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Revision as of 10:32, 21 March 2017
HEX: Answers to common questions.
This FAQ is copyright 1994 by David Boll. The rules are copyright by Pete Hein, and are used here without permission. Any suggestions, questions, or comments should be sent to David Boll: dboll@vcd.hp.com
Contents
Origin/Background of Hex
Hex was invented/discovered by Pete Hein. It a connectivity game, in the same family of games as Twixt, Bridge-It, and has some afficinados among Go players as well. It is a perfect example of one of those 'a minute to learn, a lifetime to master' games.
There's really not much Hex literature that I am aware of. Martin Gardner introduced the game in one of his Mathematical Recreations column, reprinted in his first book (I think), the one with Hexaflexagons & other stuff.
All other material I've read has been in the form of proofs of first player wins or variations on Hex. I'm not familiar with any strategic discussion of the game.
Rules of Hex
Hex is a two player strategy game played on a NxN rhombus of hexagons, as illustrated below for N=4:
Top
. . . .
. . . . Right
Left . . . .
. . . .
Bottom
Players alternately mark hexes. The goal of the first player is to form a unbroken chain of his hexes that connects the top to the bottom, while the second player attempts to form an unbroken chain of her hexes connecting the left side and the right. To make subsequent diagrams clearer, the 2 players will be referred to as H)orizontal and V)ertical, with Vertical having the first move. For notational purposes, the board is indexed by letter and number, like so:
A B C D
1 . . . .
2 . . . .
3 . . . .
4 . . . .
Elementary concepts: the 2-bridge
Connectivity is the key to hex, consider the following game with V to move next:
A B C D
1 V . . .
2 . . . H
3 . H . .
4 . . . V
No matter where V moves, H has the game won! There are only four hexes that matter in this game: A4, A3, C3, and C2 and H has a response to all four of these moves. If V moves A4, H moves A3 (and vice versa), and if V moves C3, H moves C2 (and vice versa). So, hex players consider hexes such as B3 and D2 to be connected, even though they do not touch, because a connection can be trivially forced. By playing moves of this type, you can extend two rows at a time rather than one. This document will refer to hexes 'connected' in this manner as a 2-bridge.
However, your opponent is not likely to let you march across the board, forming 2-bridges at will! Usually, connections are formed more subtly, as in the following example: (H to move)
A B C D E F G H I
1 . . . . . . . . .
2 . . . . . . . V .
3 . . V . . . . V .
4 . . . V . . . . .
5 . . . . . . V . .
6 . . V H H H V . .
7 . . . . . . H . .
8 . H . V . H . . .
9 . H . . . . . . .
Although the game might appear close at first glance, V has a clear win. V is connected from the north edge to hexes C6 and G6 (if you don't see this, refer to the section on edge tactics). And, each of these hexes are one move away from being connected to hex D8. Hex D8 is connected to the south edge (clearly), and one move away from being connected to the north edge in two different ways. H can't stop both of these connections with only a single move. So, if H and V were good players, H would resign. Of course, if H and V were good players, the game wouldn't look like this, but that's beside the point!
Edge techniques
It's handy to know when a hex is connectable to an edge regardless of what the other player does. This section discusses connection to an edge from 2, 3, and 4 rows out, using a connection template. A connection template is a pattern of OPEN hexes that will allow connection even if the opponent moves first, regardless of what the opponent does.
2 rows away: This hardly needs explaining, but suppose V was one row from the bottom edge, the template looks like this:
. V
- - <- edge
If the 2 hexes below V are clear, V is connected. If H moves to one of the dashed hexes, V moves to the other. If H moves elsewhere, V should also, since this connection is guaranteed.
3 rows away: The template is the following:
A B C D
1 . . . .
2 . . V -
3 . - - -
4 - - - -
If the dashed hexes (or their mirror image) are clear, V has a connect- ion. The basic idea is that either B3 or D3 will connect, and H can't stop both. V need not defend this connection unless H moves into one of the dashed hexes.
4 rows away: This is more complicated:
A B C D E F G
1 . . . . . . .
2 . . . . . . .
3 . . . . . . .
4 . . . - V - .
5 . . - - - - -
6 . - - - - - -
7 - - - - - - -
If H moves into this region, but doesn't move to one of {D5,E5,D6,C7,D7}, V simply moves to D6 for an easy connect.
If H moves one of {D5,E5}, V moves to the other and has a valid template from 6 rows out.
If H moves to D6 or C7, V responds F5 and has 3rd row template.
