Difference between revisions of "Tom's move"

From HexWiki
Jump to: navigation, search
(Copied/adapted page from http://hexwiki.tk)
 
(Alternative connection up: Added link for "alternative connection up" theorem.)
 
(30 intermediate revisions by 7 users not shown)
Line 1: Line 1:
In this diagram [[Red (player)|Red]] wants to connect to the bottom edge.
+
== Introduction ==
  
<hex>R8 C10 Vb7 Vb6 Ve5 Vc4 Vd2 Ha8 Hb8 Hc6 Hd5 He4 Hf3 Hg2 Hh2 Hi2 Hg3 Hh3 Hi3 Hj3 Hi4 Hj4 Hj5</hex>
+
'''Tom's move''' is a trick that enables a player to make a connection from a 2nd-and-4th row [[parallel ladder]]. It can also be used to connect a 2nd row [[ladder]] using a single stone on the 4th row, or to connect a single stone on the 4th row to the edge. Its name originates from Tom Ace (player [[User:Tom239|Tom239]]), who devised it during a game against dj11, on 15 December 2002 on [[Playsite]]. This was not its first use ever, but it is how it came to be known among Hex players on Playsite.
  
It looks impossible to use the single stone on the [[fourth row]] as a [[ladder escape]]. How can it be done? By using '''Tom's move''':
+
== Description ==
  
He [[Push (ladder)|pushes]] the ladder until he is right underneath it, and then he plays at (*).
+
Suppose Red has a 2nd-and-4th row [[parallel ladder]] and the amount of space shown here:
 +
<hexboard size="5x7"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-(a2 a1 b1 f1 g1 g2)"
 +
  contents="R arrow(12):b2 a3 a4 B b3 a5 E *:d3"
 +
  />
 +
Then Red can connect by playing at "*", the so-called "Tom's move".
  
<hex>R8 C10 Vb7 Vc7 Vd7 Vb6 Ve5 Vc4 Vd2 Ha8 Hb8 Hc8 Hd8 Hc6 Hd5 He4 Hf3 Hg2 Hh2 Hi2 Sg6 Pf7 Hg3 Hh3 Hi3 Hj3 Hi4 Hj4 Hj5</hex>
+
== Usage examples ==
  
Now the situation is completely identical to that in the [http://www.hexwiki.org/index.php?title=Tips_and_tricks#Double_ladder_on_2nd_line_escape Double ladder on 2nd line Escape] (in the article [[Tips and tricks]]).
+
=== Connecting a 2nd row ladder using an isolated stone on the 4th row ===
  
[[Blue (player)|Blue's]] natural [[defense]] is to play at (+).
+
Red to move and win:
  
<hex>R8 C10 Vb7 Vc7 Vd7 Vb6 Ve5 Vc4 Vd2 Ha8 Hb8 Hc8 Hd8 Hc6 Hd5 He4 Hf3 Hg2 Hh2 Hi2 Vg6 Hf7 Vh5 Sg4 Se7 Hg3 Hh3 Hi3 Hj3 Hi4 Hj4 Hj5</hex>
+
<hexboard size="6x9"
 +
  edges="all"
 +
  coords="none"
 +
  contents="R a5 a4 d3 b2 B a6 b4 c3 d2 e1 f1 g1 h1 i1 h2 i2 i3"
 +
  />
  
Now Red´s two pieces at g6 and h5 are [[Connection|connected]] to the [[bottom edge]], and his [[ladder]] can connect to those [[piece]]s in [[Multiple threats|two different ways]], by playing in either of the starred [[Hex (board element)|cells]]. Red is therefore safely connected to the bottom edge.
+
The solution is to [[ladder handling|push]] the [[ladder]] to 3 and then play Tom's move:
 +
 
 +
<hexboard size="6x9"
 +
  edges="all"
 +
  coords="none"
 +
  contents="R a5 a4 d3 b2 B a6 b4 c3 d2 e1 f1 g1 h1 i1 h2 i2 i3 R 1:b5 B 2:b6 R 3:c5 B 4:c6 R 5:f4"
 +
  />
 +
 
 +
=== A single stone on the 4th row is connected ===
 +
 
 +
Consider a single stone on the 4th row, with the amount of space shown:
 +
<hexboard size="5x9"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-area(a1,a4,d1) h1 i1 i2"
 +
  contents="R arrow(12):d2"
 +
  />
 +
 
