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= Proposed article: Flank =
+
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 '''flank''' is a sequence of [[friendly]] [[stone]]s that are either adjacent or linked by [[bridge]]s in a certain way, and with a certain amount of space on one side, for example like this:
+
= Proposed page: Eric's move =
<hexboard size="6x11"
+
  edges="none"
+
  coords="none"
+
  visible="area(a6,d6,g5,k3,k1,i1,e3,d3,a4)"
+
  contents="R 1:a6 2:b6 3:c6 4:e5 5:f5 6:h4 7:j3 8:k3"
+
  />
+
Apart from [[ladder]]s, flanks are one of the most common "long-distance" patterns occuring in Hex. They are useful for [[climbing]], and they can be used to form large [[interior template|interior]] and [[edge template]]s.
+
  
What makes a flank useful is that its owner can use it for [[climbing]]. For example, consider the following situation, and assume the stones "B" and "J" are connected to opposite edges.  
+
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.
<hexboard size="6x11"
+
 
   edges="none"
+
The move is named after Eric Demer, who discovered it.
   coords="none"
+
 
   visible="area(a6,d6,g5,k3,k1,i1,e3,d3,a4)"
+
== Example ==
   contents="R J:a4 A:a6 b6 c6 e5 f5 h4 j3 B:k3 E *:k1"
+
 
 +
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"
 
   />
 
   />
Then Red can [[climbing|climb]] all the way from J to the cell marked "*", by a sequence of forcing moves as follows:
+
The obvious solution is for Red to pivot immediately and hold Blue to a 5th row ladder:
<hexboard size="6x12"
+
<hexboard size="5x8"
  edges="none"
+
   coords="hide"
   coords="none"
+
   edges="bottom left"
   visible="area(a6,d6,g5,k3,k1,i1,e3,d3,a4)"
+
   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"
   contents="R A:a6 b6 c6 e5 f5 h4 j3 B:k3
+
  />
            R J:a4 2:b4 4:c4 6:d6 8:e3 10:f3 12:g5 14:h2 16:i4 18:j1 20:k1
+
However, this is not optimal, because Blue has a forcing move at x, giving Blue 2nd and 3rd row ladder escapes:
            B 1:a5 3:b5 5:d5 7:c5 9:e4 11:g4 13:f4 15:i3 17:h3 19:j2 21:k2"
+
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)"
 
   />
 
   />
It is not actually necessary for Red to play moves 6, 12, and 16; Red could also skip these moves. However, they usually do not hurt and may be useful to Red by solidifying Red's position below the flank.
+
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.
  
Intruding into the flank's bridges does not help the opponent. The flank still works even if all the bridges have already been filled in:
+
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.
<hexboard size="6x12"
+
<hexboard size="5x8"
  edges="none"
+
   coords="hide"
   coords="none"
+
   edges="bottom left"
   visible="area(a6,d6,g5,k3,k1,i1,e3,d3,a4)"
+
   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
   contents="R J:a4 A:a6 b6 c6 e5 f5 h4 j3 B:k3 E *:k1
+
             E a:c2 b:d2 c:b3 d:c3 e:a4 f:b4"
             B d5 R d6 B g4 R g5 B i3 R i4"
+
 
   />
 
   />
 +
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.
  
== Definition ==
+
== Why it works ==
  
A flank can belong to Red or to Blue, and it can be oriented in any of the 6 cardinal directions of the Hex board (a cardinal direction is parallel to an edge or to the short diagonal). In addition, it can be facing up or down (the side it is facing is the side where the empty space is). For simplicity, the following definition refers to red flanks that are oriented left-to-right and facing upward.
+
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:
 +
<hexboard size="12x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  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.
  
Each flank has three distinguished points: a starting point, which we usually mark "A", an endpoint, which we usually mark "B", and a jumping-off point, which we mark "J". We can define flanks inductively as follows:
+
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.
  
