Difference between revisions of "User:Selinger"

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= Proposed article: Pivoting template =
+
I have played Hex since early 2020, and I run the [[Hex clubs|Halifax Hex Club]]. I mostly use this user page for draft articles and other random bits and pieces that aren't yet ready to go into a real HexWiki article.
  
−
A ''pivoting template'' is a kind of edge template that guarantees that the template's owner can either connect the template's stone(s) to the edge, or else can occupy a specified empty hex and connect it to the edge.
+
= Proposed page: Eric's move =
  
−
More precisely, a pivoting template is a pattern that has a stone A and an empty hex B, such that the template's owner can continuously threaten to connect A to the edge until the point where the template's owner either connects A to the edge or occupies B and connects B to the edge. To be considered a "template", its [[carrier]] should moreover be minimal with this property.
+
Eric's move is a trick that allows a player to make the best of a 3rd row [[ladder]] approaching an [[board|obtuse corner]]. It takes away the opponent's opportunity to get a 5th row ladder.
 +
 
 +
The move is named after Eric Demer, who discovered it.
  
 
== Example ==
 
== Example ==
  
−
The following is a pivoting template.
+
Consider the following situation, with Red's 3rd row ladder approaching from the right, and Red to move.
 +
<hexboard size="5x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4"/>
 +
There's not enough room for Red to [[ladder handling#Attacking|push]] one more time, as this will give Blue a 2nd row ladder:
 +
<hexboard size="5x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4 R 1:e3 B 2:d4 R 3:c3 B 4:b5 R 5:a5 B 6:b4 R 7:a4 B 8:b3"
 +
  />
 +
The obvious solution is for Red to pivot immediately and hold Blue to a 5th row ladder:
 +
<hexboard size="5x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4 R 1:d3 B 2:e3 R 3:d2 B 4:e1 E x:b4"
 +
  />
 +
However, this is not optimal, because Blue has a forcing move at x, giving Blue 2nd and 3rd row ladder escapes:
 +
A slightly better solution is the following:
 +
<hexboard size="5x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4 R 1:d4 B 2:e3 R 3:d2 B 4:e1 E x:b4 y:c3 S area(d2,a5,d5)"
 +
  />
 +
Note that Red has formed [[edge template IV2d]], still holding Blue to a 5th row ladder. Blue still has a forcing move at x, but because Red can respond at y, this only gives Blue a 2nd row ladder escape, rather than a 3rd row escape.
  
−
<hexboard size="5x9"
+
However, even this solution is not optimal for Red, as Blue still gets a 5th row ladder. It turns out that playing a different move 3 is generally even better for Red.
−
   coords="none"
+
<hexboard size="5x8"
−
   edges="bottom"
+
   coords="hide"
−
  visible="area(a5,i5,i3,h1,e1)-f1"
+
   edges="bottom left"
−
   contents="R A:e1 E B:g1"
+
   contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4 R 1:d4 B 2:e3 R 3:b2
 +
            E a:c2 b:d2 c:b3 d:c3 e:a4 f:b4"
 
   />
 
   />
 +
Move 3 is named '''Eric's move'''. If Red plays there, Blue in theory no longer gets a 5th row ladder, nor any kind of ladder. If Blue plays at a or c, Red responds at b; if Blue plays at b, Red responds at a; if Blue plays at d, Red responds at e; if Blue plays at f, Red responds at d.
  
−
Proof: Red's main [[threat]] is to [[bridge]] to c and connect to the edge by [[ziggurat]] or [[edge template III1b]]. Therefore, to prevent Red from connecting to the edge outright, Blue must play in one of the cells a,...,g.
+
== Why it works ==
  
−
<hexboard size="5x9"
+
Eric's move prevents Blue from getting a 5th row ladder along the left edge. To see why, consider the following line of play, which is one of Blue's best attempts:
−
   coords="none"
+
<hexboard size="12x8"
−
   edges="bottom"
+
   coords="hide"
−
  visible="area(a5,i5,i3,h1,e1)-f1"
+
   edges="bottom left"
−
   contents="R A:e1 E B:g1 a:d2 b:e2 c:d3 d:c4 e:d4 f:b5 g:d5"
+
   contents="B e9 f9 g9 g11 R h9 R g10 B f11 R f10 B e11 R 1:d11 B 2:e10 R 3:b9
 +
            B 4:b10 R 5:d9 B 6:e8 R 7:d8 B 8:e7 R 9:c6 S red:f1--f8"
 
   />
 
   />
 +
If we imagine that the pink cells are occupied by a line of red stones, then Red's move 9 is actually [[Tom's move]], using that line of stones as its edge. In that case, Red would connect, proving that Blue cannot in general get a 5th row ladder. Even if the pink cells are not in fact occupied by Red, the situation is still typically good for Red.
  
