Difference between revisions of "Parallel ladder"

From HexWiki
Jump to: navigation, search
(Copied/adapted page from http://hexwiki.tk)
 
(Reorganized and cleaned up the article.)
 
(16 intermediate revisions by 6 users not shown)
Line 1: Line 1:
−
A [[laddering situation]] in which the [[attacking player|attacker]] can make two [[ladder]]s on top of each other.
+
A [[parallel ladder]] is a situation in which the attacker can make two [[ladder]]s on top of each other. The attacker's ladders are connected to each other and proceed in the same direction (both left to right or both right to left). Here is a typical example:
 +
<hexboard size="5x8"
 +
  edges="bottom"
 +
  coords="none"
 +
  contents="R c1 c2 b3 B a5 c3 R 1:d2 B 2:d3 R 3:e2 B 4:e3 R 5:b4 B 6:b5 R 7:c4 B 8:c5"
 +
  />
  
−
Consider the following position with [[Red (player)|Blue]] to play.
+
== 2nd and 4th rows ==
  
−
<hex>R10 C10 Q1 Hc1 Vd2 Vd3 He3 Vf3 Vd4 Ve4 Hf4 Hg4 Ve5 Vc6 Vd6 He6 Hi6 Hc7 Vd7 Ha8 Hb8 Vc8 Hd8 Hb10</hex>
+
=== Properties ===
  
−
All of Red's pieces form a connected [[group]]. This group is [[connection|connected]] to the [[Top edge|top]]. At the bottom, Red has a [[second row]] ladder with no possible [[ladder escpae|escape]] on the left. The potential escapes on the right are inadequate. For example, suppose Red ladders to f9. Then tries to escape with
+
A parallel ladder on the 2nd and 4th rows is a situation such as the following, with Red to move. The two red stones must be connected to the top edge (although the connection is not shown here). Red has the option of pushing the 2nd row ladder or the 4th row ladder:
 +
<hexboard size="4x6"
 +
  visible="-(a1 a2 b1 b2)"
 +
  edges="bottom"
 +
  coords="none"
 +
  contents="R c1 a3 B a4 c2"
 +
  />
  
−
:5. h9 g9
+
The first essential point is that a parallel ladder can be [[ladder handling|pushed]]. If Red pushes on the 4th row, Blue does not have the option to yield, or else Blue will lose immediately.
−
:6. h8 g8
+
<hexboard size="4x6"
−
:7. h7 f7.
+
  visible="-(a1 a2 b1 b2)"
 +
  edges="bottom"
 +
  coords="none"
 +
  contents="R c1 a3 1:d1 3:c3 B a4 c2 2:d3"
 +
  />
 +
Thus, Blue has no option but to push the ladder. Then Red can push the 2nd row ladder as well.  
 +
<hexboard size="4x6"
 +
  visible="-(a1 a2 b1 b2)"
 +
  edges="bottom"
 +
  coords="none"
 +
  contents="R c1 a3 1:d1 3:b3 B a4 c2 2:d2 4:b4"
 +
  />
 +
Note that pushing a parallel ladder only works if the 4th row ladder is "ahead" of the 2nd row ladder. Once the 2nd row ladder has caught up, it is too late to push on the 4th row, as Blue can then yield, resulting in an ordinary 3rd row ladder.
  
−
<hex>R10 C10 Q1 Hc1 Vd2 Vd3 He3 Vf3 Vd4 Ve4 Hf4 Hg4 Ve5 Vc6 Vd6 He6 Hi6 Hc7 Vd7 Ha8 Hb8 Vc8 Hd8 Hb10  Vc9 Hc10 Vd9 Hd10 Ve9 He10 Vf9 Hf10 Vh9 Hg9 Vh8 Hg8 Vh7</hex>
+
The second essential point is that a parallel ladder is stronger than either a 2nd row ladder or a 4th row ladder individually. Indeed, if the attacker wants to, they have the option of only pushing the 4th row ladder (and ignoring the 2nd row ladder), or of only pushing the 2nd row ladder (and ignoring the 4th row ladder). Thus, both 2nd row ladder escapes and 4th row ladder escapes can be used to escape parallel 2nd-and-4th row ladders. However, there are some ladder escapes that work for parallel ladders, but not for individual ladders.
  
−
Now Red's only reasonable try is 8.g7 f8. Now 9.g6 loses to 9...f5 and 9.h5 loses to the forcing sequence 9...g6 10.h6 h4 11.g5 f5. All the other escape attempts likewise fail. Is Red done for?
+
The best-known example of this is [[Tom's move]], which escapes are parallel 2nd-and-4th row ladder without requiring any pre-existing pieces on the board. Tom's move only requires a certain amount of empty space. There also exist other example (besides Tom's move) of ladder escapes that work for parallel ladders, but not for individual ladders. This is discussed in more detail [[Theory of ladder escapes#Second_and_fourth_row_parallel_ladders|here]].
  
