Parallelogram boards - Revision history

Selinger: Added a paragraph on conventions for specifying board dimension.

2023-01-25T14:39:59Z

Added a paragraph on conventions for specifying board dimension.

Selinger: Changed convention for board dimensions: columns x rows.

2023-01-24T19:39:39Z

Changed convention for board dimensions: columns x rows.

Selinger: /* Size of minimal virtual connection on parallelogram boards */ Updated table

2022-07-27T01:14:06Z

‎Size of minimal virtual connection on parallelogram boards: Updated table

Selinger: /* References */ Updated citation.

2022-07-27T01:10:11Z

‎References: Updated citation.

Selinger: /* Size of minimal virtual connection on parallelogram boards */ Fixed typo and added a data point

2021-06-02T04:44:19Z

‎Size of minimal virtual connection on parallelogram boards: Fixed typo and added a data point

Selinger: Added formulas for n=1,...,4, examples, and a citation.

2021-01-19T02:49:59Z

Added formulas for n=1,...,4, examples, and a citation.

Show changes

Selinger: Copy editing.

2021-01-10T00:58:29Z

Copy editing.

Selinger: Added category

2020-12-28T21:37:02Z

Added category

Selinger: gender agreement

2020-11-26T13:06:28Z

gender agreement

Selinger: Resolved 4x9. Lower bound is 4x8, and 4 pieces is easily achievable.

2020-10-12T04:46:07Z

Resolved 4x9. Lower bound is 4x8, and 4 pieces is easily achievable.

@@ Line 5: / Line 5: @@
    />
 The problem with playing on such parallelogram boards is that the player with the shorter distance between her sides has a simple  [[Hex_theory#Winning_strategy_for_non-square_boards|winning strategy]], even when she moves second. To mitigate this, one can permit the player with the greater distance between his sides to place a certain number of stones on the board prior to the game. In particular, it has been found that Hex on a 9×7 board is a rather fair game, when Blue may start the game with two stones at once.
 == Size of minimal virtual connection on parallelogram boards ==