−
If Blue plays at a, Red responds at b and connects outright by [[edge template IV1a]].
+
However, the use of Tom's move in this argument requires quite a bit of empty space. If there is less space, or if there are additional Blue stones in this area, then Blue might still be able to do something useful.
  
−
If Blue plays at b, Red responds with a 3rd row ladder escape fork:
+
The way in which Eric's move works is essentially the same way as blocking a 5th row ladder when there's not enough space under the ladder stone; see [[Theory_of_ladder_escapes#Definition_of_ladder_4|theory of ladder escapes]].
−
<hexboard size="5x9"
+
−
  coords="none"
+
−
  edges="bottom"
+
−
  visible="area(a5,i5,i3,h1,e1)-f1"
+
−
  contents="R A:e1 E B:g1 B 1:e2 R 2:d2 B 3:c4 R 4:d3 B 5:d4 R 6:f3 B 7:e3 R 8:g1"
+
−
  />
+
  
−
If Blue plays at c, d, or f, Red responds as follows and is connected by [[Fifth_row_edge_templates#V-2-f|edge template V2f]]. If Blue plays on the right instead of 3, Red responds as if defending template V2f.
 
−
<hexboard size="5x9"
 
−
  coords="none"
 
−
  edges="bottom"
 
−
  visible="area(a5,i5,i3,h1,e1)-f1"
 
−
  contents="R A:e1 E B:g1 B 1:d3 1:c4 1:b5 R 2:e3 B 3:e2 R 4:g1"
 
−
  />
 
−
If Blue plays at e or g, Red responds at c and gets a 2nd or 3rd row ladder, which can reach B by ladder escape fork.
 
  
−
== Usage ==
+
etc.
  
−
Pivoting templates can be useful in many situations, but are especially useful in connection with [[flank]]s.
+
= Connecting parallel ladders =
  
−
[Todo: Add an example.]
+
== Connecting a 2-5 parallel ladder ==
  
−
== More examples ==
+
Like 2-4 and 3-5 parallel ladders, a 2-5 parallel ladder can also connect to the edge outright, given enough space. One way to do this is to yield to a 3-5 parallel ladder and then use [[Tom's move for 3rd and 5th row parallel ladders]]. However, there is a way to do it with much less space. In fact, the amount of space shown here is minimal:
  
−
<hexboard size="4x6"
+
<hexboard size="5x9"
−
  coords="none"
+
 
   edges="bottom"
 
   edges="bottom"
−
   visible="area(a4,f4,f1,e2,d2,d1)"
+
  coords="none"
−
   contents="R A:d1 E B:f1"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
 +
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2"
 
   />
 
   />
 +
The ladder stones are marked "↑", and Red's winning move is "1". It is Red's only winning move within this space.
  
−
<hexboard size="4x7"
+
Proof of connectedness: Red's main threats is "*", using the highlighted cells:
 +
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
 +
  visible="area(c1,a4,a5,i5,i3,g1)"
 +
  contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 *:c4 S red:(d1,c3,b4,b5,c4,c5,d3)"
 +
  />
 +
Red's other main threat is "*", connecting via [[edge template IV2b]], and only requiring 2 of the 3 cells x, y, z:
 +
<hexboard size="5x9"
 
   edges="bottom"
 
   edges="bottom"
−
   visible="area(b2,a4,g4,g2,f1,e1,d2,c1)"
+
  coords="none"
−
   contents="R A:c1 E B:e1"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
 +
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 *:c3 S red:c3,area(e2,b5,f5,f3) E x:b3 y:b4 z:d1"
 
   />
 
   />
−
 
+
The overlap consists of the cells marked "a", "b", and "c":
−
<hexboard size="3x5"
+
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
 +
  visible="area(c1,a4,a5,i5,i3,g1)"
 +
  contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 E a:c3,d3 c:c4 b:b5,c5"
 +
  />
 +
If Blue plays at "a", Red pushes the 2nd row ladder to "c" and then uses [[Tom's move]]. If Blue plays at "b", Red responds at "c" and then uses Tom's move. Finally, if Blue plays at "c", Red plays as follows:
 +
<hexboard size="5x9"
 
   edges="bottom"
 
   edges="bottom"
−
  visible="area(c1,a3,d3,e1)-d1"
 
