Hex

Feb 22 2021

Games

A Beginner’s Guide to Hex, Part II: Sample Games

In Part I of our Beginner’s Guide to Hex, we covered some critical tactical and strategic concepts that will help you get a flying start in your journey to become a strong Hex player. This time, we’ll look at a full game played on the 15×15 board between two strong players, so you can get a sense of how those principles manifest during actual play. After that, we’ll take a look at another game, this time from the Game of Y, to see how Hex principles can apply in other connection games as well.

Commented Hex Game: LG #2195354, kspttw vs Arek Kulczycki

This game was played on a 15×15 board on LittleGolem.net, the most popular place to play Hex in correspondence style. 15×15 Hex is a relatively recent addition to Little Golem, but is already proving popular. Other board sizes are also available on LG: 11×11 for quicker, more tactical games; 13×13, which is the most popular size (and is the focus of Matthew’s brilliant Hex strategy guide); and 19×19 for deeply strategic contests.

If you’d like to follow along with the game and investigate the variations Matthew recommends, you can do so via the online Hex board on MinorTriad; this site allow you to go through the game step-by-step and experiment with different moves at any point.

Move 1: Black opens at c2. This is a popular opening on the 13×13 board, but here I would classify it as being likely on the losing side. All Hex openings are either winning or losing, there are no neutral ones; the best openings are right on the border between clearly winning and clearly losing openings, in that they may give some advantage but likely won’t immediately provoke a swap. On 15×15 c2 is probably weaker than on 13×13, since on larger boards we expect that winning openings will be a bit more centralised than on smaller boards. As a consequence, White elects to not swap sides.

Move 2: The 5-4 point (five rows from your own edge, four rows from your opponent’s edge) is by far the most popular acute corner opening. This move is connected by Template A-5, and it escapes second- and third-row ladders.

Move 3-4: This is a common joseki. Black’s (3) is connected by Template A-4, and escapes second-row ladders while threatening White (2). White’s response connects back to the edge with Template A-4, while restoring their ladder escapes.

Move 5: The 5-5 point is a popular opening in the obtuse corner. Lately, 4-4 has become more popular on the 13×13 board, a development spurred by the play of very strong AlphaGo-style bots. On the larger 15×15 board, it’s an open question whether 4-4 or 5-5 is stronger.

Whether on the 4th or 5th row, having a stone in your obtuse corner is always a plus. These corners are almost always taken early in the game.

Move 6: White needs something on his southwest edge. You may be wondering why this of all spots was chosen. One principle is that you want stones that are “attacking,” in the sense that they can reach the edge through bridge moves to the side (see the dots in the diagram). Stones like this are harder to block. This is the closest a stone on the sixth row can get to the acute corner and still be attacking. If not for the black stone on c2, White would likely have played this move at f5. You want your stone as close to the corner as possible because it does more to disrupt your opponent.

Note also that (2) was “attacking” in this sense in the east corner. This is part of why the 5-4 point is so popular.

Move 7: This is a popular combination with c2. Together, these stones form Template I-5, and they can escape just about any ladder from the second to sixth row along this edge. This Template is rather difficult to attack. Note also that this move invades the “attacking zone” of White’s f7 stone.

Move 8: White plays here at the center of the board. This move is somewhat of a distant response to (7). With (7) invading the attacking zone of the f7 stone, Black could potentially block that stone now. But White doesn’t want to overcommit to this area — it’s important to spread influence around the board, especially early in the game when the potential lines of play are so fluid. This stone can support the f7 stone while also increasing White’s influence over the centre of the board and towards the northeast edge.

Move 9: Counterintuitively, in Hex it’s often stronger to play on your opponent’s edge than on your own. Having stones separated by two empty hexes on the opponent’s second row can be quite strong. Notice how White is unable to fit Template A-3 in between (7) and (9) (nor in between (7) and c2). This stone (9) is basically connected to Black’s northwest edge now, because White’s only reasonable blocks (such as at b6 or b7) allow Black to go around to the north (at c6 or c7, respectively). Ultimately, White will need to connect to the southwest edge in the area between (9) and the obtuse corner.

Moves 10-19: White can’t break the connection of b8 to the northwest edge. Instead, White allows Black to connect. But how does this help White? This is an attack I call “undermining.” Note first that the 10-12 group is connected to White’s edge (White can connect at either A or B; Black cannot block both). Because of this, the points D and C are vulnerable for Black. If White intrudes into these bridges, they threaten to connect to the edge through 10/12, so Black is forced to reply. White can therefore invade these points for valuable territory. Additionally, towards the end of the sequence, White obtained the stone (18), gaining more territory and forcing the reply (19). Lastly, the point E is also vulnerable for Black, if they want to keep b9 connected to the edge.