 +
Then Red can connect as follows:
 +
<hexboard size="5x9"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-area(a1,a4,d1) h1 i1 i2"
 +
  contents="R arrow(12):d2 B 1:d3 R 2:c3 B 3:b5 R 4:c4 B 5:c5 R 6:f3"
 +
  />
 +
Red squeezes through the [[bottleneck]] at 2, starts a 2nd row ladder at 4, then plays Tom's move at 6. Note that all of Blue's moves are forced; if Blue plays differently, Red connects outright. This is [[edge template IV1d]].
 +
 
 +
=== In a game ===
 +
Red to move:
 +
<hexboard size="11x11"
 +
  edges="all"
 +
  coords="show"
 +
  contents="R b7 c6 d4 d5 d6 d8 f5 g3 g5 h5 i4 i6 j5 k2 k3 B c5 c7 d7 e4 e5 e6 f6 g6 h4 h6 h7 i3 i7 j3 j4 j6"
 +
  />
 +
Red's d4 [[group]] is already connected to the top edge by [[edge template IV1a|edge template IV1-a]]. To connect to the bottom, Red plays as follows:
 +
<hexboard size="11x11"
 +
  edges="all"
 +
  coords="show"
 +
  contents="R b7 c6 d4 d5 d6 d8 f5 g3 g5 h5 i4 i6 j5 k2 k3 B c5 c7 d7 e4 e5 e6 f6 g6 h4 h6 h7 i3 i7 j3 j4 j6
 +
            R 1:b8 B 2:c9 R 3:a10 B 4:a11 R 5:b10 B 6:b11 R 7:c10 B 8:c11 R 9:f9"
 +
  />
 +
Now Red is connected by Tom's move. Note that d8 is already connected to Red's group by [[double threat]] at c8 and d9.
 +
 
 +
== Why Tom's move is connected ==
 +
 
 +
To compute Blue's [[mustplay region]], we consider two red [[threat]]s:
 +
 
 +
<hexboard size="5x7"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-(a2 a1 b1 f1 g1 g2)"
 +
  contents="R arrow(12):b2 a3 a4 d3 2:b4 4:c4 6:e4 B b3 a5 3:b5 5:c5 S b4 area(b5,d3,e3,e5)"
 +
  />
 +
 
 +
<hexboard size="5x7"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-(a2 a1 b1 f1 g1 g2)"
 +
  contents="R arrow(12):b2 a3 a4 d3 2:c2 B b3 a5 S c2 c3 d2 area(b5,d3,e3,e5)"
 +
  />
 +
 
 +
These leaves only blue moves in the [[ziggurat]].
 +
 
 +
<hexboard size="5x7"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-(a2 a1 b1 f1 g1 g2)"
 +
  contents="R arrow(12):b2 a3 a4 d3 B b3 a5 S area(b5,d3,e3,e5) E a:c4"
 +
  />
 +
 
 +
If Blue plays there other than at a, then Red plays at a.
 +
<hexboard size="5x7"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-(a2 a1 b1 f1 g1 g2)"
 +
  contents="R arrow(12):b2 a3 a4 d3 2:c4 B b3 a5 E *:c2 *:b4 z:b5 y:c5 x:(e5 d5 e4 d4 e3) S area(b5,d3,e3,e5)"
 +
  />
 +
In that case, Red's 2 connects back via either of the cells marked "*", and since the piece Blue just played is in only one of the x,y,z regions, Red's 2 also connects down via either of the remaining two of those three regions. 
 +
 
 +
Thus Blue's only remaining hope is to play at a.
 +
 
 +
<hexboard size="5x7"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-(a2 a1 b1 f1 g1 g2)"
 +
  contents="R arrow(12):b2 a3 a4 d3 B b3 a5 1:c4"
 +
  />
 +
 
 +
Red responds like this:
 +
 
 +
<hexboard size="5x7"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-(a2 a1 b1 f1 g1 g2)"
 +
  contents="R arrow(12):b2 a3 a4 d3 2:b4 4:e2 B b3 a5 1:c4 3:b5 E *:d1 *:c3"
 +
  />
 +
 
 +
Red's 4 is now connected to the bottom via [[Edge_template_IV2b|edge template IV2b]], and to
 +
Red's main group by double threat at the cells marked "*". Note that 2 and 3 do not actually need to be played; these moves have been included for clarity.
 +
 
