blue, it would clearly change the winner of the game.
-However, if a dead cells is currently empty, they it will remain dead no matter what colour it or other empty dead cells are filled with. In particular, a set of several empty dead cells is captured, as a whole, by both players.
+However, if a dead cell is currently empty, they it will remain dead no matter what colour it or other empty dead cells are filled with. In particular, a set of several empty dead cells is captured, as a whole, by both players.
 == See Also ==