Tag Archives: Abstract Games

Permute Update: Now available in Ai Ai!

Since my first post on my game Permute, there’s been a very exciting development.  Thanks to the efforts of Stephen Tavener — thank you, Stephen! — Permute is now playable in his wonderful abstract-gaming mega-package Ai Ai!

Ai Ai is a fantastic, and free, collection of many dozens of excellent abstract games, all playable online or against various strong AI opponents.  I’ve talked about it in my Connection Games series a few times, but I can’t emphasise enough how essential it is if you have any interest in this category of games at all.  Ai Ai includes everything from classics like Go, Chess and Draughts, to modern legends like Amazons, Havannah, Symple, and Catchup.

Ai Ai is particularly great if you like to experiment with games.  The platform is incredibly robust, and with some simple modifications to the MGL files that define the parameters of each included game, you can try out ludicrous variants of your favourite games and Ai Ai takes it all in stride.  As you can see in my post on Symple, you can play games on ludicrously large boards if you like, or modify starting positions, and so on.

Even better, Ai Ai is festooned with super-interesting analysis functions that you can use to investigate all the included games.  You can generate opening books and endgame puzzles, produce detailed statistics on game complexity, create detailed reports on branching factors throughout a typical game, and much, much more.  I used Ai Ai to generate a full report on Permute, which Stephen has uploaded to the Ai Ai website here.

A big part of the reason I was so excited to have Permute in Ai Ai is because of these analysis functions.  While my initial testing of Permute showed that the game is fun and allows interesting strategies to develop, there were a couple of lingering questions:

  1. Draws are theoretically possible on the recommended even-length board sizes (12×12 and 16×16).  How likely are draws in typical play?  Is it possible that high-level Permute play could become infested with draws?
  2. Permute does not use a balancing protocol like the swap rule we use in many other games like Hex or Havannah.  Is the game balanced enough as-is, or does the first or second player have an advantage?  Should I add a balancing protocol?
  3. Is it possible that symmetric playing strategies might break the game?

The Ai Ai report helped alleviate my concerns on these three aspects.  While of course these results shouldn’t be taken as gospel, I’m comforted by the fact that in 88,891 games played by the AI, not a single one was drawn!  On top of that, the winning chances for each side across all those games was 49.99% for Orange and 50.01% for Yellow — nearly perfectly balanced.  Finally, Ai Ai attempted to win with various mirroring strategies, but lost every game in those instances.  Permute might still prove to have issues on these fronts when attacked with superhuman neural-net AI, or super-strong humans, but at least I can rest assured that the game doesn’t break too easily.

Playing Permute in Ai Ai

When you load up Ai Ai, you can find Permute in the ‘Combinatorial 2020’ category, which you can find in a folder if you go to the File menu and click ‘Choose Game…’.  Once it loads up you’ll be presented with a dialog box to choose a few options:

  • Resign when hopeless?  This means that the AI will determine when it has no chance to win, and will resign at that point rather than playing on.  This is a very convenient feature, though for new players it might be worth playing a few games without it on, so that you see games all the way through to the finish.
  • Alternate setup?  This allows you to choose the alternate starting position with a 2×1 chequerboard pattern rather than the standard chequerboard.
  • Board size:  Here you can choose the size of the board, ranging from 8×8 to 24×24.  The default is 12×12, which is a good size to start playing on.  When you want a deeper, longer game, I’d go for 16×16.

After choosing your options, you’ll see something like this:

permute-screenshot1

Here I’ve loaded up a 16×16 game with the standard chequerboard setup.  If this is your first time starting Ai Ai, you may find the default will be for you, the human player, to play as Orange and the AI to play as Yellow, but you can change this to Human vs Human or AI vs Human or AI vs AI using the AI menu.

Stephen has implemented a very handy system for making moves in Ai Ai that uses mouse-dragging to determine which direction your twists will go.  To make a clockwise twist, locate the 2×2 face you want to twist, and click and drag from the top-left of that face to the bottom-right; to make a counterclockwise twist, drag from the bottom-right to the top-left.  After that, just click on one of your just-twisted pieces to bandage it, and there you go — your first Permute move!  If at any time you need a reminder of how the moves work, just click the Rules tab on the right side of your Ai Ai window.

Once you get used to the input method you’ll find Ai Ai is an incredibly convenient and flexible way to play the game.  By changing the AI thinking time in the AI menu, you can tailor your opponent to your skill level.  Beware, Ai Ai can be very strong if you give it lots of time!  To give you an idea of what Ai Ai plays like on higher thinking times, here’s a sample AI vs AI game played with ten seconds of thinking time per move:

This game was quite a good one, a close back-and-forth battle.  As is typical from the AI, the game was fought initially in the corners, and once territories took shape there, both sides extended into the centre to battle for dominance there.  This seems a good way to open a game of Permute in general — territory is easier to secure along the corners and edges with fewer bandaged pieces required, and once some gains have been made in those areas the protected groups can be used as a base to stake a claim on the centre of the board.

Just for kicks, here’s another sample game played on a 24×24 board, this time with 5 seconds of thinking time per move:

As readers of this blog will know, I generally love playing abstract games on larger boards anyway, but I particularly love playing Permute on big boards.  There’s something extremely satisfying about seeing these huge chequerboard patterns gradually coalescing into interestingly-shaped blocks of colour.  On the larger boards there are tantalising hints of fascinating strategies lurking in the distance; as you’ll see in the game above, the AI battled itself across the whole board, and intriguing local battles eventually linked together into larger contests as the game evolved.  Playing on a physical board this size might be a bit challenging, not just in terms of space but also in terms of keeping track of group sizes, but since Ai Ai takes care of both those problems, I highly recommend trying some bigger boards when you have time!  In truth 16×16 will stay my recommendation for tournament play, but I can say for sure that 20×20 and 24×24 have real potential, and the resulting games still take less turns than a game of 19×19 Go to play out, given that each move affects a decent-sized chunk of the board.

What’s next?

I hope the info above might convince you to give Permute a try using Ai Ai.  This program is essential for any fan of strategic games regardless, and the implementation of Permute is just perfect.  The AI plays a tough game, and you can easily experiment with larger board sizes and the alternate start position.  As you can probably tell, I’m hugely excited to have Permute available on Ai Ai, and I’m enormously thankful to Stephen Tavener for taking the time to implement it!

