Difference between revisions of "Swap rule"

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Because the [[first player]] has a considerable advantage if he is allowed to make his first move without restrictions, the '''swap rule''' was devised. It states that one player first makes a move, and then the [[second player]] decides who plays with which colour. The swap rule is also called the "Pie rule", since it resembles the ''You cut, I choose'' rule when sharing a pie between two children. The swap rule can be implemented in two ways, as follows. (Assume that the colours are red and blue, with red moving first.)
+
The '''swap rule''' states that after Red plays the first move, Blue decides whether to swap colours or not. If Blue swaps colours, Blue becomes Red and Red becomes Blue. Whichever player ends up being Blue makes the second move and then the game continues as usual.
  
−
# The first player places a red piece in any hex on the board. Then the other player can either make a move with the blue pieces, in which case he becomes blue, or he can state that he wants to be red. After this the game continues without any more swapping.
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== Reason for the swap rule ==
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# The first player makes a move with the red pieces. The other player either makes a move with the blue pieces, or makes a [[piece swap]]. This means to remove the red piece from the board, and place a blue piece at the hexagon which is the mirror image of the hexagon in which the red piece was placed, with respect to the board's long diagonal.
+
  
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In face-to-face play the first option is most practical, since it is easier to change colours of the players that removing and adding pieces on the board. It is also less error-prone. On [[Online playing|game sites]] on the Internet the second version is more common, presumably because the colours are determined at the start of the game, and it is easier to change the board position than the colour designation.
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When playing Hex without the swap rule, the [[first player]] has a considerable advantage. The swap rule was devised to make the game more even. Namely, if the first player plays a move that is too strong, the second player will swap and be in a strong position. And if the first player plays a move that is too weak, the second player will not swap (and therefore also be in a strong position). Therefore, the swap rule creates an incentive for the first player to play a move that is as fair as possible.
  
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When playing with the swap rule, the second player theoretically has a forced win. However, the second player's advantage is much smaller than the advantage of being the first player when playing without swap. Moreover, in either cases, the first player can study the move that he'll play before the game even starts. This provides an incentive for playing as the first player even when the game is played with the swap rule.
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The swap rule is sometimes also called the "pie rule", since it resembles the well-known "you cut, I choose" method for fairly dividing a pie between two people. Namely, one person cuts the pie in two, and the other person chooses which piece to eat. Here, the incentive for the first person is to make the two pieces as equal as possible.
  
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== Hex combined with swap rule ==
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Since each opening move is theoretically either winning or losing, there exists no opening move that is exactly fair. For this reason, the second player theoretically has a forced win when playing with the swap rule. However, the second player's advantage is very small, and certainly much smaller than the first player's advantage would be when playing without the swap rule.
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According to some strong hex players, when a position is roughly balanced, having an extra stone on the board usually makes the game easier to play. This means that whenever one is in doubt about a move, it's a good idea to swap it. The first player should also take this into account by playing a first move that is probably losing under perfect play. This idea has been popularized by "lazplayer" at littlegolem/igg.
+
  
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== A more general swap rule ==
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== Implementations of the swap rule ==
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Instead of placing just one piece, the first player can place any number of red and blue pieces, and state which color has the next move. Let's assume that the first player has placed N stones in the board. The second player then can start playing hex with one color of his choice or, if he fears that the other player has an excessive advantage due to home preparation, he can swap roles with first player, remove all stones from the board and place at most N-1 stones as he wants. This rule has been first proposed by "lazplyayer" at littlegolem/igg, and it seems to be an improvement over the "simple" swap rule in current use.
+
  
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==The rule==
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Swapping can be implemented in two ways, as follows. As usual, we assume that the players are Red and Blue, with Red going first.
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After [[Red]] played the first move [[Blue]] player is asked wether she wants to swap or not, that is to invert the colors or to keep on this way.
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[[Swap_rule|Detailled article]].
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==Where to swap==
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# '''"Swap sides":''' The players perform the swap by switching colours: Red becomes Blue and Blue becomes Red. After the swap, it is Blue's turn.
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The use of the swap rule is to force the first player to play not too good moves. Therefore it is intersting to find out what are the moves that should be swapped (the best moves), what are moves that should not be swapped (the worse moves), what are the moves for which no answer is known yet.
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# '''"Swap pieces":''' The players perform the swap by switching pieces. This means the initial red piece is replaced by a blue piece in the mirror image position, where the mirroring takes place with respect to the board's long diagonal. For example, a red piece at a3 becomes a blue piece at c1. The players do not switch colours: Red stays Red and Blue stays Blue. After the swap, it is Red's turn.
  