−
  contents="R A:c1 E B:e1"
 
−
  />
 
−
 
−
<hexboard size="5x7"
 
 
   coords="none"
 
   coords="none"
−
  edges="bottom"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
−
   visible="area(a5,g5,g1,d1,b3)-f1"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 B 2:c4 R 3:b4 B 4:b5 R 5:e3"
−
   contents="R A:e1 E B:g1"
+
 
   />
 
   />
 +
This move isn't exactly a version of Tom's move, but it does for a 2-5 ladder what Tom's move does for a 2-4 ladder.
  
−
== Weak pivoting templates ==
+
== Connecting a 2-6 parallel ladder ==
  
−
There is another notion similar to a pivoting template, but slightly weaker. In a ''weak pivoting template'', we merely require that the template's owner can guarantee to either connect A to the edge or occupy B and connect B to the edge, but we drop the requirement that the owner can "continuously threaten to connect A to the edge until" that point. Typically this means that after the player occupies B, the opponent can still choose whether to let the player connect A or B to the edge.
+
2-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:
−
 
+
<hexboard size="6x12"
−
The following are examples of weak pivoting templates:
+
−
 
+
−
<hexboard size="5x8"
+
−
  coords="none"
+
 
   edges="bottom"
 
   edges="bottom"
−
  visible="area(a5,h5,h1,f1,c2,b3)-g1"
 
−
  contents="R A:f1 c2 E B:h1"
 
−
  />
 
−
 
−
<hexboard size="5x10"
 
 
   coords="none"
 
   coords="none"
−
  edges="bottom"
+
   visible="area(d1,d3,b4,a5,a6,l6,l4,j2,f1)"
−
   visible="area(c2,c3,a5,j5,j3,h1,f1,e2)-d2"
+
   contents="R arrow(12):d1,a5 B d2,a6 R 1:f2"
−
   contents="R A:c2 E B:e2"
+
 
   />
 
   />
 +
The basic idea is that this yields to 2-5, and then Red can use the previous trick.
  
−
Weak pivoting templates are sufficient to form a connection when combined with a [[flank]]. However, there are some contexts where a proper pivoting template would connect, but a weak pivoting template does not. The following is an example of this:
+
== Connecting a 3-6 parallel ladder ==
−
<hexboard size="9x9"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R h2 g2 f3 f5 c6 B b5 d4 e5 i5 i8
+
−
            S area(a9,h9,h5,f5,c6,b7)-g5"
+
−
  />
+
−
The highlighted area is a weak pivoting template, but with Blue to move, the position is losing for Red. On the other hand, if we use a proper pivoting template in the analogous situation, the position is winning for Red:
+
−
<hexboard size="9x9"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R h2 g2 f3 f5 B d5 b8 c6 e4 b7 i5 i8
+
−
            S area(b9,h9,h5,e5,c7)-g5"
+
−
  />
+
  
−
== See also ==
+
3-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:
  
−
* [[Climbing]]
+
<hexboard size="6x13"
−
 
+
  edges="bottom"
−
= Proposed article: Mustplay region =
+
   coords="none"
−
 
+
   visible="area(c1,a4,a6,m6,m4,k2,g1)"
−
Informally, a ''mustplay region'' for a player is a set of cells in which the player must move to avoid losing immediately. Mustplay analysis is an important tool for analyzing Hex positions, because it can help narrow down the number of possibilities a player must consider.
+
   contents="R arrow(12):c1,a4 B c2,a5 R 1:d2"
−
 
+
−
== Example ==
+
−
 
+
−
Consider the following position, with Blue to move:
+
−
<hexboard size="7x7"
+
−
   coords="show"
+
−
   edges="all"
+
−
   contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3"
+
 
   />
 
   />
  
−
To determine Blue's mustplay region, Blue should consider the possible ways in which Red could make a connection if it were ''Red's'' turn.  These are called Red's ''threats''. Red has (at least) the following threats:
+
== Remarks ==
−
 
+
−
* If Red plays at e4, Red is [[virtual connection|connected]] by two templates, namely [[edge template II]] and [[edge template IV2d]]. The [[carrier]] of Red's connection is the set of all cells that are required for the connection, and is highlighted: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
+
−
            R *:e4 S red:(e1--d1--d4--f4 area(f4,f7,c7))"
+
−
  />
+
−
 