* Base case: A single red stone, together with the indicated space, is a flank. In this case, the stone marked "B" is both the starting point and the endpoint. The jumping-off point "J" is also shown. <br>F0: <hexboard size="3x1"
+
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]].
  float="inline"
+
  edges="none"
+
  coords="none"
+
  contents="R B:a3 E J:a1"
+
  />
+
  
* Induction step: A flank can be extended with any of the following patterns:<br>F1: <hexboard size="3x2"
 
  float="inline"
 
  edges="none"
 
  coords="none"
 
  contents="R -:a3 +:b3"
 
/> F2: <hexboard size="4x3"
 
  float="inline"
 
  edges="none"
 
  coords="none"
 
  visible="area(a2,a4,b4,c1,c3,b1)"
 
  contents="R -:a4 +:c3"
 
/> F3: <hexboard size="4x3"
 
  float="inline"
 
  edges="none"
 
  coords="none"
 
  visible="area(a2,a4,b4,c1,c3,b1)"
 
  contents="R -:a4 +:c3 B b3 R b4"
 
/><br>Here, "−" denotes the previous endpoint, and "B" denotes the new endpoint. The orientation of these patterns matters, i.e., they cannot be rotated.
 
  
Here is an example of the flank obtained by starting with F0 and then extending with F1, F1, F3, F1, F2, F3, and F1. We always use "A" to denote the starting point and "B" to denote the endpoint of the flank:
+
etc.
<hexboard size="6x11"
+
  edges="none"
+
  coords="none"
+
  visible="area(a4,a6,d6,g5,k3,k1,i1,e3,d3)"
+
  contents="E J:a4 R A:a6 b6 c6 B d5 R d6 e5 f5 h4 B i3 R i4 j3 B:k3"
+
  />
+
We can also use algebraic notation to denote flanks. For example, we write F0+F1+F1+F3+F1+F2+F3+F1 for the above flank.
+
  
== Capped flank ==
+
= Connecting parallel ladders =
  
A flank is '''capped''' if it has been extended past its endpoint "B" with one of the following patterns:
+
== Connecting a 2-5 parallel ladder ==
C1: <hexboard size="3x2"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
  visible="area(a1,a3,b2,b1)"
+
  contents="R B:a3 b2"
+
/> C2: <hexboard size="3x2"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
  visible="area(a1,a3,b2,b1)"
+
  contents="R B:a3 b1"
+
/> C3: <hexboard size="3x2"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
  visible="area(a1,a3,b2,b1)"
+
  contents="R B:a3 b1 B a2 R b2"
+
/> C4: <hexboard size="3x3"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
  visible="-c3"
+
  contents="R B:a3 c2 c1"
+
/>
+
Here, "B" denotes the original endpoint of the flank. Other cap patterns are also possible; the above C1&ndash;C4 are just some common examples of caps.
+
  
Here are some examples of capped flanks. In each case, the flank's starting point "A" and original endpoint "B" are shown.
+
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:
  
F0+F1+C1:
+
<hexboard size="5x9"
<hexboard size="3x3"
+
   edges="bottom"
   edges="none"
+
 
   coords="none"
 
   coords="none"
   visible="-c3"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="E J:a1 R A:a3 B:b3 c2"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2"
 
   />
 
   />
F0+F2+C2:
+
The ladder stones are marked "↑", and Red's winning move is "1". It is Red's only winning move within this space.
<hexboard size="4x4"
+
 
   edges="none"
+
Proof of connectedness: Red's main threats is "*", using the highlighted cells:
 +
<hexboard size="5x9"
 +
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(a2,a4,b4,d2,d1,b1)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="E J:a2 R A:a4 B:c3 d1"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 *:c4 S red:(d1,c3,b4,b5,c4,c5,d3)"
 
   />
 
   />
F0+F2+F2+F3+F2+C1:
+
Red's other main threat is "*", connecting via [[edge template IV2b]], and only requiring 2 of the 3 cells x, y, z:
<hexboard size="6x9"
+
<hexboard size="5x9"
   edges="none"
+
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(a6,b6,g4,i2,i1,g1,d2,b3,a4)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="E J:a4 R A:a6 c5 e4 f4 B:h3 i2"
+
   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":
The point of capped flanks is that if Red plays at the jumping-off point "J" of any capped flank, Red can [[strong connection|connect]]:
+
<hexboard size="5x9"
 
+
   edges="bottom"
<hexboard size="6x9"
+
   edges="none"
+
 
   coords="none"
 
   coords="none"
   visible="area(a6,b6,g4,i2,i1,g1,d2,b3,a4)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R A:a6 c5 e4 f4 B:h3 i2
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 E a:c3,d3 c:c4 b:b5,c5"
            R 1:a4 3:b6 5:c3 7:d5 9:e2  11:f2 13:g4 15:h1
+
            B 2:b5 4:a5 6:d4 8:c4 10:e3 12:g3 14:f3"
+
 