Hopefully this won’t be the end of exciting news for Permute.  I’ve been speaking with some very talented designers about the game, and earlier today I received a beautiful concept for a purpose-built physical game set for Permute that just blew me away.  Abstract games are a bit of a risk for publishers compared to more accessible, flashier board games with fancy bits, but nonetheless I do intend to keep investigating if this game could be realisable physically.  In the worst-case scenario, perhaps we could offer 3D-printed game sets for fans to purchase, if publishers don’t want to take a chance on it.

In any case, I hope you’ll download Ai Ai and give Permute a shot!  Let me know how you get on with it.  Keep an eye on these pages for more updates on the game, and hopefully some strategic tips and tricks as I gradually become less awful at it 🙂

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Permute: A Game About Twisting Things

As some of you are aware, one of my hobbies besides games is solving twisty puzzles, also known as 3D rotational puzzles.  The most famous example is the legendary 3x3x3 Rubik’s Cube, but since that set the world alight some decades ago a fascinating community of twisty-puzzle designers has emerged, producing some truly outrageous puzzles.  Here’s a few examples from my collection: 

So, as challenging as the Rubik’s Cube is, these days you can get puzzles that quite simply put it to shame.  I love the challenges presented by these amazing puzzles, and in recent months I’ve been trying to develop a way to bring the joy of twisty-puzzling into the world of abstract strategy gaming.

A new core behaviour: the twist

The key properties of twisty puzzles that makes them so challenging is the way in which the twistable faces of the puzzle interact with one another.  Any time you twist a face on the Rubik’s Cube, or any of the monstrosities above, you are forced to disrupt some of the work you’ve already done.  This creates a feeling of tension and danger when you’re first learning to solve a new puzzle; you’re acutely aware that at any moment, a wrong move or two could re-scramble the puzzle and essentially send you back to the beginning of the solve.

I wanted to capture this feel in the form of a two-player abstract game, so I began to cast about for examples of games that used twisting mechanics to shuffle pieces around.  Probably the most famous example in abstract games is Pentago:

Pentago Game from Mindtwister USA, Black-Natural/Solid Birch: Amazon.co.uk:  Toys & Games

In Pentago, players place marbles on the board and rotate the clever 3×3 sub-boards in an attempt to build a line of five of their pieces before the opponent.  The board rotation does create an enjoyable feeling of chaos in the game, but I had to immediately dismiss this idea for my game.  In a Pentago-type game with rotatable sub-boards, the sub-boards don’t actually disrupt one another; the relationships between stones can shift as they rotate around, but the sub-boards can’t actually scramble each other, as the faces do on a Rubik’s Cube.

I soon realised that the best way to replicate the behaviour I wanted would be to allow the players themselves to define the axes of rotation.  This wouldn’t really be possible with a physical board, though — how could you build a board where any sub-board of a certain size on it could twist?  

Instead, players would select an area on the board — a 2×2 or 3×3 subsection — and rotate the pieces within it, as if the board section below them had rotated like the face of a Rubik’s Cube.  This would capture exactly what I wanted: rotations could overlap with one another, allowing pieces to get twisted around and then re-twisted and scrambled up in other newly-created ‘faces’!

Then I embarked on a series of experiments to work out how best to implement these face-twists.  My first impulse was to allow players to rotate 3×3 sections of pieces, since the 3×3 Rubik’s Cube is so iconic.  However, I soon found that, while it was definitely fun, for a serious game 3×3 twists were simply too confusing.  The board state changed so much on each turn that trying to build strategic plans felt a bit fruitless.

I finally decided on 2×2 faces as the sweet spot — four pieces were still moving every turn, creating interesting situations on the board, but there wasn’t so much disruption that calculating future moves became impossible.  The core twisting behaviour of Permute was born:

Permute-twist-demo

Here Yellow selects a 2×2 ‘face’ of pieces and twists them 90 degrees clockwise.  At the start of the move, neither player had orthogonally-connected groups on the board; at the end of the twist, both players have two groups of three.

This behaviour would allow for the possibility of disrupting groups with further twists, which was another key concept of the game for me:

Permute-twist-demo response-01

After the move above, Orange strikes back by twisting a face just to the south of Yellow’s last move.  By twisting that face clockwise, Orange wrecks Yellow’s bottom-right group and boosts his own upper-right group from three connected pieces to six!

From here the overall shape of the game fell into place in my head almost automatically:

  • I wanted the players to focus on permuting pieces around the board, without additives like placing additional pieces or removing them through capture.  That meant the board should start already full of pieces.
  • The most interesting task to do with 2×2 twists would be to connect groups, and this would also mirror the act of ‘solving’ coloured pieces on a Rubik’s Cube.  I could keep the game tactically spicy by restricting connectivity to only horizontal or vertical; this would ensure that players could slice groups in two with twists that changed connectivity to diagonal only.
  • If the goal of the game is to build the largest orthogonally-connected group of pieces, then the fairest start position would be one where not a single piece of either side is connected orthogonally — a chequerboard pattern.
  • To ensure that players had to keep the whole board in mind and not just fight over the biggest chunk of pieces, the Catchup scoring mechanism — where if the largest groups are tied, then the player with the biggest second-largest group would win; and if those are tied, then check the third-largest, etc. — would be perfect.  That would ensure players would also need to build and preserve secondary groups, in case scoring went to the wire, and would prevent the game descending into a non-stop back-and-forth slap-fight over the largest group without opportunities to play distant strategic moves.

The game already felt nearly done!  I tested out the chequerboard starting position and twisting mechanics on my Go board with some colourful plastic pieces, and I found it was easy enough to play even with physical components.  Everything felt right so far, but I still had a problem:  how to get players to stop twisting?

Bandaging

A clear issue with the game at this point was a lack of termination.  Players could endlessly twist pieces back out of position, preventing their opponents from making any serious headway.  I needed a way for moves to have some finality, and create permanent changes in board state.  That’s when I decided to take a break and play some Slyde:

slyde16-10s-1

In Slyde, players take it in turns to swap one of their pieces with a horizontally or vertically adjacent neighbour of their opponent’s colour.  After the swap, the active player’s piece becomes pinned in place and can’t move for the rest of the game (and the opponent can’t swap with it). 

This was exactly the kind of thing I need for Permute!  Since a twist moves four pieces, and up to three of them could be of the active player’s colour (twisting four would be meaningless so I excluded that as a possibility), then a player’s move could consist of two parts: a twist in either direction, followed by fixing one of their pieces in place permanently.