+
−
* If Red plays at e5, then Red is connected via two copies of [[edge template II]] and two [[bridge]]s, as shown: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
+
−
            R *:e5 S red:(e1--d1--d4 area(d4,e4,f5,f7,e7,d5))"
+
−
  />
+
−
 
+
−
* Alternatively, if Red plays at e5, Red is also connected via [[edge template II]] and [[edge template III2e]], as shown: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
+
−
            R *:e5 S red:(e1--d1--d4 area(d4,e4,f6,f7,c7))"
+
−
  /> While the last two connections both use a Blue stone at e5, they have different carriers.
+
−
 
+
−
* If Red plays at d5, Red is connected via a 3rd row [[ladder]], using f6 as a [[ladder escape]]. In this case, the carrier consists of the path the ladder will take and the space required for the ladder escape: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
+
−
            R *:d5 S red:(e1--d1--d5 area(d5,f5,f7,c7))"
+
−
  />
+
−
 
+
−
Blue's mustplay region consists of those empty cells that are in the carriers of all of Red's known threats. Therefore, Blue's mustplay region consists of the cells d1, e1, e5, e6, e7, and f7.
+
−
<hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
+
−
            S blue:(d1,e1,e5,e6,e7,f7)"
+
−
  />
+
−
 
+
−
Note that this does not mean that all of d1, e1, e5, e6, e7, and f7 are winning moves for Blue, or even that Blue has any winning moves at all. Rather, what it means is that all ''other'' moves are losing. In other words, if Blue has any winning moves at all, they must be in Blue's mustplay region. Blue must now consider each of the six moves d1, e1, e5, e6, e7, and f7 and check if any of them are winning, or barring that, which one of them is least likely to be losing.
+
−
 
+
−
To help narrow down Blue's choices even further, it helps to consider [[captured cell|captured]] and [[dominated cell|dominated]] cells.  In the above example, d1, e1, e7, and f7 are captured by Red, and therefore, Blue should not play there. This leaves Blue with e5 and e6 as the only possible moves to consider. It so happens that e5 is winning and e6 is losing. Therefore, considering the mustplay region has helped Blue identify the only possible winning move. Blue will play e5 and win the game.
+
−
 
+
−
== Definition ==
+
−
 
+
−
From the point of view of a player, a ''threat'' is a [[virtual connection]] between the opponent's board edges that the opponent can create in a single move. The ''carrier'' of the threat is the set of cells (empty or not) that are required for the virtual connection to be valid. The player's mustplay region is determined as follows:
+
−
 
+
−
* Identify as many threats as possible.
+
−
 
+
−
* Determine the intersection of the carriers of all of these threats.
+
−
 
+
−
* With respect to the chosen set of threats, the ''mustplay region'' is the set of empty cells in that intersection.
+
−
 
+
−
== Properties ==
+
−
 
+
−
* All moves outside a player's mustplay region are losing. Moves within the mustplay region may be winning or losing.
+
−
 
+
−
* If a player's mustplay region is empty, the player is losing.
+
−
 
+
−
* If there are no winning moves in a player's mustplay region, the player is losing.
+
−
 
+
−
* The mustplay region is not unique. By considering more opponent threats, a player may arrive at a smaller mustplay region.
+
−
 
+
−
== Example: no winning move ==
+
−
 
+
−
If there are no winning moves in a player's mustplay region, the player is losing. To illustrate this, consider the following position, with Blue to move.
+
−
<hexboard size="5x5"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R b3 c2 d2 e1 B e2 c3 a4 a3"
+
−
  />
+
−
Red's main threats are:
+
−
* d3, connecting via a [[ziggurat]]: <hexboard size="5x5"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R b3 c2 d2 e1 B e2 c3 a4 a3
+
−
            R *:d3 S red:(area(e5,b5,d3,e3) d2 e1)"
+
−
  />
+
−
* b4, connecting via [[edge template II]]: <hexboard size="5x5"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R b3 c2 d2 e1 B e2 c3 a4 a3
+
−
            R *:b4 S red:(a5--b5--b3--c2--d2--e1)"
+
−
  />
+
−
* c4, connecting via [[edge template II]] and a [[double threat]]: <hexboard size="5x5"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R b3 c2 d2 e1 B e2 c3 a4 a3
+
−
            R *:c4 S red:(c5,b5,c4,d3,b4,b3,d2,c2,e1)"
+
−
  />
+
−
The only empty cell in the carrier of all three threats is b5, hence Blue's mustplay region consists of b5. This means that all moves except possibly b5 are losing for Blue.
+
−
<hexboard size="5x5"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R b3 c2 d2 e1 B e2 c3 a4 a3
+
−
            S blue:(b5)"
+
−
  />
+
−
Unfortunately for Blue, b5 is also losing, because if Blue plays b5, Red can win as follows:
+
−
<hexboard size="5x5"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R b3 c2 d2 e1 B e2 c3 a4 a3
+
−
            B 1:b5 R 2:b4 B 3:a5 R 4:d4"
+
−
  />
+
−
Therefore Blue has no winning moves at all and is losing the game.
+
−
 
