Dr Eric Silverman

As you all have probably figured out by now, I really enjoy complicated board games — dense modern board games with tons of special components, 500-year-old Shogi variants with hundreds of pieces, all that stuff.  But I also have a great fondness for games on the other end of the scale: elegant abstract games with minimal rules and maximal depth.

Now an oft-cited example of this category might be Go — it’s certainly an elegant game, with rules that are easy to summarise yet a level of depth nearly unrivalled in board games.  But Go is also hard to understand, in that the goal is clear — secure more territory than your opponent — but working out how to get there is hard.  Most beginning players, myself included, are completely flummoxed by the empty board at the start of the game, and have no idea where to start.  And at the end of the game, it’s very difficult for newbies to figure out when the game is actually over!  There’s a reason a common proverb for beginning Go players is ‘lose a hundred games as fast as possible’ — building familiarity with the basics takes time and repetition.  It’s worth it, though.

But what I’m going to talk about here are games that are so simple as to be almost elemental, as in, it’s hard to imagine games with rules simpler than these.  For my money the best examples of these types of games are in the category of connection games.  In a connection game, players vie to be the first to connect key points on the board with their pieces — a simple goal, easy to see and easy to understand.  But underneath that these games offer surprising depths of strategy and tactics.

Now, the current bible for connection games is the book by Cameron Browne called — wait for it — Connection Games, which summarises the genre beautifully and includes rules and examples of play for numerous games.  It’s a great book that I certainly can’t compete with, so in this brief series of posts I’m just going to give you some details of my picks for the best games of this type, along with some useful resources and links to where you can play.

Hex

Any discussion of connection games has to start with Hex, the originator of the whole genre.  Strangely, despite the simplicity of these games, they weren’t around until quite recently.  Hex has a tangled history — now unravelled in the recent, and excellent, book Hex, Inside and Out: The Full Story — so I won’t attempt to summarise it all here.  The game was invented by Danish mathematician Piet Hein in 1942, and was initially called Polygon.  Hein sought to create a game that reflected his interests in topological properties of the plane and the four-colour theorem, and was stuck on this idea for some time, as any attempts to build his imagined game on a square grid didn’t work, as the players could easily become deadlocked.  Eventually he realised that a hexagonal grid would prevent this issue, and thus Polygon was born:

The Polygon board, an 11×11 rhombus composed of hexagons.

The rules of Polygon are incredibly simple:

• Players turns placing a single symbol of their chosen type — star or circle — in any empty hexagon on the board.  Once placed, symbols don’t move and can never be removed.
• The first player to connect the sides of the board marked with their symbol with an unbroken line of their symbols is the winner.

Easy, right?  But once he started playing, Hein realised the game was far more complicated than the rules suggested.  Soon after he launched the game in the Danish magazine Politiken with the board, rules and a call for challenging Polygon puzzles from readers.  It wasn’t long before pen-and-paper Polygon pads were selling like hotcakes all over Denmark, and the game became a bonafide hit.

The famous first-ever Polygon puzzle.  The circle player has the move.  How can they win?

Eventually Hein sold a 12×12 version of Polygon with a very nice wooden board called Con-Tac-Tix, which enjoyed some small success as well — and in fact you can still buy a version of this today from Hein’s grandson.  But the game didn’t really take off around the world until later, when famous mathematician John Nash (of A Beautiful Mind fame) rediscovered the game in 1948.

When Nash started sharing his discovery with colleagues at Princeton, the game rapidly gained adherents.  They often called it Nash, for obvious reasons, but legend has it some called it John instead — not because of Nash’s name, but as a nod to the fact they played it on the hexagonal tiles of the bathroom in the department!  Nash became Hex when Parker Brothers tried to market the 11×11 game with that name.  Around the same time Nash was attempting to market the game and was quite upset to discover he’d been scooped.  He wasn’t aware at that time that Piet Hein had in fact scooped him several years earlier anyway.

In any case, the game became an object of enthusiastic study by Nash and his colleagues, and they made numerous interesting discoveries about its properties.  Hex was largely just an object of interest for academics for the most part, as Parker Brothers’ attempt to sell the game didn’t amount to much.  A few years later the mathematician Martin Gardner played a pivotal role in the eventual worldwide popularisation of the game — his 1957 Scientific American column on Hex brought the game to a whole new audience.

