Difference between revisions of "Tom's move"

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(Alternative connection up: Added link for "alternative connection up" theorem.)
 
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== Introduction ==
 
== 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 break through 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, just how it came to be known among Hex players on Playsite.
+
'''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.
  
 
== Description ==
 
== Description ==
  
 
Suppose Red has a 2nd-and-4th row [[parallel ladder]] and the amount of space shown here:
 
Suppose Red has a 2nd-and-4th row [[parallel ladder]] and the amount of space shown here:
<hexboard size="5x8"
+
<hexboard size="5x7"
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-(a2 a1 b1 f1 g1 g2)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 E *:e3"
+
   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".
 
Then Red can connect by playing at "*", the so-called "Tom's move".
Line 41: Line 41:
 
   coords="none"
 
   coords="none"
 
   visible="-area(a1,a4,d1) h1 i1 i2"
 
   visible="-area(a1,a4,d1) h1 i1 i2"
   contents="R d2"
+
   contents="R arrow(12):d2"
 
   />
 
   />
  
Line 49: Line 49:
 
   coords="none"
 
   coords="none"
 
   visible="-area(a1,a4,d1) h1 i1 i2"
 
   visible="-area(a1,a4,d1) h1 i1 i2"
   contents="R d2 B 1:d3 R 2:c3 B 3:b5 R 4:c4 B 5:c5 R 6:f3"
+
   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.
+
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 ===
 
=== In a game ===
Line 57: Line 57:
 
<hexboard size="11x11"
 
<hexboard size="11x11"
 
   edges="all"
 
   edges="all"
   coords="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"
 
   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"
 
   />
 
   />
Line 63: Line 63:
 
<hexboard size="11x11"
 
<hexboard size="11x11"
 
   edges="all"
 
   edges="all"
   coords="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
 
   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"
 
             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.
+
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 ==
 
== Why Tom's move is connected ==
  
Let us compute Blue's [[mustplay region]]. Red has several main threats:
+
To compute Blue's [[mustplay region]], we consider two red [[threat]]s:
  
<hexboard size="5x8"
+
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   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
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   visible="-(a2 a1 b1 f1 g1 g2)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 R 1:c4 B 2:c5 R 3:e4 S area(d3,c4,c5,e5,e3)"
+
   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="5x8"
+
<hexboard size="5x7"
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-(a2 a1 b1 f1 g1 g2)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 R 1:c4 B 2:c5 R 3:d4 B 4:d5 R 5:f4 S area(e3,c4,c5,f5,f3)"
+
   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="5x8"
+
<hexboard size="5x7"
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-(a2 a1 b1 f1 g1 g2)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 R 1:d2 S area(d2,d4,c5,f5,f3,e2)"
+
   contents="R arrow(12):b2 a3 a4 d3 B b3 a5 S area(b5,d3,e3,e5) E a:c4"
 
   />
 
   />
  
<hexboard size="5x8"
+
If Blue plays there other than at a, then Red plays at a.
 +
<hexboard size="5x7"
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-(a2 a1 b1 f1 g1 g2)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 R 1:d4 S area(d2,c4,c5,d5,e3,e2)"
+
   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. 
  
<hexboard size="5x8"
+
Thus Blue's only remaining hope is to play at a.
 +
 
 +
<hexboard size="5x7"
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-(a2 a1 b1 f1 g1 g2)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 R 1:d4 S area(d2,c4,d5,f5,f3,e2)"
+
   contents="R arrow(12):b2 a3 a4 d3 B b3 a5 1:c4"
 
   />
 
   />
  
<hexboard size="5x8"
+
Red responds like this:
 +
 
 +
<hexboard size="5x7"
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-(a2 a1 b1 f1 g1 g2)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 R 1:d4 S area(d2,c4,c5,f5,f3,e2)-d5"
+
   contents="R arrow(12):b2 a3 a4 d3 2:b4 4:e2 B b3 a5 1:c4 3:b5 E *:d1 *:c3"
 
   />
 
   />
  
Blue's [[mustplay region]] consists of the intersection of the carriers of these threats, which means that Blue's only hope is to play at 1.
+
Red's 4 is now connected to the bottom via [[Edge_template_IV2b|edge template IV2b]], and to
<hexboard size="5x8"
+
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.
   edges="bottom"
+
 
 +
== 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"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-area(a1,a5,e1)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 B 1:d4"
+
   contents="R d3 B e2 f2 j2 k3 g1--j1"
 
   />
 
   />
Red responds like this:
+
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="5x8"
+
<hexboard size="6x11"
   edges="bottom"
+
   edges="bottom right"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-area(a1,a5,e1)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 B 1:d4 R 2:c4 B 3:c5 R 4:f2 E *:e1 *:d3"
+
   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.
  
The group containing 4 is now connected to the bottom via [[Edge_template_III2b|edge template III2-b]], and to
+
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.
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.
+
 
 +
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 ==
 
== 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.
 
Tom's move also works when the hex marked "a" is not empty, provided that "b" connects to Red's main group.
<hexboard size="5x8"
+
<hexboard size="5x7"
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-a1--c1 g1 h1 h2"
+
   visible="-(a2 a1 b1 f1 g1 g2)"
   contents="R a1--a4 b2 b4 c2 B a5 b3 b5 c3 R e3 E a:d1 b:e1"
+
   contents="R arrow(12):b2 a3 a4 d3 B b3 a5 E a:c1 b:d1"
 
   />
 
   />
  
Line 149: Line 171:
 
   edges="bottom"
 
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="-g2 h2 h3 f1--h1"
+
   visible="-g2 h2 h3 f1--h1 -area(a1,a5,c1)"
   contents="R b4 b5 c3 B a6 b6 c4 R e4 E b:e2 R d1 B a:d2 R c2"
+
   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 ==
 
== See also ==
  
* [[Tips_and_tricks#2nd and 4th row parallel ladder escape|2nd and 4th row parallel ladder escape]]
+
* [[Parallel ladder]]
 +
* [[Edge template IV1d]]
 +
* [[Fifth_row_edge_templates#V-2-m|Edge template V2m]]
  
 
[[category:ladder]]
 
[[category:ladder]]
 
[[category:Advanced Strategy]]
 
[[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