That would accomplish what I needed — each move would have some finality, but since only one piece would be fixed in place, groups would still be in constant danger of disruption without further moves to shore them up.  Giving players a choice of which pieces to fix in place added an additional strategic element to the game, enabling players to try to optimise their twist/fix combo to achieve the best result in terms of securing territory and/or denying territory to their opponent.

With this final element now in place, I had a complete game — the initial position, goal, end condition and moves were all set.  I decided to call the piece-fixing ‘bandaging’, a term derived from twisty puzzles.  Bandaged puzzles have certain pieces glued together so that in some positions certain moves would be blocked; the term also refers to states in some puzzles where twists in certain directions are blocked.  The term comes from the fact that bandaged puzzles were made in the early days by using Band-Aids to stick pieces together on the Rubik’s Cube.

Playtesting

Now that the rules were set, I started playtesting the game, first with trial matches against myself.  The game seemed roughly balanced in my tests on 9×9, 10×10 and 12×12 board setups.  The core twist/bandage dynamic was enjoyable and gave each player’s turn a couple of interesting decisions to make, and each move felt like a tradeoff between securing territory and sacrificing future mobility, which was just the kind of feel I wanted.

The final test was a playtest match against Phil, which we did via a convoluted setup involving sharing my Adobe Illustrator screen over Google Meets.  Phil is quite good at most games he tries, so I felt confident he’d be able to tell if the game was obviously broken pretty quickly.  We had an enjoyable match, and true to form, Phil took a convincing win:

Phil told me that while it took a bit to get used to the twisting aspect, he could see that there was room for interesting strategies to develop, and he felt engaged by the action throughout the game.  At that point I felt it was an appropriate time to share the game with the wider world and get some more feedback, so I typed up the final rules and put together a thread on the BoardGameGeek Abstract Strategy forum.

The Rules

Here are the final rules, as presented on BoardGameGeek (well, tided up a bit):

The basics: Permute is a game about twisting things, inspired by twisty puzzles like the Rubik’s Cube. The name comes from one of the two main things we can do with pieces in a twisty puzzle: permute them (shuffle their positions); or orient them (change their facing). In this game players take it in turns to rotate 2×2 sets of pieces (‘faces’) on the board, in an attempt to bring pieces of their colour together in larger groups. Once a face has been twisted, part of it is locked in place (‘bandaged’) and can’t be twisted again. When no more twists are possible, the game is over and the players’ largest groups of pieces are scored. To win the game, you must permute your pieces so that they form the largest connected group, and deny your opponent the chance to do the same!

The rules: Play proceeds on a square board with a 9×9 grid (or larger). At the start of the game, all squares are filled with alternating Yellow and Orange stones in a chequerboard pattern.

Definitions:

Face: a 2×2 subset of the board surface. A face may not extend off the board.

Bandaged Stone: a stone with a token, sticker, or other marker on it that indicates it may not be twisted again.

Bandaged Face: a face containing one or more bandaged stones. A bandaged face cannot be twisted.

Twist: a move in which all the pieces in a face are translated around that face simultaneously 90 degrees in either a clockwise or counterclockwise direction, as if rotating the face of a 2×2 Rubik’s Cube.

Group: a group is a set of same-coloured stones connected orthogonally. The value of a group is the number of same-coloured stones it contains.

Orange plays first. The swap rule can be used – after Orange’s first move, Yellow may choose either to play their first move or change their colour to Orange.

Players then take it in turns to twist one non-bandaged 2×2 face containing at least one of their colour stones 90 degrees clockwise or anticlockwise. Once a face has been twisted, the player who twisted it must select one of their stones in that face and place a token on it, thereby bandaging it.  Faces containing a bandaged stone cannot be twisted.  Faces consisting entirely of one colour cannot be twisted either, so this is not a way to pass a turn (but mono-colour faces can be disrupted by twists of neighbouring faces, of course).

The game ends when no more twists can be made. At this point scores are compared. The player with the highest-valued group wins; if both players’ largest groups are equal in size, then compare the second-largest, then the third-largest, and so on until a winner is determined.  If the board is even-sided and the scores are somehow equal all the way down, then the game is a draw, but this should be very unlikely (and outright impossible on odd-length boards).

Translation for non-gamers

That looks like a lot of rules, but really it’s a pretty simple game!  There are two players, Orange and Yellow; Orange plays first.  Each turn, the active player must select a 2×2 sub-section of the board (a ‘face’) and rotate the pieces in it 90 degrees clockwise or counterclockwise, just as if they were rotating the face of a 2×2 Rubik’s Cube.  Once the twist is done, they must choose one piece of their colour in that face and bandage it; once a piece is bandaged, it can’t ever be twisted again.  

As the players make more and more twists and bandaging moves, gradually the board will get more and more constricted.  Since faces with bandaged pieces in them can’t be twisted, moves will be blocked and players will start to have secure territories built up.  Once no more moves are possible at all, players count up their largest groups of pieces of their colour; a group is a set of pieces that are connected horizontally or vertically, diagonal connections don’t count!  See the pictures from the game between Phil and myself for a scoring example.

The player who built up the largest group of their colour wins the game.  If both players’ largest groups are the same size, then compare the second-largest groups of each player, and the largest of those two groups wins.  If those are still tied, then check the third-largest, and so on.  

So, winning a game of Permute means you have to bring your pieces together into connected groups, but because twists can disrupt so much of the board, you have to work hard to protect them!  That means bandaging pieces strategically, to hopefully prevent your opponent from tearing apart everything you’ve worked so hard to build.  Once you play for a bit, you’ll start to see ways to build your groups while simultaneously blocking or disrupting your opponent, and that’s when you’ll start to really enjoy what Permute has to offer.

Alternate starting positions

The default chequerboard starting position works well, which is why I chose that as the ‘official’ starting position in the rules.  However, during testing, Phil had suggested the possibility of an alternate starting position that might be easier on the eyes.  We worked out that a chequerboard pattern of 2×1 blocks could work well, and had another advantage in that early-game twists would immediately create some bigger connections, which could be helpful for new players who may have more trouble seeing groups right away:

In the discussion on BGG, Steven Metzger pointed out that playing on a 13×13 board would forbid the possibility of draws, and would also mitigate a possible first-mover advantage by giving the second player a stone advantage:

F2L-13x13 -- NEW start position --Orange-Yellow-01

Ultimately I’m not sure that draws will be much of a problem anyway, as maintaining precise parity across every group down the size order would be pretty unlikely, but it’s good to have the option.  Plus in a matchup between two players of uneven strength, giving the weaker player the side with extra stones on the board in this setup could help them be competitive.

