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Hex
A game of connections
The Beginning
• Invented independently by Piet Hein in 1942 and John
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Nash in 1948.
Played on a grid of hexagons. Standard size for
tournaments is 11x11 or 14x14.
Object of the game is to make a continuous chain of
your colour to connect your sides of the board.
Simple Rules
– Play anywhere: Very wide game tree
– Swap Rule: Helps balance the game
• Early Game Proofs
– First Player Win
– No Ties
The Board
Virtual Connections
• What is a virtual connection?
– Subgame: goal is to connect two board
positions, not necessarily the edges.
– A guarantee that for a given subgame, even if
the opponent plays first, you can still win the
subgame.
– An edge to edge virtual connection for a
player means they win!
Basic Virtual Connections
• Simplest Basic Virtual
Connection is the “two
bridge”
– The two blue pieces here
form a two bridge, and the
lower piece forms a two
bridge with the edge.
• In diagrams, we
represent virtual
connections by colouring
all the pieces required to
maintain the connection.
Edge Connection Templates
2 out connection
• A good example of more complex
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virtual connections are edge
templates.
These templates are virtual
connections from the edge to a
piece on the board.
The farther out the piece, the
more pieces that are needed to
connect it to the edge.
Very useful to know for learning
how to play Hex better!
4 out connections 
3 out connections
Almost Virtual Connections
• A subgame in which if we
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get one free move, we
play one piece to create a
new full connection.
The piece that creates
the full connection is
called the key of the
almost connection.
In our diagrams, a blue
key is coloured cyan and
a red key is coloured
magenta.
Complex Virtual Connections
• The advantage to virtual
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connections is the
number of pieces
required to win a game
on any board is much
smaller.
Sometimes, you can win
with only a handful of
pieces on the board!
Consider this board
position. Blue responds to
Red’s rather poor followup move …
• And Blue wins, because he already has a full connection from edge
to edge!
Discovering Virtual Connections
Recursive Virtual Connections
• By applying three rules, we can find some virtual
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connections
The connections found are a subset of all
connections, but they are still very useful
These rules were formalized in two papers
written by Vadim Anshelevich
As each connection is discovered, it is added to
a data structure for use by later rules
Every two adjacent positions have a trivial
connection joining them
The And Rule
• Assume we have
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Connection from A-B
Connection from B-C
B is our colour
A-B and B-C do not
intersect
• Then the union of A-B
and B-C is a
connection from A-C
The And Rule – Empty Intersection
• The two connections
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must have an empty
intersection
If they do not, one
opponent move can
challenge both
components
The Almost Rule
• Assume we have
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Connection from A-B
Connection from B-C
B is empty
A-B and B-C do not
intersect
• Then the union of AB, B-C, and B is an
almost connection
from A-C
The Or Rule
• Assume we have
– At least two Almost
connections from A-B
– A subset of these
connections has an
empty intersection
• Then, the union of
this subset is a
connection from A-B
The Or Rule - continued
Recursive Virtual Connections
Trivial Connections
AND Connections
ALMOST Connections
OR Connections
When no new connections are found, we are done.
Applications of Virtual
Connection Information
• We know of three good uses for virtual
connection information
– Proving a win for a player
– Evaluation Function
– Limiting the search tree
Proving a win for a player
• If a player has
an edge-toedge
connection,
they have a
guaranteed
win.
Evaluation Function
• In 1953, Claude Shannon and E.F. Moore
made a Hex-playing machine, based on
electrical resistances
• Vadim Anshelevich describes a method of
using electrical resistances in his Hex
papers.
Evaluation Function
• Each virtual connection is a wire between
two positions, with a resistance based on
the size of the connection, and the state
of the endpoints.
• A voltage is applied to the edges. The
resulting current that passes through the
system is the value of the evaluation
function.
Limiting the search tree
• On smaller boards (7x7 and less) one or
both players will often have almost
connections joining edge to edge within
the first few moves
• If the opponent does not play within this
connection, then the player can play the
‘key’ of the almost connection, forming a
full connection.
Limiting the search tree
Mustplay Regions
• If a player has multiple almost edge-edge
connections, then their opponent must
play a move that disrupts all such
connections.
• The opponent must then play a move in
the intersection of all of the player’s
almost edge-edge connections.
Mustplay Region for D5
Mustplay Regions
• Since a player is forced to play in their
mustplay region, we can eliminate parts of
the search tree.
• On smaller boards (7x7 and lower) most
lines of play have mustplay regions within
the first few moves
• On larger boards, mustplay regions will
only develop in the midgame.