@@ Line 1: / Line 1: @@
-Hex is usually played on a rhombic ''n''×''n'' board, but one can also try playing it on ''n''×''m'' parallelogram boards, where ''n'' is the number of rows, ''m'' the number of columns, and ''n'' ≠ ''m''. For example, here is a board of size 3×7:
+Hex is usually played on a rhombic ''n''×''n'' board, but one can also try playing it on ''n''×''k'' parallelogram boards, where ''n'' is the number of columns, ''k'' the number of rows, and ''n'' ≠ ''k''. For example, here is a board of size 7×3:
 <hexboard size="3x7"
    coords="none"
    edges="all"
    />
-The problem with playing on such parallelogram boards is that the player with the shorter distance between her sides has a simple  [[Hex_theory#Winning_strategy_for_non-square_boards|winning strategy]], even when she moves second. To mitigate this, one can permit the player with the greater distance between his sides to place a certain number of stones on the board prior to the game. In particular, it has been found that Hex on a 7×9 board is a rather fair game, when Blue may start the game with two stones at once.
+The problem with playing on such parallelogram boards is that the player with the shorter distance between her sides has a simple  [[Hex_theory#Winning_strategy_for_non-square_boards|winning strategy]], even when she moves second. To mitigate this, one can permit the player with the greater distance between his sides to place a certain number of stones on the board prior to the game. In particular, it has been found that Hex on a 9×7 board is a rather fair game, when Blue may start the game with two stones at once.
 == Size of minimal virtual connection on parallelogram boards ==
-On a board of size ''n''×''m'', one may ask what is the minimum number of stones Blue must place on the board prior to the game to guarantee a Blue win. Equivalently, one can ask what is the size of the minimal [[virtual connection]] between Blue's edges on an otherwise empty board.
+On a board of size ''n''×''k'', one may ask what is the minimum number of stones Blue must place on the board prior to the game to guarantee a Blue win. Equivalently, one can ask what is the size of the minimal [[virtual connection]] between Blue's edges on an otherwise empty board.
-The answer is known for certain small values of ''n'' and/or ''m'':
+The answer is known for certain small values of ''n'' and/or ''k'':
 {| class="wikitable" style="text-align: center;"
 ! ×
-! scope="column" | 1  || 2  || 3  || 4  || 5  || 6  || 7  || 8  || 9   || 10  || 11  || 12 || Formula (if known)
+! scope="column" | n=1  || 2  || 3  || 4  || 5  || 6  || 7  || 8  || 9   || 10  || 11  || 12 || Formula (if known)
 |-
-! scope="row" | 1
+! scope="row" | k=1
-| 1  || 2  || 3  || 4 || 5  || 6  || 7  || 8  || 9   || 10  || 11  || 12 || ''m''
+| 1  || 2  || 3  || 4 || 5  || 6  || 7  || 8  || 9   || 10  || 11  || 12 || ''n''
 |-
 ! scope="row" | 2
-| 0  || 1  ||2 || 2  || 3 || 4 || 4 || 5 || 6  || 6  || 7  || 8 || ⌈⅔''m'' − ⅔⌉
+| 0  || 1  ||2 || 2  || 3 || 4 || 4 || 5 || 6  || 6  || 7  || 8 || ⌈⅔''n'' − ⅔⌉
 |-
 ! scope="row" | 3
-| 0  || 0  || 1  || 2 || 3 || 3 || 4 || 5 || 5 || 6 || 7 || 7 || ⌈⅔''m'' − 1⌉
+| 0  || 0  || 1  || 2 || 3 || 3 || 4 || 5 || 5 || 6 || 7 || 7 || ⌈⅔''n'' − 1⌉
 |-
 ! scope="row" | 4
-| 0  || 0  || 0 || 1 || 2 || 2 || 3 || 4 || 4 || 5 || 6 || 6 || ⌈⅔''m'' − 2⌉
+| 0  || 0  || 0 || 1 || 2 || 2 || 3 || 4 || 4 || 5 || 6 || 6 || ⌈⅔''n'' − 2⌉
 |-
 ! scope="row" | 5
-| 0  || 0 || 0 || 0 || 1 || 2 || 2 || 3 || 3 || 4 || 5 || 5 || ⌈⅔''m'' − 3⌉, m ≥ 7
+| 0  || 0 || 0 || 0 || 1 || 2 || 2 || 3 || 3 || 4 || 5 || 5 || ⌈⅔''n'' − 3⌉, n ≥ 7
 |-
 ! scope="row" | 6
@@ Line 42: / Line 42: @@
 == Examples of minimal virtual connections ==
-=== Boards of size 1×''m'' ===
+=== Boards of size ''n''×1 ===
-For boards of size 1×''m'', it is obvious that Blue's virtual connection requires ''m'' pieces, because if any cell is left empty, Red will win in one move.
+For boards of size ''n''×1, it is obvious that Blue's virtual connection requires ''n'' stones, because if any cell is left empty, Red will win in one move.
 <hexboard size="1x7"
    edges="all"
@@ Line 51: / Line 51: @@
    />
-=== Boards of size 2×''m'' ===
+=== Boards of size ''n''×2 ===
-For boards of size 2×''m'', the size of Blue's minimal virtual connection is ⌈⅔''m'' − ⅔)⌉. Here, ⌈''x''⌉ denotes the ''ceiling'' of ''x'', i.e., the smallest integer ≥ ''x''. Examples of such minimal virtual connections are shown for ''m'' = 2, 3, 4, 5, 6, 7. The pattern continues for larger ''m''.
+For boards of size ''n''×2, the size of Blue's minimal virtual connection is ⌈⅔''n'' − ⅔)⌉. Here, ⌈''x''⌉ denotes the ''ceiling'' of ''x'', i.e., the smallest integer ≥ ''x''. Examples of such minimal virtual connections are shown for ''n'' = 2, 3, 4, 5, 6, 7. The pattern continues for larger ''n''.
 <hexboard size="2x2"
    float="inline"
@@ Line 86: / Line 86: @@
 />
-=== Boards of size 3×''m'' ===
+=== Boards of size ''n''×3 ===
-For boards of size 3×''m'', where ''m'' ≥ 3, the size of Blue's minimal virtual connection is ⌈⅔''m'' − 1⌉. Examples of such minimal virtual connections are shown for ''m'' = 3, 4, 5, 6, 7. The pattern continues for larger ''m''.
+For boards of size ''n''×3, where ''n'' ≥ 3, the size of Blue's minimal virtual connection is ⌈⅔''n'' − 1⌉. Examples of such minimal virtual connections are shown for ''n'' = 3, 4, 5, 6, 7. The pattern continues for larger ''n''.
 <hexboard size="3x3"
    float="inline"
@@ Line 116: / Line 116: @@
 />
-=== Boards of size 4×''m'' ===
+=== Boards of size ''n''×4 ===
-For boards of size 4×''m'', where ''m'' ≥ 4, the size of Blue's minimal virtual connection is ⌈⅔''m'' − 2⌉. This is proved, using tools from combinatorial game theory, in the paper [https://arxiv.org/abs/2101.06694 "On the combinatorial value of Hex positions"]. Examples of such minimal virtual connections are shown for ''m'' = 4, 5, 6, 7, 8, 9. The pattern continues for larger ''m''.
+For boards of size ''n'', where ''n'' ≥ 4, the size of Blue's minimal virtual connection is ⌈⅔''n'' − 2⌉. This is proved, using tools from combinatorial game theory, in the paper [https://arxiv.org/abs/2101.06694 "On the combinatorial value of Hex positions"]. Examples of such minimal virtual connections are shown for ''n'' = 4, 5, 6, 7, 8, 9. The pattern continues for larger ''n''.
 <hexboard size="4x4"
    float="inline"
@@ Line 150: / Line 150: @@
    contents="B c2 d3 f2 g3"
 />
-Interestingly, in addition to a virtual connection by [[bridge]]s, there is another pattern of such minimal connections when ''n'' = 4. It uses the same number of stones, but has a larger [[carrier]].
+Interestingly, in addition to a virtual connection by [[bridge]]s, there is another pattern of such minimal connections when ''k'' = 4. It uses the same number of stones, but has a larger [[carrier]].
 <hexboard size="4x4"
    float="inline"