Moves 20-26: Now White begins attacking the vulnerable points, starting at (20). After Black saves the connection, White moves to the other end to attack Black’s 5-5 stone in the obtuse corner, forcing a response there. White knows that after taking the territory at the vulnerable points, they will ultimately need to connect to their southwest edge somewhere in the lower half of it, so having 22 available strengthens this area to White’s advantage. With this secured, White returns to attacking Black in the west corner with moves 24 and 26.

Moves 27-30: Clearly, White has been playing in sente for a while, dictating the direction of play ever since the attack that began on move 10. Black seems to have had enough, and rather than respond at b7 (and handing play back to White), Black tries here to take the initiative back with a block against the h8 stone. Here the play suddenly becomes a tactical affair. White first plays (28), because having these two stones parallel to the northeast edge will virtually guarantee their connection (thanks to the help of the white stones in the east acute corner). Next White plays (30).

Here White begins to utilise that territory they gained through undermining. Since move 30 is connected to g4, White’s approaches have expanded considerably. (30) threatens to connect to f7, and g4 threatens to connect to e5. Before we look at Black’s response, it would be instructive to see what happens if Black tries to block the former with 31.g6. White might then respond at 32.f5. From here, a few lines to consider:

33. d6 e7 d8 d7 b7 (forced) c8 b9 and then c10 is a ladder escape fork for White, connected to the edge with Template A-3, and back to the main group by either c9 or e9.

33. b7 d8 (connected back by either f6 or d7) c9 d10 (d10 + d12 make for the edge template L-4, hence d10 is connected to the edge) d9 and now f9 is connected to the edge and threatens to connect to either h8 via g9, or to h5 via f8.

Moves 31-42: Black attempted to block at (31) instead. The situation that follows is highly tactical. White first plays 32. d8. After Black blocks at 33. g6, White links up g4 to e5 with 33. f5. Black gains some free territory with a bridge intrusion (35-36). At this point, e5/f5 are connected to d8/f7 by either d7 or f6. If Black blocks at 37. f6, White’s connection is assured by 38. d7 (White can connect by either b7 with Template A-2 or c9 with Template A-3), so Black plays 37. d7. White connects up with f6 (although e7 would have been better, offering no intrusion points). Black takes a little more territory with move 39, but after move 41 White’s responds with 42. d10. As mentioned in the second variation above, this move (along with the stone on d12) is connected to the edge via edge Template L-4. If Black blocks at 43. d9, White plays 44. f9, which threatens either g9 or f8.

Moves 43-45: With the southwest edge lost, Black must attempt to block the northeast edge. The odds aren’t good however. Presumably Black went all in on blocking White from the southwest because Black felt the chances were better there. Black didn’t gain much in the way of territory during that sequence that could help on this side of the board, with the possible exception of the stone on h4.

The White stone on h5 isn’t yet connected to h7, but trying to block between them will just make things worse for Black (43. h6 i5 i6 and k4 can at best be held to a fifth-row ladder, heading towards the White stones in the east corner). So Black plays 43. j6. White connects up the smart way, at 44. i5. Unlike connecting via h6, this threatens the followup k4, which would start a fifth-row ladder, as well as the potential threats from the h7/h8 group. Black is forced to attempt to block both directions at once, with 45. k6. This cleanly blocks the potential of k4, but as we shall see, can’t hold off a White attack from h8.

Move 46: White bridges away from h8. This move is based on a simple concept: note that by placing the stone out so that there’s a clear line to the edge (shown by the arrows) it can’t be stopped with simple adjacent blocks (j9 j8 k8 k7 … ). This means Black will have to block this stone to a ladder, and that’s where those two white stones on l11 and k12 will come into play.

Moves 47-50: All that remains is for White to finish off the connection. After 47. j9, White will ultimately play j8, after which a fifth-row ladder will begin (Black could hold White to either a fifth- or fourth-row ladder; generally you want to hold a player to the higher row). Before that, though, White sets up the ladder escape with move 48, which threatens to connect back via a bridge to the stone on i9. Black is forced to block (Black’s choice to play 49 at j10 as opposed to i10, is the stronger block since it leaves White with slightly less space underneath). Now White plays 50. j8, and the game is over. Although the final sequence wasn’t played, let’s quickly look at how it might have played out.

The naive approach is just hold White to the fifth-row ladder. White easily connects with Template A-4.

Black might instead jump ahead with move 55 and force a bottleneck, but after move 58 White connects to the bottom with Template A-3 and back to (50) via either A or B.

Finally, Black might try to hold White to a fourth-row ladder instead, but after move 58 White’s stones are connected in the Trapezoid template, and Black has no means of blocking White from the edge.

Y Sample Game: PCM vs Matthew Seymour

Next up we have a sample game of the Game of Y. For those of you who don’t know Y, it’s actually even easier to learn than Hex:

Two players, Black and White, compete to connect all three sides of a triangular board of hexagons.
Players take it in turns to place one stone of their colour on any empty square on the board. The first player to connect all three sides of the board with a single connected group of stones wins the game.