+
−
== Applications ==
+
−
 
+
−
=== Foiling ===
+
−
 
+
−
Consider the following situation, with Blue to move:
+
−
<hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c2 b4 f2 f5 B c4 d4 d5 g3"
+
−
  />
+
−
 
+
−
Red's main threats are:
+
−
 
+
−
* e4, connecting via [[bridge]]s and a [[ziggurat]]: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c2 b4 f2 f5 B c4 d4 d5 g3
+
−
            R *:e4 S red:area(f1,e3,e6,d7,g7,g5,f5,f3,g1)"
+
−
  />
+
−
* a6, connecting via a 2nd row [[ladder]] and [[ladder escape]]: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c2 b4 f2 f5 B c4 d4 d5 g3
+
−
            R *:a6 S red:area(c1,a5,a7,f7,f5,e5,d6,b6,d1)"
+
−
  />
+
−
* a6, connecting via a 2nd row [[ladder]] and a slightly different [[ladder escape]]: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c2 b4 f2 f5 B c4 d4 d5 g3
+
−
            R *:a6 S red:area(c1,a5,a7,g7,g5,f5,e6,b6,d1)"
+
−
  />
+
−
 
+
−
Therefore, Blue's mustplay region consists of the following 5 cells:
+
−
<hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R c2 b4 f2 f5 B c4 d4 d5 g3
+
−
            S blue:area(e6,f6,d7,e7,f7) E x:e6 y:f6 z:d7 u:e7 v:f7"
+
−
  />
+
−
Of these, y, z, u, and v are losing: if Blue plays there, Red wins by responding at x. Blue's unique winning move is x. This move is also known as a [[foiling]] move, because it takes away Red's template and Red's ladder escape at the same time.
+
−
 
+
−
=== Solving Hex puzzles ===
+
−
 
+
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Consider the following puzzle, due to Eric Demer. Blue to move and win.
+
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<hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4"
+
−
  />
+
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At first, the situation seems confusing here. Blue's central stones neither seem to have a convincing connection to the left edge nor to the right one.
+
−
 
+
−
Mustplay analysis helps clarify the situation. First, let's note that Red's e3 and g3 are already very strongly connected to the top edge; Blue cannot gain anything by intruding into that connection. (In fact, Red has [[captured cell|captured]] rows 1&ndash;3). We therefore concentrate on the bottom part of the board. Within that region, Red's main threats are:
+
−
 
+
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* d4, connecting via [[edge template III2a]]: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4
+
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            R *:d4 S red:(area(c7,c5,b5,a6,a7),d4,e3)"
+
−
  />
+
−
 
+
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* f5, connecting via [[double threat]] of f6 and a 2nd row [[ladder]] at d6, for which b5 and c5 are a [[ladder escape]]: <hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4
+
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            R *:f5 S red:area(g3,g4,f7,e7,e6,d6,c7,a7,a6,b5,c5,d6)"
+
−
  />
+
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We therefore see that Blue's mustplay region consists of the following six cells:
+
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<hexboard size="7x7"
+
−
  coords="show"
+
−
  edges="all"
+
−
  contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4
+
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            S blue:(a6 b6 c6 a7 b7 c7) E x:a6 y:b6 z:c6 u:a7 v:b7 w:c7"
+
−
  />
+
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Of these, x, y, u, v, and w are losing: if Blue plays there, Red can respond at z, re-establishing both threats.
+
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The unique winning move for Blue is z. In fact, this is basically a [[foiling]] move.
+
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+
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=== Verification of templates ===
+
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+
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Mustplay analysis is also useful in the verification of templates. In that context, it is sometimes known as ''template reduction''. For example, consider [[edge template V1a]]:
+
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<hexboard size="5x10"
+
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  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
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  edges="bottom"
+
−
  coords="none"
+
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  contents="R g1"
+
−
  />
+
−
 
