Difference between revisions of "David Boll Hex FAQ"

Latest revision as of 19:08, 28 December 2020

This page is a lightly edited version of David Boll's Hex FAQ [1], written in 1994. Please note that some of the advice contained in this page is outdated; for example, modern players use the swap rule, so opening in the center is not a viable opening.

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

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:

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 Red and Blue, with Red having the first move. For notational purposes, the board is indexed by letter and number, like so:

Elementary concepts: the 2-bridge

Connectivity is the key to Hex, consider the following game with Red to move next:

No matter where Red moves, Blue has the game won! There are only four hexes that matter in this game: A4, A3, C3, and C2 and Blue has a response to all four of these moves. If Red moves A4, Blue moves A3 (and vice versa), and if Red moves C3, Blue 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: (Blue to move)

Although the game might appear close at first glance, Red has a clear win. Red 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. Blue can't stop both of these connections with only a single move. So, if Red and Blue were good players, Blue would resign. Of course, if Red and Blue 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 Red was one row from the bottom edge, the template looks like this:

If the 2 hexes below Red are clear, Red is connected. If Blue moves to one of the dashed hexes, Red moves to the other. If Blue moves elsewhere, Red should also, since this connection is guaranteed.

3 rows away: The template is the following:

If the dashed hexes (or their mirror image) are clear, Red has a connection. The basic idea is that either B3 or D3 will connect, and Blue can't stop both. Red need not defend this connection unless Blue moves into one of the dashed hexes.

4 rows away: This is more complicated:

If Blue moves into this region, but doesn't move to one of {D5,E5,D6,C7,D7}, Red simply moves to D6 for an easy connect.

If Blue moves one of {D5,E5}, Red moves to the other and has a valid template from 6 rows out.

If Blue moves to D6 or C7, Red responds F5 and has 3rd row template.

What if Blue plays D7? You may want to try to work out a connecting strategy for Red that stays within the confines of this partial board.

The key response by Red (to a D7 move by Blue) is E5. At first, this appears to be moving directly into the defense, but Red is now threatening to connect from the second row out at both F6 and C6, and Blue 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: (Red to move and win)

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 Red, we better jump in between these two links and play E7, right? Wrong! E7 is easily defensed by D9. The correct move by Red (one of 'em, anyway, there's at least 2 others) is C8. A move to C8 pretty much forces a response of C7 by Blue, and now Red plays E7.

Under this scenario, Blue's defense of D9 won't work at all, Red just moves to D8 to secure the connection. The idea is Red made a move that forced a response by Blue (If Blue responds to C8 with E7, Red 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, Blue to play:

Note that Blue 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 Red moving to I6. Blue 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), Blue loses the game. But, suppose the situation was ever so slightly different: suppose Blue has an additional piece at H2, so the board looks like this:

Now Blue can play the same ladder sequence as before - but this time, Blue '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. Likely to get a response of H4 by Red. If Blue then tries G5, Red replies H5 - and Blue is being locked out of the right edge.
• b: Weak. Red plays F5 or E5, and is one row closer to home.
• c: Now Red responds with E5
• d: Could be good, provided Red can't connect with G4. I've never tried to work this out. I3 also looks to be a decent response for Red, 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 Red and one move by Blue, 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 Red, so perhaps placing a Red 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 Red and Blue, Red moves first. Blue is a bit weak in the opening, but the mid game is wide open after a questionable move by Red. Red 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 Red.
        F5  Blue chooses a weaker line of defense, G3 is usual. 
  2. H4     Red presses the attack on the north edge.
        I1  Blue plays the "classic" defensive move.  Note that G5 may look
            tempting here, but Blue is in trouble after H5.
  3. E4     Red shifts focus to the right side, threatening, but not estab-
            lishing, a link to the north.
        E3  Blue's piece at I1 allows this defense, but F1 might have been
            better.
  4. G2     Blue can stop Red's connection to E4 or H4, but not both!
        G3  
  5. I3     Red is now connected from the north.  J2 or H2 both get the link,
            and Blue can't stop both.
        E9  Blue shifts sides and tries to play defense on the south.
  6. H7     Nice, aggressive, offensive and defensive move.
        G10 Blue's game is all defense at this point.
  7. I9     Red connects to the south, choosing to fight for the link in the
            middle rather than along the edge.
        J2  Blue's first sign of life on offense.  Blue threatens H2 or I2, which
            links Blue from the right out to F5...
  8. H2     But, Red 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 Blue piece at H2,
  9. J5     Red tries to block,
        I4  And Blue saves the link.
 10. G7?    Not a terrible move, but D9 was probably better.  Red links up in the
            center with this move.
        G6! If Red responds with a knee-jerk F7 to save the link, Blue looks
            pretty good after G5.
 11. D6!    A nice reply to Blue's strong move.  Red de-fuses F5, and extends from
            E4, which could end up being connected to the north.
        G5  Blue makes the link anyway.  Perhaps Blue would have been better off
            moving on the south side here.
 12. F3     Red keeps the sure thing to the north.
        F7  Blue is now connected from the right out to F7/E9.
 13. C9     Once again, Red gets a link and plans to fight for connection in
            the middle.
        D5! A very sneaky threat.  If Red plays a knee-jerk E5, Blue plays D9,
            and forces Red to choose which side of C9 to play defense on.
            If north, Blue attacks south at B11, with some north ladder help
            at D5, and east ladder help at G10.  If south, C8 wins for Blue.
 14. C6!    But Red sees right through all that!  This move strengthens ties
            to C9, and allows Red to stop Blue even if Blue makes the E5 link.
        G1  Blue tries to throw some doubt in Red's ability to stop Blue from 
            connecting from D4 to the left.
 15. F2 F1  Red forces two ladder steps, which restores ability to stop
 16. E2 E1  Blue after Blue's threat of E5.
 17. H1     Now this is safe for Red.
        C8  Blue's last gasp at a win.  Inexact play by Red could lead to a
            ladder east then north to Blue'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; Blue has run out of tricks.
            Blue resigns.