Hex

1
Hex

• A game of connections

2
The Beginning

• Invented independently by Piet Hein in 1942 and
  John 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

3
The Board
4
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!

5
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.

6
Edge Connection Templates
2 out connection

• A good example of more complex 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!

3 out connections
4 out connections ?
7
Almost Virtual Connections

• A subgame in which if we 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.

8
Complex Virtual Connections

• The advantage to virtual 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
  Reds rather poor follow-up move

9

• And Blue wins, because he already has a full
  connection from edge to edge!

10
Discovering Virtual Connections
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Recursive Virtual Connections

• By applying three rules, we can find some virtual
  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

12
The And Rule

• Assume we have
• 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

13
The And Rule Empty Intersection

• The two connections must have an empty
  intersection
• If they do not, one opponent move can challenge
  both components

14
The Almost Rule

• Assume we have
• Connection from A-B
• Connection from B-C
• B is empty
• A-B and B-C do not intersect
• Then the union of A-B, B-C, and B is an almost
  connection from A-C

15
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

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The Or Rule - continued
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Recursive Virtual Connections
Trivial Connections
AND Connections
ALMOST Connections
OR Connections
When no new connections are found, we are done.
18
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

19
Proving a win for a player

• If a player has an edge-to-edge connection, they
  have a guaranteed win.

20
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.

21
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.

22
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.

23
Limiting the search tree
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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 players almost
  edge-edge connections.

25
Mustplay Region for D5
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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.