That’s it! In Hex, players must connect two specific sides of the board that share their colour, while in Y all three sides are relevant to both players. As we shall see, that fact can alter some of the tactics and strategies you may have learned from Hex, but broadly speaking your Hex knowledge is a great help in Y as well.

This game was played between PCM (Black) and Matthew Seymour (White) on iggamecenter. The board is size-14, which is relatively small for Y but still big enough for a challenging game. Matthew has annotated the game for us below:

Move 1: We’re playing with the swap rule, so Black (PCM) opens along the edge.

Move 2: White (Matthew) responds with a more central move.

Move 3: Connected left via the A-5 edge template, but the difficulty will be connecting to the bottom.

Move 4: Blocking Black’s stones from the bottom edge.

Move 8: Connected with the B-3 template.

Move 10: A blunder! e9 would have been better (winning I think) than f10, with template C-5 facing the left edge and move 4 helping guarantee the 8-2 group’s connection to the south. As it stands, 2-10 is connected south with C-5 and 2 isn’t fully connected to the left.

Moves 11-15: This block sets up a ladder with the bottleneck formation.

: Move 16.

: The plan: after (20) in this variation, White is in control.

Move 16: Ladder escape. The plan here is that after b7, White plays d6 d7 f7, and now White is connected to all three edges.

: Move 17.

: Move 18.

Move 17: Counter-threat, threatening the connection between (8) and the edge.

Move 18: 16-18 is connected to the right edge through the M-4a template, and connected to the central group (14-8) via either d7 or the ladder on the left.

Moves 19-21: Black first blocks the d7 route, then blocks between the ladder and the escape on move 21.

: Played in the game.

: Playing (22) at a6 instead would have kept the connection on the left and still threatened to connect on the right.

Move 22: Here I blunder the game away! I was concerned the 20-8 group might lose its connection to the right that I had through either 8 or the 16-18 group. It looked like Black had blocked off the 16-18 group, so I had to save it via (8). I missed the winning move 22. a6. Then, if 23. i10 I could simply play a5 and I would be connected to all three edges — in other words, it would have kept the double threat alive for connecting to the right, while also connected the group to the left.

Instead, I saved the connection to the right, but now Black can now cut me off from the left at a6. I missed this “obvious” move because (I think) I’m so used to playing Hex. 21 is connected to the left via A-2, and in Hex there’s no reason to ever invade A-2 because the two empty hexes are captured. But of course, in Y, the edge is shared by both players, so these hexes are NOT captured.

Move 23: Forced. Black blocks White from the left edge.

Moves 24-30: Ladder, followed by a break. The black group (1-29) is connected to the left and right edges. Black needs only to connect it to the bottom to win.

Moves 31-36: 4th-row ladder, followed by a bottleneck. White has no hope however, as the 7-5 group will help escape the ladder.

Moves 37-43: There are many ways to escape the ladder, but Black elects to go with this approach. More straightforward would have been e13 f14 f13 g14 g12 and then Black can play either h13 (with A-2) or j12 (with A-3). As played, (38) is the only reasonable reply to (37) (further left on this row, Black plays e11; further right on this row, Black plays g13; for plays on row 12, Black uses (37) as a second row ladder escape). (39) and (40) accomplish nothing but there’s no harm. After (41), Black can play either h13 (with A-2) or j12 (with A-3). White blocks the former, so Black plays the latter. White resigns.

: The winning Y — all of Black’s highlighted stones are connected either directly or through templates (the red hexes), and this chain connects all three sides. White has no way to stop it and no opportunities to make an earlier connection themselves, so they resigned.

So, there we have it — a quick but well-played game of 15×15 Hex, and a tricky game of Y that shows off some of the quirks of Hex’s cousins in the connection-game world. We hope these give you some useful ideas about how to apply the core concepts of Hex strategy to your own play.

Let us know in the comments what you think, and if there are other subtleties to Hex (or Y, for that matter) that you’d like to hear more about, perhaps we may do some more posts in the future.

In the meantime, enjoy, and good luck with your journey toward becoming a strong Hex player!

Tagged Connection Games, Game of Y, Games, Hex, Matthew Seymour

Feb 15 2021

3 Comments

Games

A Beginner’s Guide to Hex

Some time ago I talked a bit about Hex on this blog, discussing its history and how it kickstarted the entire connection game genre. Since then, a few readers have asked for a bit more detail on how to actually play Hex. So, for this post I have teamed up with Matthew Seymour, author of the brilliant Hex: A Strategy Guide, and we have put together this beginner’s guide to Hex strategy and tactics.

Below we will introduce you to basic tactics, templates, openings, and strategic considerations. Each section is very brief but will give you enough to get you started on improving your Hex play; after you tackle each section, please continue your studies with Matthew’s guide, which has much more detail and numerous examples of each critical concept.