What if H plays D7? You may want to try to work out a connecting strategy for V that stays within the confines of this partial board.
The key response by V (to a D7 move by H) is E5. At first, this appears to be moving directly into the defense, but V is now threatening to connect from the second row out at both F6 and C6, and H can't stop both of them.
There are several other 4 templates, but this one is the most compact and symmetrical.
Advanced strategy: Forced moves, ladders, etc.
We discussed earlier how a 2-bridge is a fundamental connection concept in Hex. As we saw, if a player moves in one of the two link hexes, the other player moves to the other link hex to maintain the connection. Interetingly enough, however, there are cases where moving into your opponent's 2-bridge can be a good move - because the hex is valuable to you later. Here's a rather contrived example: (V to move and win)
A B C D E F G H I
1 . . . . . . . . .
2 . . . . . . . . .
3 . . . . . . . . .
4 . . . . V . . . .
5 . . . V . V . . .
6 . . . . . . . . .
7 . . . H . H . . .
8 . H . . . . . . .
9 . . . . . . . . .
From our study of edge connection techniques, we know that the hexes D7 and B8 are effectively connected to the left, and F7 is connected to the right. So, as V, we better jump in between these two links and play E7, right? Wrong! E7 is easily defensed by D9. The correct move by V (one of 'em, anyway, there's at least 2 others) is C8. A move to C8 pretty much forces a response of C7 by H, and now V plays E7.
Under this scenario, H's defense of D9 won't work at all, V just moves to D8 to secure the connection. The idea is V made a move that forced a response by H (If H responds to C8 with E7, V wins with C7), and the hex C8 turns out to be a valuable one later on. Moves such as this are termed forcing moves, and understanding them is important.
Another important concept in hex is the ladder. Ladders occurs when both players place their pieces along one row or column. They're usually forced by one of the two players, and they usually occur near an edge. Here's an example, H to play:
A B C D E F G H I
1 . . . . . . . . .
2 . . . . . . . . .
3 . . . V . . . . .
4 . . . . . . . . .
5 . . . . . . . . .
6 . . V . . . V . V
7 . . . H . . H . .
8 . H . . H . . . V
9 . . . . . . . . .
Note that H is connected from the left almost all the way across, but was one hex shy of having a valid 3-template on the right side with hex G7, prior to V moving to I6. H could force a ladder here with a sequence like (H6-I5-H5-I4-H4-I3-H3-I2-H2-I1) - but there would be no point in it. Worse, after a continuation of (I7-H7), H loses the game. But, suppose the situation was ever so slightly different: suppose H has an additional piece at H2, so the board looks like this:
A B C D E F G H I
1 . . . . . . . . .
2 . . . . . . . H .
3 . . . V . . . . .
4 . . . . . . . . .
5 . . . . . . . . .
6 . . V . . . V . V
7 . . . H . . H . .
8 . H . . H . . . V
9 . . . . . . . . .
Now H can play the same ladder sequence as before - but this time, H 'ladders down' to the piece at H2 to win the game.
I'll close this section with some strategic generalities about hex:
1) Always look for useful forcing moves.
2) When trying to connect to an edge, it is usually better to get an
unassailable link to the edge first, then try to connect to the link.
3) Play defense first, offense second - even when ahead.
4) Remember the potential of ladders, and learn to see them as a single
move.
5) If you move first, you will win with perfect play. Another way to
look at this is: Unless you make a mistake, you have a winning line
(as the first player). Find it. Often the knowledge that you 'should'
have a win helps you find it.
6) Know the fundamentals of opening and edge connection.
The opening
If you're the first player, the opening move is easy: open in the center hex. The second person has a few semi reasonable choices, marked with small letters in the diagram below:
A B C D E F G H I J K
1 . . . . . . . . . . .
2 . . . . . . h . . . .
3 . . . . . . f g . . .
4 . . . . . d c e . . .
5 . . . . . a b . . . .
6 . . . . . V . . . . .
7 . . . . . . . . . . .
8 . . . . . . . . . . .
9 . . . . . . . . . . .
10 . . . . . . . . . . .
11 . . . . . . . . . . .
a: Likely to get a response of H4 by V. If H then tries G5, V replies
H5 - and H is being locked out of the right edge.
b: Weak. V plays F5 or E5, and is one row closer to home.
c: Now V responds with E5
d: Could be good, provided V can't connect with G4. I've never tried to
work this out. I3 also looks to be a decent response for V, and both
plays are strong defensive moves as well.
e: Like c, only weaker
f: The 'classic' defensive move. Recall that moving in this position is
the key to the best defense from 4 rows out, it's also typically the
best position from which to defend on a wide open board.
g: Weak, leaves the left side of the board too wide open.
h: Seems to be a good defensive play, but I've never used it or seen it.