−
No! Red can create a sufficient escape by making use of a '''parallel ladder'''. In the original position Red plays 1.e7. How can [[Blue (player)|Blue]] stop Red from connecting to the bottom? d9 lets Red [[bridge|two-chain]] from e7 to f8 connecting to the bottom;  e9 and e10 allow d9 which is connected to the bottom and threatens to connect to Red's big group through c9 and e8; d10 loses to e8, f9 ([[Forced move|forced]]), c10; hence, Blue is forced to play the parallel ladder move 1...e8.  It is simplest for Red to repeat this and ladder to f7 forcing the 2...f8 response.
+
Even without a ladder escape, a parallel ladder is awkward to defend against and will often give an advantage to the attacker. For example, a parallel ladder gives the attacker good [[climbing]] opportunities. Also, a parallel ladder is no worse than the lower ladder plus a [[Switchback#Switchback_threat|switchback threat]].
  
−
<hex>R10 C10 Q1 Hc1 Vd2 Vd3 He3 Vf3 Vd4 Ve4 Hf4 Hg4 Ve5 Vc6 Vd6 He6 Hi6 Hc7 Vd7 Ha8 Hb8 Vc8 Hd8 Hb10  Ve7 He8 Vf7 Hf8</hex>
+
=== Example ===
  
−
Now Red now goes back to the [[second row]] [[ladder]] and tries to escape. What have we gained by preceding this with the parallel ladder moves? When trying to escape, the [[threat]] to connect to d7-e7-f7 is stronger than the previous weak threat to connect to d7. This [[Multiple threats|extra threat]] will let us push our escape chain farther up the board and in this case, just far enough to win the game.
+
Consider the following position with [[Red]] to play.
  
−
Red's winning sequence is long but rather simple because every one of Blue's replies is forced. As before, Red ladders to f9 and escapes with 7. h9. Play continues 7...g9 8.h8 g8 9.h7 g7 10.h6 g6 11.h5. Red is threatening to play g5 with the double winning threats f5 and f6. But if Blue [[blocking|blocks]] this, say with 11...g5, then Red continues 12.i3 i2 13.h3 h2 and 14.g3 completes the [[win]].
+
<hex>R10 C10 Q1
 +
Hc1
 +
    Vd2
 +
      Vd3 He3 Vf3
 +
      Vd4 Ve4 Hf4 Hg4
 +
            Ve5
 +
    Vc6 Vd6 He6        Hh6 Hi6
 +
      Hc7 Vd7
 +
Ha8 Hb8 Vc8 Hd8
  
−
It is possible to use this trick off from one row farther back; i.e. with ladders on [[Third row|row 3]] and [[Fifth row|5]] but this occurs far less frequently and one has to examine some additional defensive possibilities. Consider the following position.
+
      Hb10</hex>
  
−
<hex>R10 C10 Q1 Vd5 He5 Vd6 Ve6 Hb7 Vc7 Hd7 Hb9<x/hex>
+
All of Red's pieces form a connected [[group]]. This group is [[connection|connected]] to the top. At the bottom, Red has a second row [[ladder]] with no possible [[ladder escape]] on the left. The potential escapes on the right are inadequate. For example, suppose Red breaks the ladder at e9 and then tries to [[climbing|climb]]:
  
−
Red has just played e6 trying the parallel ladder trick. With the closer ladder on [[Second row|row 2]], we saw that Blue was forced to respond with the parallel ladder play e7. But here Blue has two additional possibilities e8 and c9 (the only other possibility where Red doesn't have a way to force his group to connect to the [[Bottom edge|bottom]] is c10. But Red can respond with f8 and now Blue has nothing better than e7, g6).
+
<hex>R10 C10 Q1 Hc1 Vd2 Vd3 He3 Vf3 Vd4 Ve4 Hf4 Hg4 Ve5 Vc6 Vd6 He6 Hh6 Hi6 Hc7 Vd7 Ha8 Hb8 Vc8 Hd8 Hb10 N:on Vc9 Hc10 Ve9 Hd9 Vf7 He7 Vg5 Hf5 Vh4 Hh3</hex>
  
−
e8 yields a second row ladder after d8, e7, c8, c10, d9. The play c9 also leads to a second row ladder after the likely f7, f8, e8 (d9 is met by e7) d10. In the latter case, Red could again try the parallel ladder trick by playing g7. Of course, the presence of other pieces in the area can change the possibilities.
+
At this point Red fails to connect. Is Red done for? No! Red can escape the parallel ladder using [[Tom's move]]. Red plays like this:
 +
 
 +
<hex>R10 C10 Q1 Hc1 Vd2 Vd3 He3 Vf3 Vd4 Ve4 Hf4 Hg4 Ve5 Vc6 Vd6 He6 Hh6 Hi6 Hc7 Vd7 Ha8 Hb8 Vc8 Hd8 Hb10 N:on Vc9 Hc10 Vf8 He9 Vd9 Hd10 Vg7 He8 Vf6 Sf5 Se7</hex>
 +
 
 +
Note that all of Blue's moves are forced. If Blue moves anywhere but 4, Red will easily connect to the edge. 3 and 7 are connected to the bottom edge by [[Edge template IV2b]], so that 8 is also forced. Now Red is connected by [[double threat]] at the two cells marked "*".
 +
 