   />
 
   />
 
+
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:
Todo: point out how this generalizes a 2nd row ladder, with the flank generalizing the "edge" and the cap generalizing a ladder escape.
+
<hexboard size="5x9"
 
+
   edges="bottom"
== Interior templates from capped flanks ==
+
 
+
There are several ways of constructing [[interior template]]s from capped flanks.  
+
 
+
=== Method 1 ===
+
 
+
The simplest method is to add a Red piece to the jumping-off point "J". Since this [[strong connection|connects]] to the rest of the flank. Such a group can be viewed as a (potentially very large) interior template.
+
 
+
Many of the named interior templates are of this form. This is the case for the [[crescent]], [[trapezoid]] (in more than one way), [[scooter]], [[bicycle]], as well as the [[long crescent]] and various [[long trapezoid]]s.
+
<hexboard size="3x3"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-c3"
+
  contents="R J:a1 A:a3 B:b3 c2"
+
/> <hexboard size="3x3"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-c3"
+
  contents="R J:a1 A:a3 B:b3 c1"
+
/> <hexboard size="3x3"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-c3"
+
  contents="R J:a1 A:a3 c2 c1"
+
/> <hexboard size="4x4"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="area(b1,a2,a4,b4,d2,d1)"
+
  contents="R J:a2 A:a4 B:c3 d2"
+
/> <hexboard size="4x4"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="area(b1,a2,a4,b4,d2,d1)"
+
  contents="R J:a2 A:a4 B:c3 d1"
+
/> <hexboard size="3x6"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-f3"
+
  contents="R J:a1 A:a3 b3 c3 d3 B:e3 f2"
+
/> <hexboard size="3x6"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-f3"
+
  contents="R J:a1 A:a3 b3 c3 d3 B:e3 f1"
+
/> <hexboard size="3x6"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-f3"
+
  contents="R J:a1 A:a3 b3 c3 B:d3 f1 f2"
+
/>
+
Here is a larger template constructed by the same method.
+
<hexboard size="6x9"
+
   edges="none"
+
 
   coords="none"
 
   coords="none"
   visible="area(a6,b6,e5,g4,i2,i1,g1,b3,a4)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R J:a4 A:a6 c5 d5 f4 B:h3 i2"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 B 2:c4 R 3:b4 B 4:b5 R 5:e3"
 
   />
 
   />
 +
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.
  
=== Method 2 ===
+
== Connecting a 2-6 parallel ladder ==
  
Another way to construct interior templates from flanks is to combine a capped flank and the mirror image of a capped flank so that they overlap at the point "J", schematically like this:
+
2-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:
<hexboard size="3x4"
+
<hexboard size="6x12"
  float="inline"
+
   edges="bottom"
   edges="none"
+
 
   coords="none"
 
   coords="none"
   visible="area(c1,a3,b3,d1)"
+
   visible="area(d1,d3,b4,a5,a6,l6,l4,j2,f1)"
  contents="E *:(c1--a3) J:d1 R A:b3"
+
   contents="R arrow(12):d1,a5 B d2,a6 R 1:f2"
/> + <hexboard size="3x2"
+
  />
  float="inline"
+
The basic idea is that this yields to 2-5, and then Red can use the previous trick.
  edges="none"
+
  coords="none"
+
  contents="E *:(b1--b3) J:a1 R A:a3"
+
/> = <hexboard size="3x5"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
  visible="area(c1,a3,e3,e1)"
+
   contents="E *:(c1--a3) *:(e1--e3) J:d1 x:c3 R A:b3 A:d3"
+
/>
+
Here, the hex "J" remains empty. The point is that if Blue plays at "x", Red plays at "J", and vice versa.  
+
  
Several of the named interior templates are of this form. This is the case for the [[span]], the [[box]], the [[shopping cart]], and the [[long span]]:
+
== Connecting a 3-6 parallel ladder ==
  