Basic Tactics

The edges

Edge play in Hex is obviously critical, since in order to win we must connect our two edges across the entire board. Initially, connecting a chain of stones successfully to the edge seems like a baffling enterprise — an adept opponent can bend and twist your attempted connections away from the goal, and it can be difficult to understand how to set yourself up for a strong connection.

Edge templates are extremely useful tools for understanding edge play. In our previous discussion of Hex, we met the bridge, a configuration of two stones that proves to be virtually connected even though the stones aren’t adjacent. Similarly, edge templates show us configurations of stones that, with correct play, are guaranteed to result in connect to the edge. Templates indicate patterns where, even if your opponent has the first move, you will always be able to connect successfully. Your opponent may intrude on your template, but so long as you defend your template at every step, you will succeed.

: The bridge: the two black stones on the left are connected. If the opponent intrudes at either cell between these two stones, Black can simply respond in the other cell to save the connection, as in the example to the right.

This means however that templates are always just a move away from breaking. If you have two overlapping templates, and your opponent plays a move that intrudes on both, you can only potentially save one of them.

Here are some examples of basic edge templates:

Our first edge template is the very basic A-2 template, consisting of a stone on the second row. If White tries to block at A, Black can respond at B (second diagram), and vice versa. Either way, White has no means of stopping black. This template is very similar to the bridge template.

One of the most useful templates is template A-3, consisting of a single stone on the third row. This template comes up very frequently. We’ll analyse the situation by considering Black’s possible threats. On one hand, he could play at A and connect to the edge with template A-2. Or, he could play at B, and connect by a chain of a bridge and template A-2. The important part here is that these threats don’t overlap. If White plays at A, or one of the two hexes below it, Black simply plays at B and connects by that route. And if White plays in any of the 5 hexes on the right, Black plays at A and connects by that route.

We can succinctly convey this information in what’s called a pivot diagram. The two small dots indicate Black’s alternative moves. We can see that the rightmost dot connects back to the top stone with a bridge. Both dots are connected to the edge by one of two possible moves (template A-2).

Lastly, let’s look at Template A-4, on the fourth row:

Similar to A-3, Black has two threats: to play adjacent at A, connecting to the edge with A-3 (left), or to bridge to the side at B, also connecting with A-3 (right).

These alternatives can get around most possible blocks by White. But there’s one issue, the point C. Both the threats above overlap at this point. So Black needs a response to a block at this point. The solution is shown below (note that if White played (3) on the left side, Black could bridge to the right instead).

There are a large number of edge templates. You can find an excellent catalogue here or here.

Interior templates: connecting chains

We’ve already encountered one interior template — a template for connections in the centre of the board, away from the edges: the bridge. As you might expect, other templates exist for more complex configurations of interior stones. Knowing these templates is very helpful indeed, as once you achieve such a configuration on the board and recognise that template, you can play elsewhere, knowing that you’ll be able to successfully defend that template against later intrusion.

Some well-known interior templates are the Wheel, the Crescent, the Span, and the Trapezoid.

: Four simple interior templates. Clockwise from top left: the Wheel, the Trapezoid, the Span, the Crescent.

: Responding to play when White tries to intrude. Note how Black has easy responses to every intrusion that save the connection, either through direct adjacency or bridges.

Defending at a distance

We’ve talked a lot so far about connecting — positive, attacking play. But what to do if our opponent has us on the back foot, and a deadly connection is looming? How can we stop them?

First, we should remember that as satisfying as that positive, attacking play may be, Hex is what is called in combinatorial game theory a hot game. This means that it is always beneficial to make a move in Hex, and no stone of ours on the board is ever a detriment to us. On top of that, one player must always win in Hex, so if we successfully block the opponent from any possible connection between their edges, we’ve in fact won the game. So don’t neglect defensive play — in Hex, it’s precisely as powerful as attacking play, and will win you the game just as effectively! Defensive moves in Hex are also offensive moves.

Having said that, defence in Hex can be a delicate affair. As we’ve seen in numerous examples, stones can be connected even if they aren’t adjacent, and skilled players can move across the board at high speed, staying connected the whole way. Therefore, if we attempt to block simply with adjacent blocks — playing right next to opposing stones — a skilled opponent can easily bend around us. Likewise, if we block at a distance but misjudge the situation, the opponent may still get round us by using bridges to cover enough ground to do an end-run around our defences.

In practice we may need to combine adjacent blocks with more distant blocks in many cases. The adjacent blocks restrict the opponent’s options for bridging forward, while the distant blocks contain those advances:

: Caption: A combined block. White (1) leads Black to attempt to run around with (2), only for White to catch them with (3), preventing a further bridge forward.

The classic block short-circuits the opponent from afar, allowing the defender to respond whether the opponent advances directly forward or takes a more oblique approach:

: Caption: A classic block. White blocks at a distance at (1), and when Black encroaches with a bridge to (2), White can block further advances with (3).