 +
 
 +
== 3rd and 5th rows ==
 +
 
 +
It is also possible to have a parallel ladder on the 3rd and 5th rows, such as this:
 +
<hexboard size="6x8"
 +
  edges="bottom"
 +
  coords="none"
 +
  contents="R c1 c2 b3 B a5 c3 R 1:d2 B 2:d3 R 3:e2 B 4:e3 R 5:b4 B 6:b5 R 7:c4 B 8:c5"
 +
  />
 +
In this case, the defender has additional possibilities besides pushing. If the defender [[ladder handling|yields]] from the 5th row ladder, the result is a 2nd row ladder with [[Switchback#Switchback_threat|switchback threat]]:
 +
<hexboard size="6x8"
 +
  edges="bottom"
 +
  coords="none"
 +
  contents="R c1 c2 b3 B a5 c3 R 1:d2 B 2:d3 R 3:e2 B 4:e4 R 5:b4 B 6:b5 R 7:d4 B 8:e3 R 9:c4 B 10:c6 R 11:d5"
 +
  />
 +
There are a few other things the defender can do, but all of them result either in the attacker connecting or getting a 2nd row ladder with switchback threat. This is discussed in more technical detail [[Theory of ladder escapes#Third_and_fifth_row_parallel_ladders|here]].
 +
 
 +
There is a version of [[Tom's move]] for 3rd-and-5th row parallel ladders, but it requires a large amount of space.
 +
 
 +
[[category: Ladder]]
 +
[[category: Advanced Strategy]]
 +
[[category: Definition]]

Latest revision as of 00:36, 29 December 2021

A parallel ladder is a situation in which the attacker can make two ladders on top of each other. The attacker's ladders are connected to each other and proceed in the same direction (both left to right or both right to left). Here is a typical example:

13245768

2nd and 4th rows

Properties

A parallel ladder on the 2nd and 4th rows is a situation such as the following, with Red to move. The two red stones must be connected to the top edge (although the connection is not shown here). Red has the option of pushing the 2nd row ladder or the 4th row ladder:

The first essential point is that a parallel ladder can be pushed. If Red pushes on the 4th row, Blue does not have the option to yield, or else Blue will lose immediately.

132

Thus, Blue has no option but to push the ladder. Then Red can push the 2nd row ladder as well.

1234

Note that pushing a parallel ladder only works if the 4th row ladder is "ahead" of the 2nd row ladder. Once the 2nd row ladder has caught up, it is too late to push on the 4th row, as Blue can then yield, resulting in an ordinary 3rd row ladder.

The second essential point is that a parallel ladder is stronger than either a 2nd row ladder or a 4th row ladder individually. Indeed, if the attacker wants to, they have the option of only pushing the 4th row ladder (and ignoring the 2nd row ladder), or of only pushing the 2nd row ladder (and ignoring the 4th row ladder). Thus, both 2nd row ladder escapes and 4th row ladder escapes can be used to escape parallel 2nd-and-4th row ladders. However, there are some ladder escapes that work for parallel ladders, but not for individual ladders.

The best-known example of this is Tom's move, which escapes are parallel 2nd-and-4th row ladder without requiring any pre-existing pieces on the board. Tom's move only requires a certain amount of empty space. There also exist other example (besides Tom's move) of ladder escapes that work for parallel ladders, but not for individual ladders. This is discussed in more detail here.

Even without a ladder escape, a parallel ladder is awkward to defend against and will often give an advantage to the attacker. For example, a parallel ladder gives the attacker good climbing opportunities. Also, a parallel ladder is no worse than the lower ladder plus a switchback threat.

Example

Consider the following position with Red to play.

abcdefghij12345678910

All of Red's pieces form a connected group. This group is connected to the top. At the bottom, Red has a second row ladder with no possible ladder escape on the left. The potential escapes on the right are inadequate. For example, suppose Red breaks the ladder at e9 and then tries to climb:

abcdefghij1234567891010987651432

At this point Red fails to connect. Is Red done for? No! Red can escape the parallel ladder using Tom's move. Red plays like this:

abcdefghij12345678910978315426

Note that all of Blue's moves are forced. If Blue moves anywhere but 4, Red will easily connect to the edge. 3 and 7 are connected to the bottom edge by Edge template IV2b, so that 8 is also forced. Now Red is connected by double threat at the two cells marked "*".


3rd and 5th rows

It is also possible to have a parallel ladder on the 3rd and 5th rows, such as this:

13245768

In this case, the defender has additional possibilities besides pushing. If the defender yields from the 5th row ladder, the result is a 2nd row ladder with switchback threat:

1328597461110

There are a few other things the defender can do, but all of them result either in the attacker connecting or getting a 2nd row ladder with switchback threat. This is discussed in more technical detail here.

There is a version of Tom's move for 3rd-and-5th row parallel ladders, but it requires a large amount of space.

Retrieved from "https://www.hexwiki.net/index.php?title=Parallel_ladder&oldid=7608"