However, it’s not immediately clear how to replicate the alternative 2×1-chequered start position on an odd-length board; Phil had some ideas about this which could work, but the setup would be more awkward on a physical board.  We’ll keep trying though, eventually we’ll find a good alternative.

Permute on MindSports

I was generally pleased by the reaction on the BGG forums; most posters seem interested in the game, and had some good suggestions about the visuals.

Most exciting for me was that Christian Freeling, a designer I’ve spoken about quite a bit in these pages, was immediately positive about the game.  This meant a lot to me, not just because I’m a fan of several of his games, but also because he’s got a very strong intuitive sense about whether a game will work or not; for him to say that he felt “it is immediately obvious that it works (without endless modifications)” gave me a big boost in confidence.  

Christian is also the proprietor of MindSports, a website that hosts all of his games for online and AI play, as well as some games from outside contributors.  Lucky for me, Christian and Ed van Zon decided to implement Permute on MindSports, so now anyone can play Permute against the AI or against other people (via the MindSports Players Section)!

This was tremendously exciting for me — not only is Permute now playable easily in a digital format, but it’s sat in the MindSports website right below Catchup and Slyde!  As I described above, these two games gave me inspiration I needed to get Permute to its final form, and both are really excellent games, so I feel privileged to be sharing a page with them.

I’ve spent the weekend making some YouTube videos about Permute and writing this post, so I haven’t yet dived into online play, but I did have a couple of matches against the AI.  The AI isn’t super strong but it’s still a fun time and a great way to learn the game:

Now that my first promotional push for the game is completed, I’m happy to accept challenges for games on MindSports, so please let me know if you fancy a game 🙂

Where next?

I’m really happy with how Permute turned out, and as people are playing it here and there I’ve had some great feedback on it.  That being the case I’m not planning to make any further changes to it, beyond perhaps adjusting the starting position if computer analysis finds a strong advantage for either player or something.

However, the core twisting mechanism does have lots of potential for future development.  I have two new twisty experiments I’m working on right now: a four-colour twisty game on a hexagonal grid; and a square-grid game where players only twist, and no bandaging happens.  The latter is a difficult design challenge, so if you have thoughts about it feel free to air them in the BGG discussion thread on the topic!

Twisty experiment -- game 1-01

The initial test of the idea in that thread (shown above) has some potential, but definitely needs some work.  In this game, players only twist 2×2 faces, and pieces become fixed in place (‘solved’) when they join a group of pieces connected to three or more neutral edge pieces.  There are some other ideas in the thread that I think are worth investigating too, and ultimately I think some synthesis of these concepts will produce a good game.  However I’m going to let all this simmer in the back of my head for awhile, and keep most of my attention on enjoying Permute for now.

In the meantime, I hope some of you out there will give Permute a try!  Go check out MindSports, have some games against the AI, and get in touch if you want to have a game with me.  I hope that some more strong players will have a go at the game, and that soon we may see some interesting tactical and strategic concepts develop.

I’ll do some follow-up posts on Permute in the future and show off some sample games with interesting play, so please look forward to that.  At some point too I’ll reveal Permute’s other twisty siblings once they’re in good shape 🙂 

If you’re dying for more Permute content, please do check out my YouTube videos: I have a short intro to Permute with some sample moves; a longer intro with a full sample game against the AI; and finally a video introducing Catchup and Slyde alongside the wonderful Ai Ai game-playing platform.

So, give the game a shot and let me know what you think!  Perhaps I’ll see you on MindSports.  Before I go, I wanted to say another heartfelt thanks to Christian and Ed for putting Permute up on MindSports, and to Nick Bentley and Mike Zapawa for creating Catchup and Slyde respectively, without which Permute might have just stayed as a weird twisty concept in my head and never become a playable game.  

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Symple: a game that matters

UPDATE 1 MAY 2020: Added ‘Playing Symple over the board’ section, and downloads for the Ai Ai .mgl files for large/oblong Symple boards and HexSymple.

Way back in Connection Games III: Havannah and Starweb, I praised designer Christian Freeling’s games but expressed a bit of skepticism regarding his list of six ‘games that matter’:

Christian has invented a tonne of well-regarded games over the years, and he has his own opinions on the most essential ones — namely Grand Chess, Dameo, Emergo, Sygo, Symple and Storisende.  Although I’m not sure I can agree with most of them, personally speaking — that list is mostly games I certainly admire, design-wise, but don’t particularly enjoy playing.

I still stand by most of that — I do admire all of those games, and still I’m not a super-fan of several of them.  But I’m here today to tell you that I was wrong, in fact very wrong, about two of them:  Symple and Sygo.  Today I’ll tell you about Symple, designed by Christian Freeling and Benedikt Rosenau, and in a future post I’ll introduce its descendant Sygo as well.

Full credit for this change of heart must go to David Ploog, who sent me a draft of an excellent article he’s writing for Abstract Games Magazine about games featuring many moves per player turn, with Symple being the star of the piece.  His explanation of the game massively piqued my interest, so I started exploring it via Stephen Tavener’s Ai Ai software — which I’ve recommended many times already, you really should download it!

What I discovered is that Symple is not just an ingenious piece of invention, it is that most elusive kind of ingenious invention — one that you see in action and think ‘how did no one think of this before?’  The core of it is easy to grasp, yet in play it surrounds you with staggering complexity while still remaining manageable.  Having been obsessed with it now for a little while, playing many games, and analysing many others with the AI, I’m convinced that this game is something truly special.  Had it been invented hundreds of years ago, I think today it’d be sat alongside Chess, Go and Shogi as one of the great traditional games.

Enough gushing — from here I’ll explain the game, show off a few example games, and maybe gush a bit more here and there.  In deference to David’s article, which will explain strategic concepts to you far more clearly and expertly than I could, I’ll shy away from detailed playing tips and simply direct you to play it on Mindsports or via Ai Ai and explore it until David’s article is out.