In either case, in order to defend we need to evaluate the opponent’s options for onward connection, and place our stones in anticipation of those options. If we defend reactively, and follow the opponent around right at their heels, then we’ve no hope of survival. If we instead constrict their choices and contain their subsequent advances, then we may just keep them at bay.

Ladders

Often, when approaching an edge of the board, you’ll end up in a situation like this:

White wants to connect and Black can’t allow it, so Black blocks at (3). White can’t connect right away but can force Black to carry on blocking all along the edge. This series of back-and-forth adjacent plays is called a ladder. White is completely in control here; Black has to respond to every ladder stone White plays, otherwise White’s connection is assured.

Go players will be familiar with sente — the concept of maintaining the initiative, by making moves that force an immediate response from the opponent. When you have sente you are in control of the game; you are making profitable moves, and all your opponent can do is match you, stone for stone, unable to direct play to their advantage. Sente is just as important in Hex as it is in Go, and ladders are one common manifestation of it.

However, you have probably noticed that if White continues playing the ladder here, it’s Black that ends up connecting across the board:

To make the ladder profitable for them, White needs to incorporate some additional tactical plays. If White had an additional stone in place to form a ladder escape, then when the ladder reaches that stone, they can connect to the edge with ease.

Of course, when a ladder is already forming, taking a turn to place a ladder escape stone simply dooms the ladder. So players will often place ladder escape stones during the opening phase of the game, to allow for profitable ladder play later.

Another option is to place a stone that is both a ladder escape and a threat to connect by another route. Your opponent will be forced to block either the threat or the ladder, and then you can connect by the other means. In the diagram below, Black plays a stone at 1. This can escape the ladder, but it also threatens to connect via A (with two bridges). White can’t block both approaches, so Black will connect. This is called a ladder escape fork.

If you have no ladder escapes or forks available, you’ll have no choice but to “break the ladder”, as Black does with move 7 in the diagram below (note that 7 is connected with Template A-3). In the acute corner, this tends to reverse the roles: notice how now it’s White who’s the attacking player with a ladder.

Strategic Considerations

Openings

The opening in Hex is an interesting moment, as the first player needs to consider not just what is the best move to play, but what is the best move to play that won’t get swapped. In Go or Chess, you can play your opening move without fear of that move suddenly becoming your opponent’s opening, but not so with Hex!

As the second player, you need to do the opposite calculation: has my opponent played a move that, with perfect play, would give them a winning advantage? If so, I should swap; if not, I can safely play on.

On smaller boards, Hex has been solved, meaning that we know the precise outcome of any given opening move. That leads to diagrams like these:

: Opening results for 7×7 Hex. The colour of the stone on each cell indicates the winner, if Black plays their first move to that cell. (recreated from Hayward et al., 2004)

: Opening results for 8×8 Hex (winner if Black plays first). (Hendersen, Arneson and Hayward 2009).

: Opening values for 9×9 Hex (winner if Black plays first). (Pawlewicz and Hayward 2014)

Above are three diagrams showing the ultimate winner, with perfect play, of opening moves played at every cell on the board. There’s an important trend to notice here — the winning openings for any given board size are not straightforwardly extendable to larger boards! While we can see a general theme that opening moves in the centre are stronger than those on the edges, the specific outcomes of those edge cells change as we change the board. That means that on the boards we humans play Hex on — from 11×11 upward — not only do we not have these convenient maps of what moves win or lose, but we cannot use the opening maps from smaller boards as a definitive indication of the outcome of any opening on the bigger ones.

Matthew’s guide focusses on the 13×13 game, and for openings on that board, he’s produced a swap map that can help guide you in the opening. The cells with black dots are Matthew recommendations for good opening moves for Black. When you are the second player, if your opponent opens anywhere in the shaded area, you should swap — those moves have the potential to give a winning advantage, so you’re better off taking that stone for yourself. If your opponent plays outside that shaded zone, let them carry on — you can possibly do better by playing your own opening.

For larger board sizes, like 15×15 and 19×19, we don’t yet have enough games played at a high level to put together a reasonable swap map. However, we can make some reasonable inferences about good opening moves; in particular, opening in the obtuse corners seems a good way to go on all board sizes.

These are very simple principles, but should be enough to get you started. One thing to bear in mind is that we humans are far from perfect play, even on 11×11, so both sides are likely to make mistakes, not just in the opening but throughout the game. So our goal at this stage should be simply to ensure that our opening doesn’t obviously disadvantage us; we don’t need to fret too much about whether a particular move is 100% winning or losing.

Playing in the corners

The Hex board has two types of corners — acute and obtuse — that have different properties. Corners are the only parts of the board where your stones can both strengthen your own position and weaken your opponent’s, and for that reason, players tend to play stones in the corners early in the game. Typically you will want to play stones in at least one corner on each of your edges during the opening.