Less offensive potential than f.
So, if both players move typically, the opening will be 1. F6 G3, or some rotated version of this. After this, it's hard to say. D5 is tempting, it threatens to connect to F6, and is in the 'classic' defensive position with respect to G3. D4 looks OK also - it establishes a edge link at the cost of weaker defense on G3. H3 and I3 have possibilities as well, they both establish a link and play defense on the weaker side of G3. I tend to lean toward D4 and I3; one of my playing partners (who is a stronger player than me) seems to prefer D5.
Opening variations: The advantage of the first move is quite strong, and
sometimes you want to neutralize this advantage. The following are some
ways in which this can happen:
1) The first player cannot open on the main diagonal. I've only played this
version twice as the opener, one time I opened as close to the center
hex as possible, and another time I opened at F4 (on a 11x11). The
first player still has the advantage.
2) The often-discussed 3 move equilization: One player constructs a
position with two moves by V and one move by H, and the second player
chooses who plays which side. If the opening position is carefully
constructed, the game is quite playable.
3) Even simpler: one move equilization. It has been proved that opening in
the acute corner is a losing move for V, so perhaps placing a V near
an acute corner, then letting the other player choose sides would be
workable in Hex. I've tried this once, and it seemed OK.
Sample Game (11x11 board)
The following is an annotated transcript of a hex game. Follow along with this on a board, and you could learn all kinds of stuff. The players are H and V, V moves first. H is a bit weak in the opening, but the mid game is wide open after a questionable move by V. V ends up preserving the win by carefully maintaining the winning line throughout some tricky positions.
Moves Comments
1. F6 The center hex, the strongest opening move for V.
F5 H chooses a weaker line of defense, G3 is usual.
2. H4 V presses the attack on the north edge.
I1 H plays the "classic" defensive move. Note that G5 may look
tempting here, but H is in trouble after H5.
3. E4 V shifts focus to the right side, threatening, but not estab-
lishing, a link to the north.
E3 H's piece at I1 allows this defense, but F1 might have been
better.
4. G2 H can stop V's connection to E4 or H4, but not both!
G3
5. I3 V is now connected from the north. J2 or H2 both get the link,
and H can't stop both.
E9 H shifts sides and tries to play defense on the south.
6. H7 Nice, aggressive, offensive and defensive move.
G10 H's game is all defense at this point.
7. I9 V connects to the south, choosing to fight for the link in the
middle rather than along the edge.
J2 H's first sign of life on offense. H threatens H2 or I2, which
links H from the right out to F5...
8. H2 But, V shuts the door on that!
H6 This piece is linked to the right, even though it's not the
usual 4th row template, thanks to the H piece at H2,
9. J5 V tries to block,
I4 And H saves the link.
10. G7? Not a terrible move, but D9 was probably better. V links up in the
center with this move.
G6! If V responds with a knee-jerk F7 to save the link, H looks
pretty good after G5.
11. D6! A nice reply to H's strong move. V de-fuses F5, and extends from
E4, which could end up being connected to the north.
G5 H makes the link anyway. Perhaps H would have been better off
moving on the south side here.
12. F3 V keeps the sure thing to the north.
F7 H is now connected from the right out to F7/E9.
13. C9 Once again, V gets a link and plans to fight for connection in
the middle.
D5! A very sneaky threat. If V plays a knee-jerk E5, H plays D9,
and forces V to choose which side of C9 to play defense on.
If north, H attacks south at B11, with some north ladder help
at D5, and east ladder help at G10. If south, C8 wins for H.
14. C6! But V sees right through all that! This move strengthens ties
to C9, and allows V to stop H even if H makes the E5 link.
G1 H tries to throw some doubt in V's ability to stop H from
connecting from D4 to the left.
15. F2 F1 V forces two ladder steps, which restores ability to stop
16. E2 E1 H after H's threat of E5.
17. H1 Now this is safe for V.
C8 H's last gasp at a win. Inexact play by V could lead to a
ladder east then north to H's link to the east edge.
18. E7 The only win-preserving move in the area.
D8 Forced.
19. E8 Threatens to connect at D9 or F8; H has run out of tricks.
H resigns.