 +
== Pushing the 4th row ladder first ==
 +
 
 +
Sometimes, there is not enough room to play Tom's move right away, but enough room can be created by first pushing the 4th row ladder. For a typical example, consider the following situation. It is Blue's turn, and Red wants to connect her stone to the bottom edge.
 +
<hexboard size="6x11"
 +
  edges="bottom right"
 +
  coords="none"
 +
  visible="-area(a1,a5,e1)"
 +
  contents="R d3 B e2 f2 j2 k3 g1--j1"
 +
  />
 +
If Red tries to play Tom's move immediately, it doesn't work, because 8 does not connect back to Red's main group.
 +
<hexboard size="6x11"
 +
  edges="bottom right"
 +
  coords="none"
 +
  visible="-area(a1,a5,e1)"
 +
  contents="R d3 B e2 f2 j2 k3 g1--j1 B 1:d4 R 2:c4 B 3:b6 R 4:c5 B 5:c6 R 6:f4 B 7:e5 R 8:g3 B 9:e4"
 +
  />
 +
What Red can do instead is start by pushing the 4th row ladder twice.
 +
<hexboard size="6x11"
 +
  edges="bottom right"
 +
  coords="none"
 +
  visible="-area(a1,a5,e1)"
 +
  contents="R d3 B e2 f2 j2 k3 g1--j1 B 1:d4 R 2:e3 B 3:e4 R 4:f3 B 5:f4 R 6:c4 B 7:b6 R 8:c5 B 9:c6 R 10:d5 B 11:d6 R 12:e5 B 13:e6 R 14:h4 B 15:g5 R 16:i3 E *:g4 *:h2"
 +
  />
 +
Of course this only works if after pushing the ladder, there is still enough room for Tom's move.
 +
 
 +
It is not actually necessary to push the 2nd row ladder (moves 6&ndash;13 can be omitted), but they have been included for clarity.
 +
 
 +
Note that when Red pushes the 4th row ladder, Blue cannot [[ladder handling|yield]], as this would give Red a [[ladder escape fork]] for the below 2nd row ladder. Also, as explained in more detail in the article on [[parallel ladder]]s, the 4th row ladder must be pushed ''before'' the 2nd row ladder has caught up to it. If Red starts by first pushing the 2nd row ladder, then it is too late to push the 4th row ladder.
 +
 
 +
== Variants ==
 +
 
 +
=== Alternative connection up ===
 +
 
 +
Tom's move also works when the hex marked "a" is not empty, provided that "b" connects to Red's main group.
 +
<hexboard size="5x7"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-(a2 a1 b1 f1 g1 g2)"
 +
  contents="R arrow(12):b2 a3 a4 d3 B b3 a5 E a:c1 b:d1"
 +
  />
 +
 
 +
For example, Tom's move works in this situation:
 +
<hexboard size="6x8"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-g2 h2 h3 f1--h1 -area(a1,a5,c1)"
 +
  contents="R arrow(12):b4 b5 c3 B a6 b6 c4 R e4 E b:e2 R d1 B a:d2 R c2"
 +
  />
 +
 
 +
This is a special case of a [[Theorems_about_templates#Alternative_connection_up|general theorem]].
 +
 
 +
=== Tall variant ===
 +
 
 +
If "d" is empty, there is a variant of Tom's move that does not require a connection via "b", or even for "b" to be empty; it merely requires "c" and "e" to threaten to connect to Red's main group. An example is this situation:
 +
<hexboard size="6x8"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-h2 h3 g1--h1 -area(a1,a5,c2,d1)"
 +
  contents="R arrow(12):b4 b5 c3 B a6 b6 c4 R e4 E c:f2 d:g2 e:e3 R e1 B b:e2 R d2"
 +
  />
 +
In this version of Tom's move, Red's 4 is different than usual (the usual move 4 does not work here). As before, moves 2 and 3 don't need to be played, but make it easier to see how the connection works.
 +
<hexboard size="6x8"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-h2 h3 g1--h1 -area(a1,a5,c2,d1)"
 +
  contents="R arrow(12):b4 b5 c3 B a6 b6 c4 R x:e4 R e1 B e2 R d2
 +
            B 1:d5 R 2:c5 B 3:c6 R 4:f2 E y:f3"
 +
  />
 +
Note that "x" is connected to Red's main group without requiring "y", and "4" is also connected to Red's main group without requiring "y". (However, Red cannot guarantee to connect both "x" and "4" to her main group without requiring "y"). If Blue tries to cut Red off from the edge, Red responds as follows:
 +
<hexboard size="6x8"
 +
  edges="bottom"
 +
  coords="none"
 +
  visible="-h2 h3 g1--h1 -area(a1,a5,c2,d1)"
 +
  contents="R arrow(12):b4 b5 c3 B a6 b6 c4 R x:e4 R e1 B e2 R d2
 +
            B 1:d5 R 2:c5 B 3:c6 R 4:f2 E y:f3 B 5:e5 R 6:g4"
 +
  />
 +
Now no matter how Blue plays, Red can connect to the edge by a sequence of forcing moves. The situation is analogous to the [[Interior_template#The_hammock|hammock template]].
 +
 