<hexboard size="3x4"
+
3-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-a1 d3"
+
  contents="R a2 R A:a3 R A:c3 R d2 E J:c1 x:b3"
+
/> <hexboard size="3x4"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-a1 d3"
+
  contents="R b1 R A:a3 R A:c3 R d1 E J:c1 x:b3"
+
/> <hexboard size="3x4"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-a1 d3"
+
  contents="R b1 R A:a3 R A:c3 R d2 E J:c1 x:b3"
+
/> <hexboard size="3x6"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-a1 f3"
+
  contents="R a2 A:a3 A:c3 d3 B:e3 f2 E J:c1 x:b3"
+
/>
+
Here is a larger example:
+
<hexboard size="6x15"
+
  coords="none"
+
  edges="none"
+
  visible="area(c4,c6,i6,m4,n3,n2,k2,g4,f3,d3)"
+
  contents="R c4 B:c5 d6 A:e6 A:g6 h6 j5 l4 B:m4 n3 E x:f6 J:g4"
+
/>
+
  
=== Method 3 ===
+
<hexboard size="6x13"
 
+
   edges="bottom"
A third way to construct interior templates from flanks is to combine a capped flank with a capped flank rotated by 180 degrees,  schematically like this:
+
<hexboard size="3x2"
+
  float="inline"
+
   edges="none"
+
 
   coords="none"
 
   coords="none"
  contents="E *:(a1--a3) J:b3 R A:b1"
+
   visible="area(c1,a4,a6,m6,m4,k2,g1)"
/> + <hexboard size="3x2"
+
   contents="R arrow(12):c1,a4 B c2,a5 R 1:d2"
  float="inline"
+
   />
  edges="none"
+
  coords="none"
+
  contents="E *:(b1--b3) J:a1 R A:a3"
+
/> = <hexboard size="3x4"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
  contents="E *:(a1--a3,d1--d3) J:b3 R A:b1 E J:c1 R A:c3"
+
/>
+
or like this:
+
<hexboard size="3x2"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
  contents="E *:(a1--a3) J:b3 R A:b1"
+
/> + <hexboard size="3x2"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
  contents="E *:(b1--b3) J:a1 R A:a3"
+
/> = <hexboard size="4x5"
+
  float="inline"
+
  edges="none"
+
  coords="none"
+
   visible="area(a2,a4,c4,e3,e1,c1)"
+
   contents="E *:(a2--a4,e1--e3) J:b4 R A:b2 E J:d1 R A:d3"
+
/>
+
If Blue plays at one of the hexes marked "J", Red can play at the other to keep the group connected.
+
 
+
Of the named interior templates, the [[parallelogram]] and the [[wide parallelogram]] are of this form:
+
 
+
<hexboard size="3x4"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="-a1 d3"
+
  contents="R a2 A:b1 A:c3 d2 E J:c1 J:b3"
+
/> <hexboard size="4x5"
+
  float="inline"
+
  coords="hide"
+
  edges="none"
+
  visible="area(c1,a3,a4,c4,e2,e1)"
+
  contents="R A:b2 a3 A:d3 e2 E J:d1 J:b4"
+
/>
+
 
+
But of course, it is again possible to constuct infinitely many examples. Here is a larger example:
+
 
+
<hexboard size="6x10"
+
  coords="none"
+
  edges="none"
+
  visible="area(b4,b5,d5,h4,j2,j1,h1,d2,c3)"
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   contents="R A:f4 g4 B:i3 j2 A:e2 B:c3 b4 E J:e4 J:f2"
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/>
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== Edge templates from capped flanks ==
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ADD EXAMPLES.
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== Usage example ==
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FROM A GAME.
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== Remarks ==
  
== 3rd row ladders along flanks ==
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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.
  
Above, we pointed out that climbing along a flank is analogous to a 2nd row ladder. It is similarly possible to climb along a flank at a greater distance. In other words, there is an analog of a 3rd row ladder along a flank. This requires slightly more space, and if the ladder is to connect, it requires a different kind of cap (or ladder escape).
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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.
  
ADD EXAMPLE.
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Also, as noted on the page [[Tom's move for 3rd and 5th row parallel ladders]], 3-5 parallel ladders can't always be pushed as 3-5 parallel ladders; the defender has the option to downgrade it to a 2nd row ladder with switchback threat. The resulting 2nd row ladder may not connect. Therefore, Tom's move may not be available if the ladder starts some distance from where there is space.

Latest revision as of 12:00, 27 May 2024

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.

Also, as noted on the page Tom's move for 3rd and 5th row parallel ladders, 3-5 parallel ladders can't always be pushed as 3-5 parallel ladders; the defender has the option to downgrade it to a 2nd row ladder with switchback threat. The resulting 2nd row ladder may not connect. Therefore, Tom's move may not be available if the ladder starts some distance from where there is space.