How to play Symple

Symple is best played on a Go board, using standard black and white Go stones.  I’d recommend the full 19×19 Go board (or even larger — more on that later), although for the first few games a 13×13 board or even 9×9 would be a good idea, to get used to the core concepts.  Here are the rules:

  1. Each player chooses a colour, Black or White — White always goes first.  Before starting, players should agree on an integer value P, which will affect the scoring at the end of the game.
  2. Key definition: a group is a set of horizontally and/or vertically adjacent stones (orthogonally adjacent, in other words) of the same colour.  A single stone is also a group.
  3. Making moves: on their turn, a player must do one of the following (players may not pass their turn):
    1. Plant: place one stone anywhere on the board that is not adjacent to any stones of the same colour, which creates a new group
    2. Grow: add one stone to every possible group of their colour on the board, by placing a stone on a vacant point horizontally or vertically adjacent to that group.  If a group has no vacant horizontally or vertically adjacent points, then that group may not be grown.
  4. Balancing mechanism: once per game, if neither player has yet made a growth move, Black may grow all of their groups and then plant one new group in the same turn.
  5. Move restrictions: if a group grows and the added stone touches another group, then both groups are considered to have grown (meaning the group the new stone touched now can’t grow this turn).  However, two groups may grow in such a way that only the two new stones are now adjacent.
  6. Scoring: the game ends once the board is completely full.  At that point, both players count up the total number of their stones on the board, and the number of separate groups they have.  Their final score is the total number of their stones, minus points for every group they have on the board.

To summarise, in Symple players seek to claim territory on the board for their groups of stones to grow in by first playing planting moves, then growing those groups all at once with subsequent growth moves.  Creating groups early in the game is important in order to claim territory and secure space for future growth, but creating a new group takes an entire turn for just one stone placement; conversely, during a growth turn, a player may place a huge number of stones in one turn, sometimes 10 stones or more during a 19×19 game.  The scoring system gives players a penalty of P points for each group of their stones at the end of the game, meaning that connecting one’s groups is paramount.

Growth turns give players an enormous array of choices.  A typical early-game growth turn might look something like this:

symple19-growth-example2

Here White is taking a growth turn, as depicted in the Ai Ai software.  In Ai Ai, groups that have a growth move already chosen are faded out, as in the top left of the image, and new stones are indicated with a ‘+’.  In this instance White has taken advantage of the two-stone separation between the two groups on the upper left to grow both groups in such a way that they are now connected.  Around the other groups, the green asterisks indicate where legal growth moves can be played.  At the end of this turn, White will have played up to a total of eight stones, one for each group.  Note that White could end up only playing seven, if they elect to grow one of the two top-right groups into connection with the other; that would then block the second group from growing that turn, and from then on they would be a single group.

Playing a fistful of stones in one turn is initially intimidating, but these massive multi-moves naturally keep one’s mind focussed on strategy over tactics.  You will find yourself considering the optimal directions of growth to restrict your opponent, facilitate your expansion, and develop opportunities for later connections between groups.  The growth mechanism makes the game feel organic and flowing; more than single stones, you’re manipulating amorphous, amoebic groups that ooze and coalesce across the board.  The feel of this game in play is unlike any other abstract strategy game I’ve played before.

Examples of play

Let’s look at some examples of completed games, to get a better idea of how the scoring system works.  A finished game of Symple looks something like this:

symple19-p10-10s-sample1

A finished game of Symple on a 19×19 board (P = 10).  Black wins, 128 points to 83.

In this game, the players agreed to play with P set to 10 points.  Black finished with 5 groups totalling 178 stones, for a final score of 178 – (5 * 10) = 128; White finished with 10 groups totalling 183 stones, for a final score of 183 – (10 * 10) = 83.  Black solidly outmanoeuvred White here, connecting more groups together to significantly reduce their point penalty and take the win.  Note White’s unfortunate 1- and 2-stone groups on the bottom right — these alone took 20 points from White’s score!

This GIF shows off the whole game:

symple19-p10-10s-sample-game1

The endgame in Symple can be quite challenging and subtle, as in this close game:

symple19-15s-sample1

A finished game of Symple on a 19×19 board (with P = 10).  White wins, 79 points to 62.

Here both players finished with 10 groups, but White managed a win.  By restricting Black’s ability to grow certain groups earlier in the game, White eventually forced Black to play stones into isolated squares in the late stages of the game, causing significant extra scoring penalties that secured the game for White.  Here’s the complete game in GIF form:

symple19-15s-sample-game1

In Symple, managing your growth carefully and strategically is very important, as the final score difference may end up coming down to ensuring that one’s final stone placements aren’t forced to be new, point-draining groups.

The balancing mechanism provides some great tension in the early game, as well.  Black has one opportunity to grow and plant in the same turn, but White knows this, of course, and can short-circuit that chance by playing a growth move earlier than expected.  But growing too early can be too committal, losing an important opportunity to plant a new group in key territory.  This dynamic provokes a pleasing little game of chicken, as both players try to suss out their opponent’s rhythms and strike at the right time — “should I do my double move now, or will White wait another turn to do their first growth turn?”

On the whole, Symple has a great flow to it, and every phase of the game feels consequential.  In the opening, players plant all over the board, attempting to claim space for future growth while impeding the opponent’s opportunities for later connection.  All the while the will-they-won’t-they tension of the balancing mechanism lurks in the background.  In the middlegame, players switch over to growth moves and their groups extend their tendrils across the board, competing with opposing groups for territory.   In the endgame, Symple turns ‘cold’, as players turn from aggressive expansion to cautious growth to avoid getting hemmed in, while trying to force their opponent into positions where they’re forced to plant a stone somewhere unfortunate.  The final result is shaped by key moments in each of those phases, making the whole experience feel cohesive and dynamic.

Playing Symple over the board

Playing Symple is very easy when using a computer program or web-based implementation, since the software will track group sizes and scores for you.  When playing on a real board, however, a bit more effort is required to keep track of things.

A single turn in Symple can require a lot of individual moves for each player, and one can easily get confused as to which group has already been grown.  Most players recommend using a second, easily distinguishable type of token or stone to mark your intended growth moves, and replacing them with normal stones once you’ve decided on all your moves for that turn.  That allows you to think about each growth move without getting confused about which group can still grow.  If you’re using a Go set and want to maintain the austere aesthetic of black and white stones, then consider using Chinese-style Go stones with one flat side as your markers, and Japanese-style double-convex stones for regular plays.

The other aspect is scoring, which in Symple involves a lot of counting.  However, the game continues until the board is completely full, so as David Ploog pointed out in a BGG discussion, scoring can be done quite efficiently: simply count the number of groups for each player, then remove all stones of one of the colours from the board, and count the stones for the other colour.  Since you’ve recorded the numbers of groups, you can freely rearrange the remaining stones into an easily-countable shape, too.  That’s all the info you need to then calculate the scores for both players.  Using this method, scoring a game of Symple shouldn’t take any longer than scoring a Go game.