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On every size [[a1_opening|a1]], b1 and symmetrical moves are known losing moves (except 2 x 2, where b1 wins, see [[small boards]]).
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In face-to-face play, the "swap sides" method is most practical, since it is easier for the players to switch colours than to remove and add pieces on the board. This is especially true when playing with pencil and paper. It is also less error-prone. On [[Online playing|online game sites]], the "swap pieces" method is more common, presumably because the colours are determined at the start of the game, and it is easier to change the board position than the colour designation. This also makes the game record more readable, since it is always clear which player was "Red" and which was "Blue".
  
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*The red marked hexes are to be swapped.
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== When to swap ==
  
−
*The blue marked hexes are not to be swapped.
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The decision whether to swap is an important one. Accidentally swapping a weak move, or accidentally failing to swap a strong move, is bad for the second player.  
  
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*The star marked hexes are average moves, so the game should be balanced with or without swap.  
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Different players have different preferences for which moves to swap. It is generally agreed that moves near the center of the board are far too strong and should be swapped, whereas moves on the first player's own edge (except in the obtuse corner) are too weak and should not be swapped. a1 and b1 are provably losing and should never be swapped. a3–a8, a10, a11, c2, and c10 (on an 11 × 11 board) are relatively balanced, and whether or not to swap them depends on the player's preferences. a9 is fairly weak and should probably not be swapped.  
  
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Hex is not strongly solved on big sizes, therefor the result of [[optimal play]] is not known for every cells.  
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To get an idea of which opening moves are winning, it is useful to study the situation for [[small boards]]. For boards up to size 9 × 9, the winning opening moves are known. While the winning opening moves have not been solved for boards of size 10 × 10 or greater, it is reasonable to extrapolate from the smaller board sizes. On the [[small boards]], the red hexes should be swapped, and the blue hexes should not be swapped. Fair opening moves are probably the ones that are near the boundary of red and blue.  
  
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=== Small Sizes ===
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Some strong players have suggested that when a position is roughly balanced, having an extra stone on the board usually makes the game easier to play. This means that whenever there is doubt about a move, it might be a good idea to swap it. However, the first player can take advantage of this behaviour by playing a first move that is probably losing under perfect play. This idea has been popularized by "lazplayer" at littlegolem/igg.
  
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See [[Small boards]] to know which cells lead to victory with optimal play. You might however assume that you and your opponent do not play optimally, and decide to play tricky losing moves in order to trap your opponent !
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Another consideration is not to use the same opening move all the time. Some players may be very familiar with particular openings. Playing an unfamiliar opening can confuse the opponent.
  
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=== Bigger Sizes ===
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In the following diagrams, it is suggested that an opening move played in a red cell should be swapped; those in a blue cell should not be swapped; and those in a cell marked "*" are relatively balanced. However, these diagrams reflect someone's subjective preference. Many players have different ideas.
  
