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= Proposed page: Question =
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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 '''question''' is a move that forces a player to choose between two or more available options in an area of the board. The player can either '''answer''' the question, by playing one of the available responses, or '''not answer''' and [[tenuki|play elsewhere]]. However, if the original question is sufficiently [[forcing move|forcing]], it must be answered right away.
+
= Proposed page: Eric's move =
  
Sometimes a player has a choice of accomplishing one of several different things in a given region, such as: choosing to connect one stone vs. another, connecting more strongly vs. denying the opponent a ladder escape, etc. In such situations, it is often in the player's interest to postpone the choice as long as possible, to keep their options open until they know more about what is going on on the rest of the board. By playing a question, the opponent can sometimes force them to make the choice earlier than they would have liked.
+
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.
  
Of course, in some very general sense, ''every'' move is a question, namely the question: "How will you respond to this move?". But usually the term is applied more narrowly in some region of the board that is not quite [[settled region|settled]], where one player plays a move that forces the other player to settle it, or at least simplify it, in one of several ways.
+
The move is named after Eric Demer, who discovered it.
  
== Example: U-turn ==
+
== Example ==
  
Perhaps the simplest example of a question is playing in the center of a [[wheel#U-turn|U-turn]]. The U-turn is the following position, consisting of two [[Multiple_threat#Overlapping_threats|overlapping]] bridges:
+
Consider the following situation, with Red's 3rd row ladder approaching from the right, and Red to move.
<hexboard size="3x3"
+
<hexboard size="5x8"
  edges="none"
+
   coords="hide"
   coords="none"
+
   edges="bottom left"
   visible="-a1 c3 c2"
+
   contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4"/>
   contents="R A:a2 B:c1 C:b3 E *:b2"
+
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"
With Blue to move, Red has a choice between connecting A to B or connecting A to C, but Red cannot achieve both of these things simultaneously. By playing in the cell marked "*", Blue asks the question "which or B or C do you want to connect to?"
+
   coords="hide"
 
+
   edges="bottom left"
The following is a position where asking this question is the only winning move for Blue:
+
   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"
<hexboard size="4x3"
+
   edges="all"
+
   coords="top left"
+
   contents="R a2 B:a3 C:c2 A:b4 B a1 b2 c4 E *:b3"
+
 
   />
 
   />
With Blue to move, in the upper part, ''Blue'' gets to choose whether Red connects B or C to the top edge. In the lower part, Blue can force ''Red'' to choose whether to connect B or C to the bottom edge. By playing the question at "*", Blue forces Red to make this choice. If Red chooses B, Blue blocks B on top and vice versa.
+
The obvious solution is for Red to pivot immediately and hold Blue to a 5th row ladder:
 
+
== Example: Edge template with two stones ==
+
 
+
Consider the following edge template:
+
 
<hexboard size="5x8"
 
<hexboard size="5x8"
 
   coords="hide"
 
   coords="hide"
   edges="bottom"
+
   edges="bottom left"
  visible="area(e1,a5,h5,h1)"
+
   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:e1 B:g1 E *:f1 *:f3"
+
 
   />
 
   />
This template has the curious property that, with Blue to move, Red can choose to connect either A or B to the edge, but cannot guarantee to connect them both. (See edge templates [[Fifth_row_edge_templates#V-2-g|V-2g]] and  [[Fifth_row_edge_templates#V-2-g|V-2h]]).
+
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:
Blue can ask the question "Which stone to you want to connect?" by playing at either of the cells marked "*". Red does not need to decide right away. For example, in the below diagram, if Blue intrudes at 1 and Red responds at 2, Red still retains the potential to connect either stone. But if Blue then plays at 3, then Blue threatens to cut off A at x or B at y. Red cannot defend both.
+
 
<hexboard size="5x8"
 
<hexboard size="5x8"
 
   coords="hide"
 
   coords="hide"
   edges="bottom"
+
   edges="bottom left"
  visible="area(e1,a5,h5,h1)"
+
   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)"
   contents="R A:e1 B:g1 B 1:f1 R 2:d3 B 3:e2 E x:d2 y:f3"
+
 
   />
 
   />
Similar, if the game proceeds like in the next diagram, then after move 7, Blue threatens to cut off A at x or B at y, and Red cannot save both connections.
+
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.
 
<hexboard size="5x8"
 
<hexboard size="5x8"
 
   coords="hide"
 
   coords="hide"
   edges="bottom"
+
   edges="bottom left"
  visible="area(e1,a5,h5,h1)"
+
   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 A:e1 B:g1 B 1:f1 R 2:e2 B 3:f2 R 4:g2 B 5:e4 R 6:e3 B 7:f3 E x:d4 y:g3"
+
            E a:c2 b:d2 c:b3 d:c3 e:a4 f:b4"
 
   />
 
   />
Of course, there are many other lines of play to consider. If Blue is not careful at each move, Red might be able to connect both A and B. But if Blue plays correctly, Red is eventually forced to answer the question.
+
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.  
  