 +
== Tom's move for 3rd-and-5th row parallel ladders ==
 +
 
 +
<i>Main article: [[Tom's move for 3rd and 5th row parallel ladders]].</i>
 +
 
 +
There is a version of Tom's move that works for parallel ladders on the 3rd and 5th rows. It requires a large amount of space:
 +
<hexboard size="6x12"
 +
  coords="hide"
 +
  edges="bottom"
 +
  visible="area(d1,b3,b6,l6,l4,j2,f1)"
 +
  contents="R arrow(12):c2 b3 b4 B b5 c3 R 1:e3"
 +
  />
 +
By playing at "1", Red can connect to the edge. Verifying this requires a lot of steps, but here is the basic idea:
 +
<hexboard size="6x12"
 +
  coords="hide"
 +
  edges="bottom"
 +
  visible="area(d1,b3,b6,l6,l4,j2,f1)"
 +
  contents="R arrow(12):c2 b3 b4 B b5 c3 E x:i4
 +
            R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1"
 +
  />
 +
Notice that Red's 3 is connected left by double threat at the two cells marked "*", and connected right by [[Fifth_row_edge_templates#V-2-m|edge template V2m]]. The latter template is itself based on Tom's move at "x". It works, for example, like this:
 +
<hexboard size="6x12"
 +
  coords="hide"
 +
  edges="bottom"
 +
  visible="area(d1,b3,b6,l6,l4,j2,f1)"
 +
  contents="R arrow(12):c2 b3 b4 B b5 c3
 +
            R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1
 +
            B 6:f4 R 7:g3 B 8:g4 R 9:e4 B 10:d6 R 11:e5 B 12:e6 R 13:f5 B 14:f6 R 15:i4"
 +
  />
 +
Now Red is connected by the (ordinary) Tom's move.
 +
 
 +
See [[Tom's move for 3rd and 5th row parallel ladders]] for more details.
 +
 
 +
=== Tall variant ===
 +
 
 +
Tom's move for 3rd-and-5th row parallel ladders also has a tall version. It works in the same way as the tall version of Tom's move for 2nd-and-4th row parallel ladders above, given the appropriate amount of space.
 +
 
 +
<hexboard size="7x12"
 +
  coords="hide"
 +
  edges="bottom"
 +
  visible="area(d2,b4,b7,l7,l5,j3,g2,f1,e1)"
 +
  contents="R arrow(12):e1 d2 c3 b4 b5 B b6 c4 e2 R 1:e4 B 2:d5 R 3:f2"
 +
  />
 +
 
 +
== See also ==
 +
 
 +
* [[Parallel ladder]]
 +
* [[Edge template IV1d]]
 +
* [[Fifth_row_edge_templates#V-2-m|Edge template V2m]]
 +
 
 +
[[category:ladder]]
 +
[[category:Advanced Strategy]]

Latest revision as of 21:44, 20 April 2024

Introduction

Tom's move is a trick that enables a player to make a connection from a 2nd-and-4th row parallel ladder. It can also be used to connect a 2nd row ladder using a single stone on the 4th row, or to connect a single stone on the 4th row to the edge. Its name originates from Tom Ace (player Tom239), who devised it during a game against dj11, on 15 December 2002 on Playsite. This was not its first use ever, but it is how it came to be known among Hex players on Playsite.

Description

Suppose Red has a 2nd-and-4th row parallel ladder and the amount of space shown here:

Then Red can connect by playing at "*", the so-called "Tom's move".