An eminently flexible game

Alongside the straightforward rules, unique gameplay and immense strategic depth, Symple has some practical advantages that add even more interest.  An important element of the design is that the value of the group penalty is not fixed, and players can experiment with different values.  Smaller values reduce the emphasis on connecting groups, while larger values make it even more essential.  David Ploog recommends P = 10 for 19×19 games, and in my experiments so far I agree; I’ve also played some 19×19 games at 12, 14 and 16 and have enjoyed those too.  When experimenting with different values, bear in mind that on boards with odd numbers of squares, you should use an even value for P to ensure that draws are not possible.

The enormous multi-move turns of Symple also mean that the game is incredibly scalable.  Symple plays remarkably quickly even on 19×19, since each turn can easily provide 10 or more stone placements — and these mechanics tend to emphasise strategic concerns over tactical ones, which helps to keep the game from bogging down with excessive calculation for every stone placement.  As a consequence of these unique properties the game plays well even on ludicrously large boards.  Here’s a game I played against the AI on a 37×37 board:

symple37-p18-2s-close1

A game played against the AI on a 37×37 board (P = 18).  I won as Black by only 3 points, with a final score of 434 to White’s 431.

This game was huge, but still surprisingly playable; in the middlegame we were often placing well over 20 stones per turn, so even with 1,369 squares to fill the game moved at a good pace.  The result came right down to the wire — I won by only 3 points.  If you’re bored you can watch the whole game in animated GIF form here.

I’ve definitely never played an abstract game before Symple that could take place on a board that large and remain playable and fun.  Go is one of my all-time favourite games, but play on the standard 19×19 is already very challenging; I’d never go near 37×37 Go.  Symple’s mechanics mean that the board fills quickly, and strategy reigns supreme over tactics, and so even on boards this large one doesn’t feel too hopelessly confused.

I’ve also found that Symple presents some interesting challenges on rectangular boards.  Here’s a game played on a 19×29 board:

symple-rect-19x29-p12-2s-close1

A game of Symple on a 19×29 board (P = 12).  Black wins, 117 points to 110.

This was a test game between two AIs, played when I first modified the Symple file in Ai Ai to permit rectangular boards.  Note that both players took advantage of the strange board geometry, growing huge groups horizontally across the board.  You can view the whole game in GIF form here.

Note that Ai Ai by default only supports square boards up to 19×19; I have modified the Symple.mgl file in Ai Ai to permit rectangular boards with sizes up to 37 in either dimension.  You can download the .mgl file needed in this Google Drive folder; simply add it to the ‘mgl’ subfolder within your Ai Ai folder, and it will appear in your games list.  The file you need is called Symple-rect.mgl.

All my tests of weird board dimensions have confirmed that the core mechanics of Symple are not just clever and elegant, they’re also extremely robust.  The game remains interesting even with bizarre values of P, or when played on extremely large boards or weirdly-shaped boards.

HexSymple

Speaking of weird boards, it turns out that Symple is also incredibly good on hexagonal boards, too.  Christian Freeling calls it HexSymple, and in this variant the game is played on a hexhex board (a hexagon-shaped board composed of hexagonal spaces).  The rules are identical to regular Symple.  Here’s a game played on a hexhex board with 12 hexes to a side (that’s 397 hexes in total):

hexsymple-sz12-p10-sample1

A completed game of HexSymple on a hexhex-12 board (=10).  White wins with 153 points to Black’s 124.

In this game White managed to constrain Black’s growth along the left edge and take the win.  HexSymple has a very interesting character — the board geometry means that cutting off groups is more difficult than on the square board, since all hexes have six neighbours instead of four and there are no diagonal cuts possible.  The game feels very expansive as a result, with ambitious connections snaking across the board in every direction.  Here’s the full game in GIF form:

hexsymple-sz12-p10-sample-game1

Just as in regular Symple, HexSymple is incredibly scalable, and I’ve played a few games on very large boards because I am a bit crazy for large boards.  Here’s one on a hexhex-25 board (that’s 1,801 hexes):

hexsymple-sz25-p30-2s-sample1

A completed game of HexSymple on hexhex-25 (P = 30).  White wins, 643 points to 588.

You can see here how the expanded connectivity of hexes makes truly enormous groups possible; check out White’s gigantic group stretching from the top right all the way around the board to the top left!  You can see the whole game in GIF form by clicking here.

On the whole I highly recommend HexSymple.  The board topology creates some interesting wrinkles in play, but the overall strategy remains broadly similar to regular Symple.  The result is a fascinating variant that works as a great change of pace, and stands up as a great game in its own right, too.  I haven’t yet seen a consensus on what good values of might be for different sizes of boards, but in my experience you can safely use significantly larger values than on similarly-sized rectangular boards and get a similar experience, due to the increased connectivity between hexes.

Note that HexSymple is not implemented in Ai Ai by default, but a simple modification of the regular Symple.mgl file makes it possible to play.  You can download the file you need, helpfully titled HexSymple.mgl, from this Google Drive folder.  HexSymple of course has its own dedicated page on Mindsports, and you can play the game online there too.

 

A modern classic

In the very crowded field of modern abstract strategy games, Symple (and HexSymple) are rare specimens that feel like classics.  In some alternate universe, I imagine Symple having frequent high-level tournaments, with a professional player scene, ample literature on high-level play, and an online community with millions of players.  Perhaps in the not-too-distant future this may come to pass in this reality, too.  In the end I agree with Christian — this is certainly a ‘game that matters’.

Sometime in the (relatively) near future, I’ll post a follow-up to this and introduce Sygo, a combination of Symple and Go and Othello-style piece-flipping captures that seems like it shouldn’t work, but totally does.  Like most fans of Go I’m not very keen on most Go variants, because they normally just disturb the elegant balance of simplicity and depth that makes Go so seminal.  But Sygo feels different enough to have its own character, and HexSygo even more so.

Before that, I owe you all a couple of Shogi posts which are still in the works.  I’m pleased to say that the Japanese Chu Shogi Players Association — Chu Shogi Renmei — has sent me a treasure trove of historical information on the game.  It’s all in Japanese, of course, so it will take me some time to read, but with any luck I’ll have some interesting information to report further down the line.

In the meantime, please do yourself a favour and whip out your Go set, get yourself on Mindsports, download Ai Ai, or preferably all of the above, and give Symple a try.

 

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Connection Games II: Y, Poly-Y, Star and *Star

Welcome to part II of my series of posts about games, part of my mission to keep my brain busy while I’m on strike!