== Example: Template intrusion ==
+
== Why it works ==
  
Consider the following position, with Blue to move:
+
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="11x11"
+
<hexboard size="12x8"
   coords="left bottom right"
+
   coords="hide"
   edges="left bottom right"
+
   edges="bottom left"
  visible="area(a7,a11,k11,k7)"
+
   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
   contents="R f7 g7 B d8 h9 S area(f7,c9,a11,g11,g7)"
+
            B 4:b10 R 5:d9 B 6:e8 R 7:d8 B 8:e7 R 9:c6 S red:f1--f8"
 
   />
 
   />
Note that Red is connected to the edge by [[Fifth_row_edge_templates#V-2-b|edge template V2b]], as highlighted. Blue would like to intrude into this template to gain strength either on the left or on the right.
+
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.
  
Blue would like a 4th row ladder escape on the left. But the problem is that if Blue plays at d9 or c10, Red can reconnect by playing a [[minimax|minimaxing move]] at h7, which strenghtens Red's position.
+
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.
<hexboard size="11x11"
+
 
   coords="left bottom right"
+
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]].
   edges="left bottom right"
+
 
   visible="area(a7,a11,k11,k7)"
+
 
   contents="R f7 g7 B d8 h9 B 1:d9 R 2:h7 S area(f7,e8,c11,g11,h8,h7)"
+
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:
 +
 
 +
<hexboard size="5x9"
 +
   edges="bottom"
 +
   coords="none"
 +
   visible="area(c1,a4,a5,i5,i3,g1)"
 +
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2"
 
   />
 
   />
Blue would also like a 4th row ladder escape on the right. But again, the problem is that if Blue moves at g10 or g9, Red can reconnect at g8, or by playing a minimaxing move, say at b10:
+
The ladder stones are marked "↑", and Red's winning move is "1". It is Red's only winning move within this space.
<hexboard size="11x11"
+
 
   coords="left bottom right"
+
Proof of connectedness: Red's main threats is "*", using the highlighted cells:
   edges="left bottom right"
+
<hexboard size="5x9"
   visible="area(a7,a11,k11,k7)"
+
   edges="bottom"
   contents="R f7 g7 B d8 h9 B 1:g10 R 2:b10 S area(f7,c9,a11,f11,g8,g7)"
+
   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)"
 
   />
 
   />
Neither of these outcomes is great for Blue. Instead, what Blue can do is ask the template a question:
+
Red's other main threat is "*", connecting via [[edge template IV2b]], and only requiring 2 of the 3 cells x, y, z:
<hexboard size="11x11"
+
<hexboard size="5x9"
   coords="left bottom right"
+
   edges="bottom"
   edges="left bottom right"
+
   coords="none"
   visible="area(a7,a11,k11,k7)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R f7 g7 B d8 h9 B 1:f8"
+
   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"
 
   />
 
   />
Basically, the question is: "How do you want to reconnect?" And based on the answer, Blue will be able to gain some strength on the left or on the right, without giving Red quite as much territory as would otherwise have been the case.
+
The overlap consists of the cells marked "a", "b", and "c":
 +
<hexboard size="5x9"
 +
  edges="bottom"
 +
  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"
 +
  coords="none"
 +
  visible="area(c1,a4,a5,i5,i3,g1)"
 +
  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.
  
For example, if Red reconnects at e8, then Blue can play e10:
+
== Connecting a 2-6 parallel ladder ==
<hexboard size="11x11"
+
 
   coords="left bottom right"
+
2-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:
   edges="left bottom right"
+
<hexboard size="6x12"
   visible="area(a7,a11,k11,k7)"
+
   edges="bottom"
   contents="R f7 g7 B d8 h9 B 1:f8 R 2:e8 B 3:e10 S red:(d9 d10 f9 f10 g8 g10)"  
+
   coords="none"
 +
   visible="area(d1,d3,b4,a5,a6,l6,l4,j2,f1)"
 +
   contents="R arrow(12):d1,a5 B d2,a6 R 1:f2"
 
   />
 
   />
 +
The basic idea is that this yields to 2-5, and then Red can use the previous trick.
  
Now Red's [[mustplay region]] consists of the 6 highlighted cells. If Red plays at d9, Blue gets a forcing move at b10, giving Blue a 4th row escape on the left, without Red getting g8. If Red plays at d10, g8, or g10, Blue gets a forcing move at d9, giving Blue a 4th row escape on the left without Red getting g8. If Red plays at g9, Blue defends the bridge at f9 and then plays as before. Finally, if Red plays at f10, Blue can respond at g8, getting a 4th row escape on the right. Although Red can still reconnect at b10, taking away Blue's ladder escape on the left, Red does not have the option of getting g8.
+
== 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:
 +
 
 +
<hexboard size="6x13"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="area(c1,a4,a6,m6,m4,k2,g1)"
 +
  contents="R arrow(12):c1,a4 B c2,a5 R 1:d2"
 +
  />
  
To be continued... and simplified?
+
== Remarks ==
  
== To do ==
+
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.
  
Add other illustrative examples, such as a template intrusion that forces the player to  trade-off between a stronger connection and letting the opponent get a ladder escape, etc.
+
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 point out that Josekis are essentially sequences of questions and answers.
+
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.