This page shows various patterns of interest to those who play the
Game of Hex.
For an even larger collection of patterns, see the
Hex Templates
at Hall of Hexagons.
While the templates at the Hall of Hexagons are far more numerous, the
patterns on this page have the advantage that they come with explanations
of how they are constructed. That information informs your play in response
to a template intrusion by your opponent.
The templates presented here represent threats and virtual
connections.
A threat is a pattern that represents a position in which a player
may form, in a single play, a connection or virtual connection. A threat
is not a connection, nor even a virtual connection — it is the menace
that one might be formed.
A virtual connection (or vc) is a pattern that represents
a position in which the owner of the vc can force a connection. A virtual
connection occurs when two or more threats are present, and there is no
way for an opponent to interfere with all of them simultaneously. In other
words, a virtual connection is a fork.
Threats
These are the most commonly encountered threats. Here, the illustrated
threats will connect an owned cell with an edge.
Owned cells are shown in light blue. Cells that must be empty for the
threat to exist are shown in yellow. The threatened play is shown as
a yellow cell marked with a large blue dot. Simple forks are shown
by cells containing a lowercase letter, with each tine of the same fork
having the same letter. The fact that the bottom edge is blue's destination
is emphasized by the light blue border.
Threats can appear in several orientations, and they may threaten
to connect two cells together, rather than connecting a cell to an edge.
Here are the same threats again, in different orientations, and in their
cell-connecting guise.
Like threats, virtual connections may be between an occupied cell and edge,
or they can join two occupied cells. However, I have not (yet) collected any
interior vcs, so the following virtual connections are all edge templates.
8 8 8 8 8 8
4 . 0 * O . 6
. @ 0 O O .
4 0 0 . O O 6
\ ^ / \ ^ /
8 8 8 8 8 8
4 . O * 0 . 6
. O O 0 @ .
4 O O . 0 0 6
\ ^ / \ ^ /
2/3/3
8 8 8 8 9
4 . 0 * .
. 0 0 * 6
4 0 0 0 .
\ ^ ^ /
8 8 8 8 9
4 . 0 * .
. @ 0 * 6
4 0 0 O .
\ ^ ^ / -
8 8 8 8 9
4 . O * .
. O O * 6
4 O O @ .
\ ^ ^ /
Templates with only one owned cell, or templates with multiple but adjacent
owned cells, leave no doubt about which cell is virtually connected.
When there is more than one owned cell, and they are not adjacent, it can be
ambiguous. In such cases I mark the virtually-connected cell with an at-sign
('@').
In this next template, it's pretty clear which cell is virtually connected by
the template, but I mark it anyway for the sake of consistency.
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O ! ! O O . . .
4 . O O 0 @ O O O O . 6
. O O 0 0 0 O O O O .
4 O O 0 0 0 0 O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O ! ! O O . . .
4 . O O O @ 0 O O O . 6
. O O O 0 0 0 O O O .
4 O O O 0 0 0 0 O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O O{x} O{x} O O . . .
4 . O O O O{x} O O O O . 6
. O O O O{x} O{x} O O O O .
4 O O O O{x} O{x} O{x} O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * ! . . . . 6
. . . O O{x} ! @ 0 . . .
4 . O O O 0 0 0 0 0 . 6
. O O O 0 0 0 0 0 0 .
4 O O O 0 0 0 0 0 0 0 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . 0 @ O{x} O O . . .
4 . 0 0 0 0 0 O O O . 6
. 0 0 0 0 0 0 O O O .
4 0 0 0 0 0 0 0 O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O O O O O . . .
4 . O O O O{x} O O O O . 6
. O O O O{x} O{x} O O O O .
4 O O O O{x} O{x} O{x} O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . 0 @ 0 O O . . .
4 . 0 # 0 0 # 0 O O . 6
. 0 0 0 O{x} 0 0 0 O O .
4 0 0 0 0 0 0 0 0 O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O 0 0 O O . . .
4 . O O 0 @ 0 O O O . 6
. O O # 0 0 # O O O .
4 O O 0 0 O{x} 0 0 O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O 0 @ 0 O . . .
4 . O 0 # 0 0 # 0 O . 6
. O 0 0 0 O{x} 0 0 0 O .
4 O 0 0 0 0 0 0 0 0 O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O O O O O . . .
4 . O O O O{x} O O O O . 6
. O O O O O O O O O .
4 O O O O{x} O O{x} O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
(To be completed. I don't yet know how to respond
to the three remaining attacks.)
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O O O O O . . .
4 . O O O O{x} O O O O . 6
. O O O O O O O O O .
4 O O O O O O O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O O O O O . . .
4 . O O O O O O O O . 6
. O O O O O O O O O .
4 O O O O{x} O O O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8 8
4 . . . O * O . . . . 6
. . . O O O O O . . .
4 . O O O O O O O O . 6
. O O O O O O O O O .
4 O O O O O O{x} O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
This template contains a four-way fork. The first three tines of the
fork are relatively simple; unfortunately, all three overlap in
one place, marked x. That overlap means that a fourth tine is
required that avoids x. In this case, that fourth tine uses
all of the template except for x.
8 8 8 8 8 8 8 8 8 8
4 . . O O O . . . . 6
. . * O O O O . . .
4 . 0 0{x} O O O O O . 6
. O @ O O O O O O .
4 O 0 0 O O O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8
4 . . O O O . . . . 6
. . * 0 O O O . . .
4 . O 0{x} @ 0 O O O . 6
. O O 0 0 0 O O O .
4 O O 0 0 0 0 O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
8 8 8 8 8 8 8 8 8 8
4 . . O O O . . . . 6
. . * O O O O . . .
4 . 0 @{x} 0 O O O O . 6
. # 0 0 # O O O O .
4 0 0 O 0 0 O O O O 6
\ ^ ^ ^ ^ ^ ^ ^ ^ ^ /
This completes the proof of the original template.
There is an alternate way of presenting the middle of the
foregoing proof. Instead of presenting the fourth tine of the top-level
fork and its first two subtemplates, we can view the first three tines
of the original template as forcing a play in their sole overlap. This
leads to to a series of moves where all the opponent's responses
are forced. At the end of the sequence, we arrive at the same six-way
fork that was just shown.