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The [[Theory|theoretical]] outcome is known for a few cells (for instance a1). For the other cells, stronger players' advice can give a hint on whether a move should be swapped or not. One could also try to recognize a pattern in the winning cells for small boards and extrapolate to bigger sizes.
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=== Size 11 ===
 +
<hex>R11 C11 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Hi1 Hj1 Vk1
 +
              Sa2 Sb2 Sc2 Vd2 Ve2 Vf2 Hg2 Hh2 Hi2 Vj2 Vk2
 +
              Sa3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Vi3 Vj3 Hk3
 +
                Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4 Vj4 Vk4
 +
                Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5 Vj5 Vk5
 +
                  Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6 Vj6 Vk6
 +
                  Va7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Vi7 Vj7 Vk7
 +
                    Va8 Vb8 Vc8 Vd8 Ve8 Vf8 Vg8 Vh8 Vi8 Vj8 Vk8
 +
                    Ha9 Vb9 Vc9 Vd9 Ve9 Vf9 Vg9 Vh9 Vi9 Vj9 Sk9
 +
                      Va10 Vb10 Hc10 Hd10 He10 Vf10 Vg10 Vh10 Si10 Sj10 Sk10
 +
                      Va11 Hb11 Hc11 Hd11 He11 Hf11 Hg11 Hh11 Hi11 Hj11 Hk11
 +
                       
 +
</hex>
  
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====Size 10====
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=== Size 13 ===
−
<hex>R10 C10 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Hi1 Vj1
+
−
              Sa2 Sb2 Sc2 Vd2 Ve2 Vf2 Vg2 Vh2 Vi2 Vj2
+
−
              Sa3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Vi3 Vj3
+
−
                Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4 Vj4
+
−
                Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5 Vj5
+
−
                  Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6 Vj6
+
−
                  Va7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Vi7 Vj7
+
−
                    Va8 Vb8 Vc8 Vd8 Ve8 Vf8 Vg8 Vh8 Vi8 Sj8
+
−
                    Va9 Vb9 Vc9 Vd9 Ve9 Vf9 Vg9 Sh9 Si9 Sj9
+
−
                      Va10 Hb10 Hc10 Hd10 He10 Hf10 Hg10 Hh10 Hi10 Hj10</hex>
+
−
 
+
−
==== Size 14 ====
+
 
TODO
 
TODO
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==== Size 19 ====
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=== Size 14 ===
 
TODO
 
TODO
 +
 +
== A more general swap rule ==
 +
 +
It has been suggested that one could use a more general swap rule. Under this proposal, instead of placing just one piece, the first player places any number of red and blue pieces, and state which color has the next move. Let's assume that the first player has placed N stones in the board. The second player then can start playing Hex with one color of his choice or, if he fears that the other player has an excessive advantage due to home preparation, he can swap roles with the first player, remove all stones from the board and place at most N-1 stones as he wants. This rule has been first proposed by "lazplyayer" at littlegolem/igg. However, it is not widely used, nor implemented by any [[online play|game sites]].
 +
 
==See also==
 
==See also==
  
−
[[Basic (strategy guide)#10_.C3.97_10_swap_rules|Guideline for the 10x10]], in the basic strategy guide.
+
[[Basic (strategy guide)#10_.C3.97_10_swap_rules|Guideline for 10x10 board size]], in the basic strategy guide.
  
 
==External links==
 
==External links==
  
−
[http://www.cs.cmu.edu/People/hde/hex/hexfaq/ A faq about Hex]
+
[http://www.cs.cmu.edu/People/hde/hex/hexfaq/ A FAQ about Hex]
  
−
[http://www.cs.ualberta.ca/~queenbee/openings.html A more complete site] with solution to sizes 7. Beware, the colors are inverted, vertical is blue there.
+
[http://www.cs.ualberta.ca/~queenbee/openings.html A more complete site] with solutions to size 7. Beware, the colours are inverted, vertical is blue there.
  
 
[[category:opening]]
 
[[category:opening]]
 
[[category:Basic Strategy]]
 
[[category:Basic Strategy]]
 
[[Category:Definition]]
 
[[Category:Definition]]

Revision as of 05:12, 5 December 2020

The swap rule states that after Red plays the first move, Blue decides whether to swap colours or not. If Blue swaps colours, Blue becomes Red and Red becomes Blue. Whichever player ends up being Blue makes the second move and then the game continues as usual.

Reason for the swap rule

When playing Hex without the swap rule, the first player has a considerable advantage. The swap rule was devised to make the game more even. Namely, if the first player plays a move that is too strong, the second player will swap and be in a strong position. And if the first player plays a move that is too weak, the second player will not swap (and therefore also be in a strong position). Therefore, the swap rule creates an incentive for the first player to play a move that is as fair as possible.