The corners being so important often leads to pitched battles to establish control over them, and so strong players may study corner patterns (think joseki in Go) to navigate these tactical scuffles. If you don’t have a presence in a corner and your opponent does, invading is useful in order to reduce their influence there, and corner patterns will help you to reduce that influence. Conversely, if your opponent invades your corner, you can use these patterns to settle the fight and maintain as much of your initial influence as possible. The challenge in these situations is judging when there is no more profit to be gained, and thus when it’s time to move on from the corner battle and establish yourself elsewhere.

There is a lot to discover in these corner patterns, but don’t worry too much about these early in your Hex journey; as you start to face stronger opposition and find your corner play is letting you down, refer to Matthew’s guide for detailed examples of how to fight for the corners.

: A sample corner pattern. Black jumps into the corner with (1) (the most common acute corner opening), and White responds with a close block at (2), which is also connected to the edge via Template A-4. Black (3) blocks the template while gaining some territory, while White (4) allows second-row ladder escapes. Black (5) connects to the southern edge by Template A-4.

Influence

We’ve alluded to this concept in the previous section, so now let’s expand on what influence means in Hex strategy. Stones in Hex are not just localised points — they have impact on the board around them and on other nearby stones. Every stone has the potential to connect to something or to block something else, and when placing our stones we need to consider the influence of the stones around our planned placement.

In the early stages of a Hex game, gaining influence is important. We would do well to place our stones around the board, to spread them out; this maximises the potential influence of each stone. Conversely, if we don’t spread our stones out, we may have a strong influence in a particular area but will be weak elsewhere. If we are struggling to find an effective place to play, we can look at our relative influence on different areas of the board; if we find some areas where we have low influence, those might be good places to play our next moves.

As you might expect, stones in the corners have a high degree of influence — the proximity to two edges means those stones are better able to restrict your opponent’s activity in that area and force them to work around you in a limited space. Placements in the corners also are tougher to block, and provide you with ladder escape stones for later in the game.

: The first ten moves of a Hex game between two strong players on Little Golem (https://www.littlegolem.net/jsp/game/game.jsp?gid=2000904). Note that play in the early going has focussed mainly on the corners, and both players are spreading their stones far across the board to gain influence.

The edges are somewhat less intuitive. We might feel secure playing near the centre of our own edges, as this seems a useful way to block the opponent, but in practice these kinds of placements do not provide strong influence. Instead, we should play near our opponent’s edges — this forces them to work around you and makes it harder for them to connect.

Beginning Go players often play in a style referred to as Puppy Go, where they continually play very close to every one of their opponent’s moves, following them around the board like an excitable puppy. We can easily be tempted to play Puppy Hex in a very similar way. Unfortunately this is an adorable, but poor strategy; in a Puppy Hex scenario your opponent is dictating play completely, and since you are always one stone behind they will have free choice of where to establish influence and you will always be playing catch-up. Always keep an eye on the broader board situation, and try to take the initiative when the situation allows it — don’t let your opponent drag you around by the nose!

As you become more comfortable playing in an influence-oriented style, you can start to focus on making moves that serve multiple purposes. Gaining influence is good, but gaining influence and blocking the opponent is even better! This is a challenging step, requiring you to have both tactical and strategic vision, but as you gain more experience and become able to recognise common tactical motifs, you’ll be better able to keep these in mind as you seek to expand your presence across the board as well.

As a final note, we should remember that Hex is fundamentally a scalable game — we can play Hex on any size board we like without changing the rules, but the feel of play will change. Hex on larger boards is a challenging and rewarding affair, but specific tips on those epic battles is beyond the scope of this article. However, we encourage you to try larger boards, as they by necessity will make you play in an influence-oriented style. With so much additional empty space on the board, you’ll need to learn to anticipate where battles for influence and territory will rage, long before they actually happen. That experience can help you on the smaller boards too, training you to think globally more consistently.

Territory

Territory is a critical concept to understand in Hex strategy. Think of territory as the potential your stones create for future connection; the more territory you control, the more tactical options you have for later attempts to form connections between your stones.

As a starting point, we might say that each stone creates territory in the area immediately around itself; in other words, the empty hexes immediately adjacent to it. However, as we see below, this definition falls apart fairly quickly:

These intrusions by Black gain no useful territory. In both cases, White simply blocks any onward connections, so the ‘territory’ gained (the shaded cells) offers nothing that Black didn’t already have!

If we believe that stones create territory regardless of their disposition, then we will run into situations like the above, where our stone is effectively a wasted move, as it will never actually be able to connect to anything. Instead we should restrict the definition a bit more: the territory around our stones consists of the adjacent hexes that could in theory participate in a connection. If we want to invade somewhere and gain influence from that play, we need to be certain that the placement provides useful territory; if the stone does not gain territory, then we have simply placed a stone for no real purpose. Without territory we cannot claim influence, as the enemy can simply work around us at no real cost.

: Black (1) is a useless gain of territory, as there’s no way around the White group just above it. Black simply ends up hemmed in the corner.