Usage examples

Connecting a 2nd row ladder using an isolated stone on the 4th row

Red to move and win:

The solution is to push the ladder to 3 and then play Tom's move:

51324

A single stone on the 4th row is connected

Consider a single stone on the 4th row, with the amount of space shown:

Then Red can connect as follows:

216435

Red squeezes through the bottleneck at 2, starts a 2nd row ladder at 4, then plays Tom's move at 6. Note that all of Blue's moves are forced; if Blue plays differently, Red connects outright. This is edge template IV1d.

In a game

Red to move:

abcdefghijk1234567891011

Red's d4 group is already connected to the top edge by edge template IV1-a. To connect to the bottom, Red plays as follows:

abcdefghijk1234567891011129357468

Now Red is connected by Tom's move. Note that d8 is already connected to Red's group by double threat at c8 and d9.

Why Tom's move is connected

To compute Blue's mustplay region, we consider two red threats:

24635
2

These leaves only blue moves in the ziggurat.

a

If Blue plays there other than at a, then Red plays at a.

x2xxzyxx

In that case, Red's 2 connects back via either of the cells marked "*", and since the piece Blue just played is in only one of the x,y,z regions, Red's 2 also connects down via either of the remaining two of those three regions.

Thus Blue's only remaining hope is to play at a.

1

Red responds like this:

4213

Red's 4 is now connected to the bottom via edge template IV2b, and to Red's main group by double threat at the cells marked "*". Note that 2 and 3 do not actually need to be played; these moves have been included for clarity.

Pushing the 4th row ladder first

Sometimes, there is not enough room to play Tom's move right away, but enough room can be created by first pushing the 4th row ladder. For a typical example, consider the following situation. It is Blue's turn, and Red wants to connect her stone to the bottom edge.

If Red tries to play Tom's move immediately, it doesn't work, because 8 does not connect back to Red's main group.

821964735

What Red can do instead is start by pushing the 4th row ladder twice.

24166135148101215791113

Of course this only works if after pushing the ladder, there is still enough room for Tom's move.

It is not actually necessary to push the 2nd row ladder (moves 6–13 can be omitted), but they have been included for clarity.

Note that when Red pushes the 4th row ladder, Blue cannot yield, as this would give Red a ladder escape fork for the below 2nd row ladder. Also, as explained in more detail in the article on parallel ladders, the 4th row ladder must be pushed before the 2nd row ladder has caught up to it. If Red starts by first pushing the 2nd row ladder, then it is too late to push the 4th row ladder.

Variants

Alternative connection up

Tom's move also works when the hex marked "a" is not empty, provided that "b" connects to Red's main group.

ab

For example, Tom's move works in this situation:

ab

This is a special case of a general theorem.

Tall variant

If "d" is empty, there is a variant of Tom's move that does not require a connection via "b", or even for "b" to be empty; it merely requires "c" and "e" to threaten to connect to Red's main group. An example is this situation:

bcde

In this version of Tom's move, Red's 4 is different than usual (the usual move 4 does not work here). As before, moves 2 and 3 don't need to be played, but make it easier to see how the connection works.

4yx213

Note that "x" is connected to Red's main group without requiring "y", and "4" is also connected to Red's main group without requiring "y". (However, Red cannot guarantee to connect both "x" and "4" to her main group without requiring "y"). If Blue tries to cut Red off from the edge, Red responds as follows:

4yx62153

Now no matter how Blue plays, Red can connect to the edge by a sequence of forcing moves. The situation is analogous to the hammock template.

Tom's move for 3rd-and-5th row parallel ladders

Main article: Tom's move for 3rd and 5th row parallel ladders.

There is a version of Tom's move that works for parallel ladders on the 3rd and 5th rows. It requires a large amount of space:

1

By playing at "1", Red can connect to the edge. Verifying this requires a lot of steps, but here is the basic idea:

5132x4

Notice that Red's 3 is connected left by double threat at the two cells marked "*", and connected right by edge template V2m. The latter template is itself based on Tom's move at "x". It works, for example, like this:

517329681541113101214

Now Red is connected by the (ordinary) Tom's move.

See Tom's move for 3rd and 5th row parallel ladders for more details.

Tall variant

Tom's move for 3rd-and-5th row parallel ladders also has a tall version. It works in the same way as the tall version of Tom's move for 2nd-and-4th row parallel ladders above, given the appropriate amount of space.

312

See also