The swap rule is sometimes also called the "pie rule", since it resembles the well-known "you cut, I choose" method for fairly dividing a pie between two people. Namely, one person cuts the pie in two, and the other person chooses which piece to eat. Here, the incentive for the first person is to make the two pieces as equal as possible.

Since each opening move is theoretically either winning or losing, there exists no opening move that is exactly fair. For this reason, the second player theoretically has a forced win when playing with the swap rule. However, the second player's advantage is very small, and certainly much smaller than the first player's advantage would be when playing without the swap rule.

Implementations of the swap rule

Swapping can be implemented in two ways, as follows. As usual, we assume that the players are Red and Blue, with Red going first.

  1. "Swap sides": The players perform the swap by switching colours: Red becomes Blue and Blue becomes Red. After the swap, it is Blue's turn.
  2. "Swap pieces": The players perform the swap by switching pieces. This means the initial red piece is replaced by a blue piece in the mirror image position, where the mirroring takes place with respect to the board's long diagonal. For example, a red piece at a3 becomes a blue piece at c1. The players do not switch colours: Red stays Red and Blue stays Blue. After the swap, it is Red's turn.

In face-to-face play, the "swap sides" method is most practical, since it is easier for the players to switch colours than to remove and add pieces on the board. This is especially true when playing with pencil and paper. It is also less error-prone. On online game sites, the "swap pieces" method is more common, presumably because the colours are determined at the start of the game, and it is easier to change the board position than the colour designation. This also makes the game record more readable, since it is always clear which player was "Red" and which was "Blue".

When to swap

The decision whether to swap is an important one. Accidentally swapping a weak move, or accidentally failing to swap a strong move, is bad for the second player.

Different players have different preferences for which moves to swap. It is generally agreed that moves near the center of the board are far too strong and should be swapped, whereas moves on the first player's own edge (except in the obtuse corner) are too weak and should not be swapped. a1 and b1 are provably losing and should never be swapped. a3–a8, a10, a11, c2, and c10 (on an 11 × 11 board) are relatively balanced, and whether or not to swap them depends on the player's preferences. a9 is fairly weak and should probably not be swapped.

To get an idea of which opening moves are winning, it is useful to study the situation for small boards. For boards up to size 9 × 9, the winning opening moves are known. While the winning opening moves have not been solved for boards of size 10 × 10 or greater, it is reasonable to extrapolate from the smaller board sizes. On the small boards, the red hexes should be swapped, and the blue hexes should not be swapped. Fair opening moves are probably the ones that are near the boundary of red and blue.

Some strong players have suggested that when a position is roughly balanced, having an extra stone on the board usually makes the game easier to play. This means that whenever there is doubt about a move, it might be a good idea to swap it. However, the first player can take advantage of this behaviour by playing a first move that is probably losing under perfect play. This idea has been popularized by "lazplayer" at littlegolem/igg.

Another consideration is not to use the same opening move all the time. Some players may be very familiar with particular openings. Playing an unfamiliar opening can confuse the opponent.

In the following diagrams, it is suggested that an opening move played in a red cell should be swapped; those in a blue cell should not be swapped; and those in a cell marked "*" are relatively balanced. However, these diagrams reflect someone's subjective preference. Many players have different ideas.

Size 11

Size 13

TODO

Size 14

TODO

A more general swap rule

It has been suggested that one could use a more general swap rule. Under this proposal, instead of placing just one piece, the first player places any number of red and blue pieces, and state which color has the next move. Let's assume that the first player has placed N stones in the board. The second player then can start playing Hex with one color of his choice or, if he fears that the other player has an excessive advantage due to home preparation, he can swap roles with the first player, remove all stones from the board and place at most N-1 stones as he wants. This rule has been first proposed by "lazplyayer" at littlegolem/igg. However, it is not widely used, nor implemented by any game sites.

See also

Guideline for 10x10 board size, in the basic strategy guide.

External links

A FAQ about Hex

A more complete site with solutions to size 7. Beware, the colours are inverted, vertical is blue there.

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