+
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To show that the template is valid, we must show that Blue has no way of disconnecting the template's red stone from the edge. We use mustplay analysis to reduce the number of possiblities. Red's main threats are:
+
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+
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* Connecting via a [[bridge]] to [[ziggurat]], in two different ways: <hexboard size="5x10"
+
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  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
−
  edges="bottom"
+
−
  coords="none"
+
−
  contents="R g1 R *:f3 S red:area(f2,f3,d5,g5,g2)"
+
−
  /> <hexboard size="5x10"
+
−
  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
−
  edges="bottom"
+
−
  coords="none"
+
−
  contents="R g1 R *:f3 S red:area(f2,c5,f5,f3,g2)"
+
−
  />
+
−
 
+
−
* Connecting via [[template IVa]]: <hexboard size="5x10"
+
−
  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
−
  edges="bottom"
+
−
  coords="none"
+
−
  contents="R g1 R *:f2 S red:area(e2,c3,a5,g5,g3,f2)"
+
−
  />
+
−
 
+
−
* Connecting via a [[bridge]] and [[template IVa]]: <hexboard size="5x10"
+
−
  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
−
  edges="bottom"
+
−
  coords="none"
+
−
  contents="R g1 R *:h2 S red:area(h1,d5,j5,j3)"
+
−
  />
+
−
 
+
−
* Connecting via a [[bridge]] and [[edge template III1b|template III-1-b]]: <hexboard size="5x10"
+
−
  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
−
  edges="bottom"
+
−
  coords="none"
+
−
  contents="R g1 R *:f3 S red:area(f2,c5,g5,g2)-e5"
+
−
  />
+
−
 
+
−
* Connecting via [[template IVb]], in two different ways: <hexboard size="5x10"
+
−
  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
−
  edges="bottom"
+
−
  coords="none"
+
−
  contents="R g1 R *:f2 S red:area(e2,c3,a5,h5,h3,g2)-e4"
+
−
/> <hexboard size="5x10"
+
−
  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
−
  edges="bottom"
+
−
  coords="none"
+
−
  contents="R g1 R *:g2 S red:area(f2,d3,b5,i5,i3,h2)-f4"
+
−
  />
+
−
Therefore, Blue's mustplay region consists of only three cells:
+
−
<hexboard size="5x10"
+
−
  visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
+
−
  edges="bottom"
+
−
  coords="none"
+
−
  contents="R g1 S blue:(f3 f5 d5)"
+
−
  />
+
−
To finish verifying the template, it then remains to show that each of these three moves are losing for Blue. See the article on [[edge template V1a]] for the details.
+
−
 
+
−
=== Computer Hex ===
+
−
 
+
−
Mustplay analysis is used in computer Hex to reduce the number of possibilities that must be considered for a player's next move. This drastically reduces the size of the search tree.
+
−
 
+
−
== References ==
+
−
 
+
−
Hayward, Björnsson, Johanson, Kan, Po, and van Rijswijck: [http://webdocs.cs.ualberta.ca/~hayward/papers/s7x7hex1.pdf "Solving 7x7 Hex with domination, fill-in, and virtual connections"], ''Theoretical Computer Science'' 349;123&ndash;139, 2005.
+
  
−
Hayward: [http://webdocs.cs.ualberta.ca/~hayward/papers/s7x7hex1.pdf "A puzzling Hex primer"]. In ''Games of No Chance 3'', Cambridge University Press, 56:151&ndash;162, 2009.  
+
In all three cases, for the ladder to propagate, the top ladder should be one hex further ahead of the bottom ladder than shown above. (If the bottom ladder is already caught up, the top ladder can no longer be pushed). For the 3-5 and 3-6 parallel ladders, Red doesn't necessarily have to push the bottom ladder before playing 1. However, for the 2-6 ladder, Red ''does'' have to push the bottom ladder first.
  
