Difference between revisions of "Parallelogram boards"

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(Resolved 4x9. Lower bound is 4x8, and 4 pieces is easily achievable.)
(Added formulas for n=1,...,4, examples, and a citation.)
 
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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.
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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''. For example, here is a board of size 3×7:
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<hexboard size="3x7"
 +
  coords="none"
 +
  edges="all"
 +
  />
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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.
  
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== Size of minimal virtual connection on parallelogram boards ==
  
Number of pieces head start the vertical player needs to force a win:
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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.
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The answer is known for certain small values of ''n'' and/or ''m'':
 
{| class="wikitable" style="text-align: center;"
 
{| class="wikitable" style="text-align: center;"
 
! ×
 
! ×
! scope="column" | 1  || 2  || 3  || 4  || 5  || 6  || 7  || 8  || 9  || 10  || 11  || 12
+
! scope="column" | 1  || 2  || 3  || 4  || 5  || 6  || 7  || 8  || 9  || 10  || 11  || 12 || Formula (if known)
 
|-
 
|-
 
! scope="row" | 1
 
! scope="row" | 1
| 1  || 2  || 3  || 4 || 5  || 6  || 7  || 8  || 9  || 10  || 11  || 12
+
| 1  || 2  || 3  || 4 || 5  || 6  || 7  || 8  || 9  || 10  || 11  || 12 || ''m''
 
|-
 
|-
 
! scope="row" | 2
 
! scope="row" | 2
| 0  || 1  ||2 || 2  || 3 || 4 || 4 || 5 || 6  || 6  || 7  || 8
+
| 0  || 1  ||2 || 2  || 3 || 4 || 4 || 5 || 6  || 6  || 7  || 8 || ⌈⅔''m'' − ⅔)⌉
 
|-
 
|-
 
! scope="row" | 3
 
! scope="row" | 3
| 0  || 0  || 1  || 2 || 3 || 3 || 4 || 5 || 5 ||   ||   ||  
+
| 0  || 0  || 1  || 2 || 3 || 3 || 4 || 5 || 5 || 6 || 7 || 7 || ⌈⅔''m'' − 1⌉
 
|-
 
|-
 
! scope="row" | 4
 
! scope="row" | 4
| 0  || 0  || 0 || 1 || 2 || 2 || 3 || 4 || 4 || || ||
+
| 0  || 0  || 0 || 1 || 2 || 2 || 3 || 4 || 4 || 5 || 6 || 6 || ⌈⅔''m'' − 2⌉
 
|-
 
|-
 
! scope="row" | 5
 
! scope="row" | 5
| 0  || 0 || 0 || 0 || 1 || 2 || 2 || 3 ||  ||  || (≤)5 ||  
+
| 0  || 0 || 0 || 0 || 1 || 2 || 2 || 3 ||  ||  || ≤5 ||  
 
|-
 
|-
 
! scope="row" | 6
 
! scope="row" | 6
| 0  || 0 || 0 || 0 || 0 || 1 || 2 || 2 || 3 ||  ||  || (≤)5
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| 0  || 0 || 0 || 0 || 0 || 1 || 2 || 2 || 3 ||  ||  || ≤5
 
|-
 
|-
 
! scope="row" | 7
 
! scope="row" | 7
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| 0  || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 2  || 2?  ||  ||
 
| 0  || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 2  || 2?  ||  ||
 
|}
 
|}
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== Examples of minimal virtual connections ==
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=== Boards of size 1×''m'' ===
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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.
 +
<hexboard size="1x7"
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  edges="all"
 +
  coords="none"
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  contents="B all"
 +
  />
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=== Boards of size 2×''m'' ===
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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''.
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<hexboard size="2x2"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b1"
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/><hexboard size="2x3"
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  float="inline"
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  edges="all"
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  coords="none"
 +
  contents="B b1 c2"
 +
/><hexboard size="2x4"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b1 c2"
 +
/><hexboard size="2x5"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b1 c2 e1"
 +
/><hexboard size="2x6"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b1 c2 e1 f2"
 +
/><hexboard size="2x7"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b1 c2 e1 f2"
 +
/>
 +
 +
=== Boards of size 3×''m'' ===
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 +
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''.
 +
<hexboard size="3x3"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b2"
 +
/><hexboard size="3x4"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b2 d1"
 +
/><hexboard size="3x5"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b2 d1 e2"
 +
/><hexboard size="3x6"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b2 d1 e2"
 +
/><hexboard size="3x7"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b2 d1 e2 g1"
 +
/>
 +
 +
=== Boards of size 4×''m'' ===
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 +
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''.
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<hexboard size="4x4"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B c2"
 +
/><hexboard size="4x5"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B c2 d3"
 +
/><hexboard size="4x6"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B c2 d3"
 +
/><hexboard size="4x7"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B c2 d3 f2"
 +
/><hexboard size="4x8"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B c2 d3 f2 g3"
 +
/><hexboard size="4x9"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  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]].
 +
<hexboard size="4x4"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b3"
 +
/><hexboard size="4x5"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b3 e2"
 +
/><hexboard size="4x6"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b3 e2"
 +
/><hexboard size="4x7"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b3 e2 e3"
 +
/><hexboard size="4x8"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b3 e2 e3 h2"
 +
/><hexboard size="4x9"
 +
  float="inline"
 +
  edges="all"
 +
  coords="none"
 +
  contents="B b3 e2 e3 h2"
 +
/>
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 +
== References ==
 +
 +
* P. Selinger, [https://arxiv.org/abs/2101.06694 "On the combinatorial value of Hex positions"], 2021.
 +
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[[category: Theory]]

Latest revision as of 02:49, 19 January 2021

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 nm. For example, here is a board of size 3×7:

The problem with playing on such parallelogram boards is that the player with the shorter distance between her sides has a simple 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.

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.

The answer is known for certain small values of n and/or m:

× 1 2 3 4 5 6 7 8 9 10 11 12 Formula (if known)
1 1 2 3 4 5 6 7 8 9 10 11 12 m
2 0 1 2 2 3 4 4 5 6 6 7 8 ⌈⅔m − ⅔)⌉
3 0 0 1 2 3 3 4 5 5 6 7 7 ⌈⅔m − 1⌉
4 0 0 0 1 2 2 3 4 4 5 6 6 ⌈⅔m − 2⌉
5 0 0 0 0 1 2 2 3 ≤5
6 0 0 0 0 0 1 2 2 3 ≤5
7 0 0 0 0 0 0 1 2 2
8 0 0 0 0 0 0 0 1 2 2?

Examples of minimal virtual connections

Boards of size 1×m

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.

Boards of size 2×m

For boards of size 2×m, the size of Blue's minimal virtual connection is ⌈⅔m − 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.

Boards of size 3×m

For boards of size 3×m, where m ≥ 3, the size of Blue's minimal virtual connection is ⌈⅔m − 1⌉. Examples of such minimal virtual connections are shown for m = 3, 4, 5, 6, 7. The pattern continues for larger m.

Boards of size 4×m

For boards of size 4×m, where m ≥ 4, the size of Blue's minimal virtual connection is ⌈⅔m − 2⌉. This is proved, using tools from combinatorial game theory, in the paper "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.

Interestingly, in addition to a virtual connection by bridges, there is another pattern of such minimal connections when n = 4. It uses the same number of stones, but has a larger carrier.

References