Hex remains highly popular with mathematicians and computer scientists today, as well as with gamers, as it has some fascinating properties.  For example, draws are completely impossible in Hex — no matter how inept or random the players’ moves, eventually one of the players will always make a winning connection across the board.  This result is actually a consequence of something called the Brouwer fixed-point theorem, which I won’t get into here.  We also know that a winning strategy for the first player exists, but we have no idea what it is (well, we’ve found it by brute-force computer calculation for 9×9 boards and smaller, but not on the boards we actually play on).  A quick browse of the literature on Hex will reveal some fascinating contributions from big names in maths and computer science.

The current state of play

In the years since Piet Hein’s invention of Polygon, Hex has evolved somewhat.  The classic 11×11 board is still popular, since it has a nice balance of speed of play and intricacy.  Games on the 11×11 board are over relatively quickly, yet these 121 hexagons allow for a staggering 10⁵⁶ possible board positions, 10 billion times more than the number of possible Chess positions (10⁴⁶)!

However many Hex players nowadays are using larger boards, with 13×13 and 19×19 being particularly popular. 14×14 is fairly common as well, particularly as that was John Nash’s preferred board size. In any case, larger boards push the game further into the realm of strategy rather than tactics, allowing for deeper moves with greater subtlety. Here’s how a 13×13 Hex board looks today:

A 13×13 Hex board.

And here is a 19×19 Hex board that I designed and just had printed on a 19×19 neoprene mat.  The mat is 93cm x 56cm, and the hexes are large enough for use with Go stones.

My 19×19 Hex board.

In general we’ve abandoned the circles and stars of Polygon’s heyday and opted for the two players using black and white stones to mark their hexes, with the board edges marked accordingly.  Often you’ll see blue and red stones used instead.

More importantly, now that we know that Hex gives the first player a winning advantage, we play Hex using the swap rule, an ingenious way to even things out.  When the first player places their stone, the second player may choose to play one of their colour in response, after which the game proceeds normally, or they may choose to swap colours and take that move for their own first move!

This clever rule change means that the first player must intentionally play a weaker opening move to avoid a swap, thereby mitigating their first-player advantage instantly.  In practice the strongest opening moves are in the centre of the board, as these allow for connecting stones to extend in every direction, so generally the first player will play around the edges at the start to avoid a swap.  As you might expect, the first-player advantage is somewhat diminished on larger boards, given that the impact of individual moves is smaller in general.

Side note — the swap rule is often called the pie rule as well, as it mirrors the fairest way to divide up a slice of pie between two people: one person cuts, the other chooses which slice they will eat.

Playing Hex

So, once we’ve grabbed a funky rhomboid board of our preferred size, a couple piles of stones and sat round a table to play, how does the game actually work?  Here’s a quick sample game, showing me defeating a basic computer opponent on the 11×11 board:

I played this game using a fantastic bit of free Java-based software called Ai Ai, which has numerous awesome abstract strategy games available to play with a variety of AI opponents — find it here: http://mrraow.com/index.php/aiai-home/aiai/

The play in this game was reasonably simple, but if you jump onto the most popular site for playing Hex, Little Golem, and check out the larger boards you’ll soon see that the end result of a Hex game can look pretty complicated:

A game played earlier today on Little Golem (https://www.littlegolem.net/jsp/game/game.jsp?gid=2145661)

Black resigned after 70 moves, admitting defeat.  The reason why Black resigned may not be immediately obvious; after all, Black seems to have made good progress along the left side!  However, we can start to understand how games of Hex evolve once we understand some basic positions, particularly the bridge:

An example of a bridge: White’s stones 2 and 4 can be connected no matter what Black does.

The bridge means that connection between the two relevant stones is unstoppable.  As you can see above, if Black plays at A to attempt to break apart White’s stones, White simply plays at B, and vice versa.  The bridge is a simple example of a template, a formation of stones and empty hexes that facilitates an unstoppable connection.