−
[[category:Theory]]
+
Also, the fact that these ladders all connect means that they are not really "ladders" in the usual sense; they are basically just templates. Note that unlike Tom's move (2-4 and 3-5 ladders), the connection requires no space above the height of the ladder, so the space in which the ladder would normally travel is already enough space to connect it.
−
[[category:Intermediate Strategy]]
+
−
[[category:Computer Hex]]
+

Latest revision as of 02:02, 17 October 2023

I have played Hex since early 2020, and I run the Halifax Hex Club. I mostly use this user page for draft articles and other random bits and pieces that aren't yet ready to go into a real HexWiki article.

Proposed page: Eric's move

Eric's move is a trick that allows a player to make the best of a 3rd row ladder approaching an obtuse corner. It takes away the opponent's opportunity to get a 5th row ladder.

The move is named after Eric Demer, who discovered it.

Example

Consider the following situation, with Red's 3rd row ladder approaching from the right, and Red to move.

There's not enough room for Red to push one more time, as this will give Blue a 2nd row ladder:

83176254

The obvious solution is for Red to pivot immediately and hold Blue to a 5th row ladder:

4312x

However, this is not optimal, because Blue has a forcing move at x, giving Blue 2nd and 3rd row ladder escapes: A slightly better solution is the following:

43y2x1

Note that Red has formed edge template IV2d, still holding Blue to a 5th row ladder. Blue still has a forcing move at x, but because Red can respond at y, this only gives Blue a 2nd row ladder escape, rather than a 3rd row escape.

However, even this solution is not optimal for Red, as Blue still gets a 5th row ladder. It turns out that playing a different move 3 is generally even better for Red.

3abcd2ef1

Move 3 is named Eric's move. If Red plays there, Blue in theory no longer gets a 5th row ladder, nor any kind of ladder. If Blue plays at a or c, Red responds at b; if Blue plays at b, Red responds at a; if Blue plays at d, Red responds at e; if Blue plays at f, Red responds at d.

Why it works

Eric's move prevents Blue from getting a 5th row ladder along the left edge. To see why, consider the following line of play, which is one of Blue's best attempts:

987635421

If we imagine that the pink cells are occupied by a line of red stones, then Red's move 9 is actually Tom's move, using that line of stones as its edge. In that case, Red would connect, proving that Blue cannot in general get a 5th row ladder. Even if the pink cells are not in fact occupied by Red, the situation is still typically good for Red.

However, the use of Tom's move in this argument requires quite a bit of empty space. If there is less space, or if there are additional Blue stones in this area, then Blue might still be able to do something useful.

The way in which Eric's move works is essentially the same way as blocking a 5th row ladder when there's not enough space under the ladder stone; see theory of ladder escapes.


etc.

Connecting parallel ladders

Connecting a 2-5 parallel ladder

Like 2-4 and 3-5 parallel ladders, a 2-5 parallel ladder can also connect to the edge outright, given enough space. One way to do this is to yield to a 3-5 parallel ladder and then use Tom's move for 3rd and 5th row parallel ladders. However, there is a way to do it with much less space. In fact, the amount of space shown here is minimal:

1

The ladder stones are marked "↑", and Red's winning move is "1". It is Red's only winning move within this space.

Proof of connectedness: Red's main threats is "*", using the highlighted cells:

1

Red's other main threat is "*", connecting via edge template IV2b, and only requiring 2 of the 3 cells x, y, z:

z1xy

The overlap consists of the cells marked "a", "b", and "c":

1aacbb

If Blue plays at "a", Red pushes the 2nd row ladder to "c" and then uses Tom's move. If Blue plays at "b", Red responds at "c" and then uses Tom's move. Finally, if Blue plays at "c", Red plays as follows:

15324

This move isn't exactly a version of Tom's move, but it does for a 2-5 ladder what Tom's move does for a 2-4 ladder.

Connecting a 2-6 parallel ladder

2-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:

1

The basic idea is that this yields to 2-5, and then Red can use the previous trick.

Connecting a 3-6 parallel ladder

3-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:

1

Remarks

In all three cases, for the ladder to propagate, the top ladder should be one hex further ahead of the bottom ladder than shown above. (If the bottom ladder is already caught up, the top ladder can no longer be pushed). For the 3-5 and 3-6 parallel ladders, Red doesn't necessarily have to push the bottom ladder before playing 1. However, for the 2-6 ladder, Red does have to push the bottom ladder first.

Also, the fact that these ladders all connect means that they are not really "ladders" in the usual sense; they are basically just templates. Note that unlike Tom's move (2-4 and 3-5 ladders), the connection requires no space above the height of the ladder, so the space in which the ladder would normally travel is already enough space to connect it.

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