Moving on from the last post about Hex, this time we’re going to explore a whole series of connection games, each by the same designer and each a clear progression from the last.  By the time we get to the final game in the series, we’ll see one of the more complicated and sophisticated connection games out there.

The Game of Y

First though, let’s start with something simple.  In fact, this game is even simpler than Hex, which scarcely seems possible!  Recall that in Hex, each player has a slightly different goal — both seek to connect across the board, but each player is connecting different sides.

In the Game of Y, created by Craige Schensted (who later renamed himself Ea Ea) and Charles Titus in 1953, players have the same goal — to connect all three sides of a triangular board made of hexagons.  To sum it up:

  1. Players take turns placing one stone of their colour in any empty hexagon on the triangular board.  Once placed, stones do not move and are never removed.
  2. The first player to connect all three sides of the board wins.  Corner hexes count as part of both sides to which they are adjacent.

And that’s it!  Winning connections spanning the three sides look kind of like the letter Y, hence the name.  Just like Hex, Y cannot have draws, one player will always win eventually.  The first player has a winning advantage here, as well, so using the swap/pie rule is recommended to alleviate this.

Y is sometimes thought to be even more elemental than Hex, given the greater purity of the win condition.  In fact, Hex can be shown to be a special case of Y, but in practice the games are pretty distinct in terms of the tactics required.

Here’s a sample game I played against a simple AI on a triangular board 21 hexes on a side; I used Stephen Taverner’s excellent Ai Ai software that comes with a plethora of great connection games:

 

y-win-1

Y benefits from playing on a larger board, given the shorter distances between sides when compared to a Hex board.

One issue with Y is that, even more so than Hex, the centre hexes are very powerful.  Whichever player controls the centre is very likely to win.  Schensted and Titus developed a number of ideas for new boards that would reduce emphasis on the centre, and eventually the ‘official’ Y board became this interesting geodesic hemisphere:

y-17-curvedThis board reduces the connectivity of the central points, giving the sides and edges greater influence on play.  Some theorise however that on this board basically every first move should be swapped by the second player, although I don’t believe there’s any hard evidence that this is true.  Kadon Enterprises sells a lovely wooden version of this board, albeit smaller than the one above, with 91 points available for stone placement. 

UPDATE:  Phil Bordelon reports in the comments below that the Kadon Y board is so small that the game feels like a trivial first-player win!  So perhaps if you want to try this board shape, play the larger one on the Gorrion server, or print the larger pattern on a mat for face-to-face play.  David Bush also wrote to say that he believes the geodesic Y board can’t be balanced with just the pie rule, and given his serious pedigree in the connection games world I think I will take his word for it!  He also says that three-move equalisation can be a solution.  In this method, one player creates a position with two Black stones and one White, with White to move, and the other then decides whether to play White or Black.

Another alternative option to the geodesic Y board is known as Obtuse-Y.  In this version of the game we play on a hexagonal board tessellated with hexagons (a ‘hexhex’ board), with three pairs of sides marked — first player to connect all three of those marked sides is the winner.  I like this version of Y since a large board in this format is more compact than a gigantic triangle of hexes, and it’s easier to have a balanced game than on the geodesic board.  I made two boards for this version, which you can find on BoardGameGeek — a hexhex-10 (10 hexes on a side) and hexhex-12.

hexhex-10_Obtuse-Y-01

My hexhex-10 board for Obtuse-Y — connect all three colours to win!  There are 271 playable hexes on this board.

In any case, Y is another simple-yet-deep experience and highly recommended.  You can play against the AI using the Ai Ai software linked above, or against humans in real time via igGameCenter, or by correspondence on Richard’s PBEM Server.  The geodesic version is only playable on Gorrion — definitely give it a try.

Finally, I want to note that Y has a ‘Misère’ version, much like Hex, where you try to force the opponent to connect all three sides before you do.  This variant of Y is called ‘Y-Not’.  I just love that.

 

Mudcrack Y and Poly-Y

The next step in Y’s evolution came when Schensted and Titus published a gorgeous little book called Mudcrack Y and Poly-Y (you can still buy it from Kadon), which contained hundreds of strange hand-drawn boards for Y players.  They intended for players to use these boards by marking spaces with their chosen colour using coloured pencils.  These weird little boards seem totally different from the normal Y triangle or geodesic hemisphere, and yet turn out to be topologically equivalent.  Here’s a sample page:

mudcrack y1

A sample page of Mudcrack Y boards.  Print them out, grab a couple of coloured pencils and give them a go!

As part of their continued quest to improve on the Game of Y, this book also reveals Poly-Y, a follow-up game intended to be a further generalisation of Y:

  1. Players take turns placing a stone of their colour on any empty space on the board.  Once placed stones cannot move and are never removed.
  2. If a player’s stones connect two adjacent sides and a third non-adjacent side, that player controls the corner between the two adjacent sides.
  3. If a player controls a majority of the corners on the board, that player wins.

As you might have guessed, once again this game permits no draws (as long as you play on a board with an odd number of corners), so one player will always win.   The pie rule is used to mitigate the first-player advantage.

Wikipedia and BoardGameGeek claim there is no ‘official’ Poly-Y board, but this isn’t correct.  On the archived version of Craige/Ea Ea’s website you can find his summary of the history of Y/Poly-Y/Star/*Star, where he says this:

“Craige tried boards with more and more corners, 5, 7, 9, 15 … . At first it seemed that the more corners the better — there were more points to contest and a beautiful global strategic picture emerged. But as the number of corners increased, of necessity the length of the edges decreased. When the edges became too short it was found that it was too easy to make a Y touching 3 consecutive edges, thus “capturing” the middle edge and the two corners bounding it. This “edge capture” tended to make the game more tactical and local, focused on quick gains along the edge, thus losing the elegant global strategic flavor . So the strategic depth increased at first as the number of corners increased, but then decreased. Finally a board with 9 corners and 7 cells along each edge was chosen as the ideal balance….  Craige chose the 208 cell board with the 7-sided regions halfway to the center as the standard Poly-Y board.”

The same document has a picture of the standard board compared to two other candidates:

poly-y-board-official

The ‘standard’ Poly-Y board is in the centre.  The highlighted spaces on each board are the heptagons, required to allow the board to have 9 corners and still consist mostly of hexagons.

Here’s an example Poly-Y game won by Black on a board with 106 spaces and five corners — note here that the other player is Grey, and the yellow spaces are unoccupied:

poly-y-sample-game2

Black controls the corners on the left side, and has blocked Grey from catching up.