If you look again at the sample games above, you’ll see several examples of bridges being used to establish connections between stones.  Using this formation is far more efficient than placing stones methodically next to one another, but the connection they provide is just as solid!  Using bridges and similar templates allows you to build connections in fewer moves.  As you learn more of these templates in Hex, you’ll be able to spot a win or a loss coming long before the final stone is placed.

By the way, now that you know what a bridge is, you should be able to solve Piet Hein’s puzzle above!

Another key concept of Hex is that defence and offence are the same thing.  Remember that in Hex one player will always win — from this we can work out that if we prevent any possible win by the opponent, that means we have to win instead!  So when playing Hex, don’t be focussed just on your own bridge-building and forget your opponent — spending your moves on blocking them still gets you closer to a win.  Sometimes the best offence is a good defence.

To get started with Hex, I suggest you just jump right in and start playing some games.  You can play Hex  on Little Golem, Richard’s PBEM Server, Amecy Games, Gorrion, Hexy.games and igGameCenter among others.  You’ll soon find that Hex is an intricate and precise game with enormous amounts of depth.  If you work on building bridges, blocking your opponent, and getting a general feel for the flow of the game, you’ll soon start to get the hang of the basics.

After losing a few times and hopefully stumbling across a win or two, go and visit Matthew Seymour’s incredibly detailed guide on Hex strategy.  His site is details key concepts like ladders and edge templates, walks you through some sample games, and provides lots of useful resources, plus everything is demonstrated through interactive diagrams!  It’s an incredible guide.  The bridge example above is a screenshot from this site, which I hope will encourage you to visit.  On the real site you can experiment and play moves on all the diagrams, which really helps cement the concepts explained in the guide.

Hex Variants

As you might expect with a game this elemental, numerous Hex variants have been devised over the years to spice things up.  There’s a tonne of these so I’ll just briefly highlight a few interesting ones:

Misère Hex:  Think of this as Opposite Hex — the first player to connect their sides of the board loses!  It’s an odd style of play to get your head around, where you need to force the opponent to connect while avoiding making progress yourself.  Interestingly, it’s been proven that the losing player has a strategy that guarantees every hex on the board will be filled before the game finishes.

Pex:  The rules here are the same as Hex, but the game is played on an unusual board — instead of hexagons, the board is tiled with irregular pentagons.  This changes the tactics significantly, given that the board spaces now have different connectivity, and makes for an interesting change of pace.   You can play Pex online at igGameCenter.

An 8×8 Pex board.

Nex:  This intriguing variant uses the standard Hex board, but alongside your White and Black stones you add neutral Grey stones.  Grey stones can’t be part of either player’s winning connection, so they are obstacles to both players.  But what makes this game brilliant is the new options available — a player’s turn now gives them two possible moves:

1. The player to move may add one stone of their colour AND one neutral stone to any empty hexes on the board, OR
2. They may swap out two neutral stones for stones of their colour, and then replace one stone of their colour with a neutral one.

This means that moves are not permanent in Nex — your stones can be recycled when the board situation changes, and seemingly innocuous neutral stones can suddenly become new threats for either side when they transform.

Just like in Hex, there are no ties and one player must win.  You can play Nex on igGameCenter.

A sample Nex game from the book Mathematical Games, Abstract Games — Black resigned.

Chameleon: Another intriguing variant that significantly changes up the play, Chameleon decouples players from colours.  In Chameleon, one player is Vertical and must make a connection of either colour from top to bottom, and the other is Horizontal and must make a connection of either colour from side to side.  On each turn a player may place a Black stone OR a White stone on the board on any empty hex.

The consequence of this is that players have to be aware of threats in the opponent’s direction from stones of either colour, making each move feel incredibly consequential!  It’s a bit of a mind-bender.  Chameleon benefits from playing on larger boards, as connections can happen too quickly on smaller ones given that players use both colours.  You can play it online using Richard’s PBEM Server.

What next?

Now that you’ve had an intro to the original connection game, you’ll be well-equipped to try your hand at Hex’s many fascinating cousins.  The basic concepts of Hex are helpful in a lot of other connection games too, although each of them adds their own unique wrinkles.

Over the next few posts, I’ll highlight some more connection games with interesting properties that are fun to play, including the Game of Y, TwixT, Havannah, ConHex, Unlur, and more.