@@ Line 28: / Line 28: @@
 |-
 ! scope="row" | 5
-| 0  || 0 || 0 || 0 || 1 || 2 || 2 || 3 || 3 ||  || ≤5 ||
+| 0  || 0 || 0 || 0 || 1 || 2 || 2 || 3 || 3 || 4 || 5 || 5 || ⌈⅔''m'' − 3⌉, m ≥ 7
 |-
 ! scope="row" | 6

@@ Line 185: / Line 185: @@
 == References ==
-* P. Selinger, [https://arxiv.org/abs/2101.06694 "On the combinatorial value of Hex positions"], 2021.
+* P. Selinger, "On the combinatorial value of Hex positions", [http://math.colgate.edu/~integers/vol22.html#g3 Integers 22:G3], 2022. Also available from [https://arxiv.org/abs/2101.06694 arxiv:2101.06694]
 [[category: Theory]]

@@ Line 19: / Line 19: @@
 |-
 ! scope="row" | 2
-| 0  || 1  ||2 || 2  || 3 || 4 || 4 || 5 || 6  || 6  || 7  || 8 || ⌈⅔''m'' − ⅔)⌉
+| 0  || 1  ||2 || 2  || 3 || 4 || 4 || 5 || 6  || 6  || 7  || 8 || ⌈⅔''m'' − ⅔⌉
 |-
 ! scope="row" | 3
@@ Line 28: / Line 28: @@
 |-
 ! scope="row" | 5
-| 0  || 0 || 0 || 0 || 1 || 2 || 2 || 3 ||  ||  || ≤5 ||
+| 0  || 0 || 0 || 0 || 1 || 2 || 2 || 3 || 3 ||  || ≤5 ||
 |-
 ! scope="row" | 6

← Older revision		Revision as of 21:37, 28 December 2020
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	\|}		\|}
		+
		+	[[category: Theory]]

← Older revision		Revision as of 13:06, 26 November 2020
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−	Hex is usually played on a rhombic n×n board, but one can also try playing it on n×m parallelogram boards, where n is the number of rows, m the number of columns, and n ≠ m. However, there is a simple [[Hex_theory#Winning_strategy_for_non-square_boards\|symmetry winning strategy]] for the player with the shorter distance between his sides, even when he moves second. To mitigate this, one can allow the player with the greater distance between ~~his~~ sides to begin the game and place a certain number of pieces at once in her first move. In particular, it has been found that Hex on a 7×9 board is a rather fair game, when the vertical player may start the game with two pieces at once.	+	Hex is usually played on a rhombic n×n board, but one can also try playing it on n×m parallelogram boards, where n is the number of rows, m the number of columns, and n ≠ m. However, there is a simple [[Hex_theory#Winning_strategy_for_non-square_boards\|symmetry winning strategy]] for the player with the shorter distance between his sides, even when he moves second. To mitigate this, one can allow the player with the greater distance between her sides to begin the game and place a certain number of pieces at once in her first move. In particular, it has been found that Hex on a 7×9 board is a rather fair game, when the vertical player may start the game with two pieces at once.

@@ Line 17: / Line 17: @@
 |-
 ! scope="row" | 4
-| 0  || 0  || 0 || 1 || 2 || 2 || 3 || 4 ||   ||  ||  ||
+| 0  || 0  || 0 || 1 || 2 || 2 || 3 || 4 || 4 ||  ||  ||
 |-
 ! scope="row" | 5