While Craige/Ea Ea endorses the 208-cell nonagon as the best Poly-Y board, in Mudcrack Y and Poly-Y they also state the game plays well on any board with the following characteristics:

  • Equal numbers of spaces along each side
  • Mostly six-sided board spaces
  • An odd number of sides (they prefer 5- and 9-sided boards)

Here’s sample 5-, 7- and 9-sided boards to print and play on:

poly-y-5-sided

5-sided Poly-Y board

poly-y-7-sided

7-sided Poly-Y board

poly-y-board9

9-sided Poly-Y board

 

Poly-Y is a clever game, and very deep; the win condition pushes players to extend their groups of stones all over the board, linking corners together through the centre to block the other player from securing their corners.  The oddly-shaped boards are also fun to play on, and give the game a certain quirky aesthetic appeal.  However, perhaps due to the rapid-fire iterations on Y produced by Schensted, Poly-Y never got the same level of recognition as Y itself.

 

Star

Schensted wasn’t quite done yet — far from it.  His next invention was Star, which ramped up the complexity of the scoring system from Poly-Y and created a game that pushes players to connect all over the board.  Star is another very deep game, and even on a small board presents a considerable challenge.

Star is played on a board of tessellated hexagons with uneven sides — in the sample small board below, you can see that three sides are five hexes long, and the other three are six hexes long; this ensures that there’s an uneven number of edge cells so that draws are impossible:

Star

A small board for Star with 75 hexes and 33 border cells.

Here’s how to play Star:

  1. Players take turns placing one stone of their colour on any empty hex.  Once placed, stones do not move and are never removed.
  2. A connected group of stones touching at least three of the dark partial hexes around the edge of the board is called a ‘star’.  Each star is worth two points less than the number of dark border hexes it touches.
  3. When both players pass or when the board is full, the player with the most points wins.
  4. As per usual, the pie rule is used to mitigate the first player advantage.

This may sound a bit opaque, but the basic gist is: form as many stars as you can, but connect them together to maximise your points.  The end result of a game of Star is an intricate web of connections snaking across the board for each player, attempting to connect and block simultaneously wherever possible.  Since the entire edge of the board is available for scoring, the whole board interior tends to come into play as well, and unlike most connection games the board tends to be nearly full when the game finishes.

Unfortunately, despite pretty much universal praise for this game it’s very difficult to find sample games of Star, so here’s the only one I could find from Cameron Browne’s book Connection Games — I can’t emphasise enough that this is a great book that you should definitely buy!

star-sample1

In this example, the edge scoring cells are marked by X’s rather than a border of partial hexagons.  Note that the completed game takes up nearly the entire board, and the pattern of connections formed is quite intricate even on this small playing area.

As with other connection games, playing on larger boards amps up the strategy.  I found these boards lurking around the Wayback Machine, do give them a try:

Star2

This board has 192 interior cells and 51 border cells.

Star3

This board has 243 interior cells and 57 border cells.

I liked these boards so much that I made a range of Star boards — sizes 8, 9, 10 and 12 (the number being the number of hexes on the longer sides).  I hope a few folks might print them out and give Star a try sometime.

Star-8_PURPLE-01

My size 8 Star board in purple

Unfortunately, despite the coolness of this game it’s been thoroughly overshadowed by its successor; Star does appear to be playable at Richard’s PBEM Server, albeit only with an ASCII interface.

 

*Star

Finally we come to the last in the line of games spawned from our old friend the Game of Y.  *Star takes yet another leap up in complexity, and to be completely honest, I don’t fully understand how this game works.  This is partly because the instructions are written in what feels like an alien language — scoring refers to things called ‘peries’ and ‘quarks’ and it’s all a bit strange.  However the abstract strategy game community praises this game nearly universally, so I remain keen to try and figure it out.

My understanding, questionable though it may be, is that the game essentially takes the core concept of Star — connect groups of edge-adjacent pieces together to maximise your points — to the next level by adding a scoring bonus for controlling corner spaces, and a significant scoring penalty (equal to twice the difference in the number of groups between the two players) for the player with the larger number of groups.  This heavily incentivises the players to connect their groups, and the end result of this is some beautiful patterns of stones snaking across the board, as in these two sample games from the manual (one tiny one and one normal-sized one):

Here’s a closeup of that awesome board:

starstar-board1

The *Star board.  The centre star can be used by either player as a connection between groups — neither player may place their stones on it.

Note that the board has thicker lines to define smaller board sub-regions, which allows players to ease themselves into the full game.  The game is popular enough to be produced in physical form by Kadon Enterprises, who make a wooden board set for *Star that I absolutely must buy at some point:

star-wood-board2starwood

How cool is that!  Someday I shall own this game, and I shall figure out exactly how to play it.

Luckily there’s a simpler game also playable on this board — Star-Ywhere the players compete to be the first to complete a connection between two adjacent sides and one side not adjacent to either of those two.  For *Star veterans there’s also Double-Star, where players place two stones per turn and the other rules remain the same; this seems like a small change but it significantly alters the play.  New tactics and strategies are necessary to cope with the new threats that are possible with two stone placements.

So there we have it — a hectic journey from the elemental Game of Y through to the complex but highly-regarded *Star, courtesy of the brilliant minds of Craige Schensted/Ea Ea and Charles Titus.  Craige/Ea Ea has stated that *Star is ‘what the other games were trying to be’, so from his perspective each game was improving on the last, and *Star is the best of the lot.

While researching and playing/trying to play these games, I’ve found that Star and *Star are frequently compared to Go, despite having connective goals rather than territorial ones.  Given the much more flexible nature of the connective goals in these two games, I can see why — instead of connecting specific sides, players define for themselves the key parts of the board as they play.  This is much more ‘Go-like’ in that the board is more of a blank slate, and does not inherently define the direction of play as much as in other connection games.  So, if you’re a Go fan and skeptical of connection games, maybe try these two.

If you’re new to connection games in general, I’d start with Hex, then Y, then Poly-Y.  You might enjoy Star and *Star more after trying some other games with more freeform connective goals, but with easier-to-grasp rules.  I’d recommend maybe trying Havannah and Starweb for that purpose — and lucky you, they’ll be in my next post 🙂

 

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Connection Games I: Hex

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 Storyso 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:

polygon-board-1

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.

polygon-puzzle-1

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:

hex13-sample

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.

Hex19-1

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:

hex11-win1

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:

hex13-sample2

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:

hex-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 HexThink 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.

pex-iggc

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.

nex-sample-game1

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.

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