: Black (1) at i8 is better — this move gains territory heading back toward the centre, where Black already has a presence and can make use of it.

Taking the initiative

As in many other abstract games, in Hex gaining the initiative is of huge importance. Recall the Puppy Hex discussion earlier — imagine if we could force the opponent to play Puppy Hex. If we can place stones with aplomb while our opponent can do nothing but respond, we can dictate the pace of play and dominate the board at our leisure.

Here we will go in-depth into some Go terms we mentioned earlier: sente and gote. In Go, when we play a stone that forces the opponent to respond — because a group is threatened with capture, for example — we say that is sente, meaning we are gaining the initiative. Our next move after the sente move is essentially free; the opponent’s response is mandatory, so our next placement can be anywhere we like, and we can use that to gain influence or territory. Conversely, the forced response the sente move creates is gote — we are forced to be the puppy for that move and play where the opponent demands.

In Hex we also have sente and gote moves. For example, we may recognise that our opponent has an edge template in play, so we may choose to intrude on that template and gain some influence. That move is sente because it demands a response; the opponent must play to save the template, otherwise that connection is lost. At that moment our opponent’s move is gote, lending us the initiative.

: A sente move in Hex. White aims to connect to their edge with (1). Black intrudes at (2), and White is forced to play at (3) to save the connection.

As we gain more experience of Hex strategy, we will be better able to identify opportunities to gain sente. At the same time, we must be mindful of our opponent’s threats, and remember that playing gote moves to save a critical connection is vital too! We should try to avoid being the Hex puppy whenever possible, but sometimes there’s no escaping it.

Tenuki

Let’s look at another situation:

Here Black is threatening to cut the stone A off from the top-right edge, and the straightforward response would be for us to save the connection and take gote, such as by responding at L3. After all, by not playing there we lose the connection.

However, in this situation we can see that White has an opportunity to make an intrusion of their own, on the other end of the board. Black’s threat depends on using the stone B to connect to the bottom-right edge of the board. White’s board situation will allow them to make other connections, even if they sacrifice the connection under attack by Black, but Black’s situation is just as fragile. In cases like this we may elect to tenuki — to play away from the threat and allow our opponent to break the connection. Instead of defending against the threat we attack elsewhere, and now they must make a choice: either save their own template, or finish ours off. If they finish ours off, they must make a second move, giving us influence elsewhere; if they take gote to save their own connection, then we have regained the initiative.

In this game, White elected to attack Black’s B stone with move (2), rather than save the connection of A to the edge. Black elected to save the connection, playing out a standard joseki sequence, leaving White with the initiative.

Tenuki is an advanced concept, and often difficult to judge. In general, you will have more opportunities for playing away from threats in the early- and middlegame, when the board is less full and there will be opportunities for other connections. In the late game, typically both players will have committed many stones to particular connections, and there is inherently less flexibility; if we ignore a threat, we are more likely to hand the win to the opponent.

The Joy of Hex

We’ve covered a lot of ground in this post — over the course of these few sections we’ve gone from the basics of the board geometry through to advanced strategic play. Yet for all that, we’ve barely scratched the surface. From here, you can move on to Matthew’s detailed guide to Hex, and dig deeper into all of these concepts. While you’re there, be sure to try out his fantastic collection of 500 Hex puzzles (also available in PDF, in Hex style and Go style) to sharpen your tactical vision. If you need help with openings, he also used over 6,000 online games on 13×13 to generate a very useful opening database.

Having said that, resist the temptation to power through all this material. Take some time with these concepts, apply them to your games, and move on only when you feel comfortable and confident. Remember too that Hex is perhaps the most famous modern abstract strategy game, but it is still very new in the grand scheme of things. Traditional games like Go, Chess and Shogi have had centuries for strategies to be developed, whereas in Hex we are all still beginners in some sense! So there is always more to discover and more to learn.

If nothing else, we hope this brief introduction will give you an appreciation for Hex’s incredible depth and nuance. Hex is a disarmingly simple game, so much so that a brand-new player may be tempted to ask ‘…that’s it?!’ when told the rules for the first time. But within that sparse framework lies a world of intricate tactical and strategic variety. This simplicity means Hex also has amazing flexibility — we can play lightning-fast blitz games on 11×11 boards, strategic masterclasses on 19×19, or mind-bending, baffling escapades as long as a game of Go on 26×26. Each one of these configurations is rich with possibility. Learning Hex also benefits you in other connection games — the tactics you learn here can transfer to other games, like the Game of Y (more on that in our next post).

Above all, we hope you have fun with the game! Go spend some time testing out your strategies online, entering tournaments, analysing games and writing about them. But alongside that, teach your friends and family (when Covid restrictions allow!), help them learn some basic tactical and strategic concepts, and show them why you love it. Every new player we bring to the game makes Hex’s future ever brighter, so the more we help others to see what the game can offer, the more enjoyment we’ll all have in the years to come.

Next moves

In the second and final part of our Hex mini-series, we will analyse a complete game of Hex in detail, and show how the concepts we’ve introduced here play out in a game between strong players.

Then we will analyse a brief game of Y, as well, to demonstrate how Hex concepts transfer to other, related games — and we’ll point out how some concepts change when we move to a different game.

Extra nerd stuff

Check out these papers if you’d like to know more about how the small-board swap maps above were generated:

SOLVING 7×7 HEX: VIRTUAL CONNECTIONS. AND GAME-STATE REDUCTION. R. Hayward, Y. Bjomsson, M. Johanson, M. Kan, N. Po, J. van Rijswijck. Department of Computing Science, University of Alberta, Edmonton, Alberta, Canada.

Solving 8×8 Hex. Henderson, Arneson and Hayward, IJCAI 2009.

9×9 Hex: Scalable Parallel Depth-First Proof Number Search. Paulewicz and Hayward, Proc. Computers and Games CG2013, Springer LNCS 8427 (2014) 138-150.

Tagged Connection Games, Games, Hex, Matthew Seymour, Strategy Guide

Feb 12 2021

9 Comments

Chess, Games, Shogi

Quick picks: interesting abstract games in brief

As some of you will be aware, as a way of keeping myself occupied during the pandemic I’ve learned how to use Adobe Illustrator to design stuff. A particularly enjoyable, if slightly odd, area of design I’ve gotten into is designing game boards for abstract strategy board games. I’ve had a good time getting to know the software and experimenting with many different designs, and now that nice neoprene game mats can be custom-printed for affordable prices, I’ve actually gone ahead and had some of my designs printed out as well. Hopefully, in some theoretical future where the pandemic is over, I can use these boards to introduce friends and colleagues to some of my favourite games.

I’ve made a lot of boards over the last year, so rather than wait until I can find the time and energy to write detailed blog posts on all of the games that go with them, I thought I’d share a few abstract strategy gems with you with just a few sentences about why they’re interesting. Each brief review includes links to full-size images of the boards I’ve made for each game, which you can print if you wish. Some of these games will get covered in detail in the future; for now, hopefully these short descriptions will entice some of you to give these games a try.

As a side note, I can output these designs in a huge number of formats — PDF, PNG, JPG, SVG, whatever — so if any of these strike your fancy but you need a different format, just let me know in the comments and I’ll upload it for you.

Catchup

Catchup is a wonderful game by Nick Bentley that I’ve mentioned briefly before, because the scoring system inspired my choice of scoring system for Permute. This is a game I’ll definitely cover in the future, as it’s incredibly easy to learn, yet within moments of starting to play you’ll realise the core strategic dilemmas at the heart of the game. Catchup is a really dynamic and exciting game, and personally I think Catchup is Nick’s best design by far.

Why it’s great: Catchup’s unique feel stems from its unusual movement protocol: each turn, you place two stones anywhere on the board, unless your opponent equalled or exceeded your score after their last move, and then you can place three stones. The winner is the player who forms the largest group of connected stones at the end of the game, so the result is a tense back-and-forth where you absolutely must connect your stones to win, but each time your biggest group becomes equal to or larger than your opponent’s, they get a much more powerful move with which to fight back.

About the boards: The board on the top left above is a standard hexhex board, seven hexes on a side, with a scoring track where players can place a stone on the number representing the size of their current largest group. The other five are variant boards with uneven sides, which an experienced Catchup player has suggested may generate more interesting play.

Chess: Supersized

These are simply enlarged chessboards — 10×10 squares and 12×12 — that I plan to print on mats and use to play large variants of Chess. Many Chess fans over the years have attempted to transport the magic of the Royal Game to larger boards, and thankfully a number of them succeeded in creating some very enjoyable variants that feel like Chess, but still have a unique personality. I’m planning to write an article in the future that will cover a bunch of large Chess variants and give you some detailed recommendations; for now, here’s a few worth checking out on both board sizes, should you fancy giving them a go.

Some recommended 10×10 Chess variants: Caissa Brittania (checkmate the Queen instead of the King!), Decimaka (hybrid of Chess and Maka Dai Dai Shogi), Elven Chess (hybrid of Chess and Chu Shogi), Grand Chess (Christian Freeling’s most famous Chess variant), Grand Shatranj (ancient Persian Chess brought to 10×10), Omega Chess (commercial variant with Wizards and Champions), Opulent Chess (Grand Chess but more my style — higher piece density, less wild tactically), Shako (Chess with Cannons and Elephants).

Some recommended 12×12 Chess variants: Chu Shogi (the best 12×12 Chess-type game, period), Gross Chess (mix of Grand Chess, Omega and Asian variants, very playable), Metamachy (fast-paced Pawns and crazy historical pieces give it a unique and fun feel), Zanzibar-XL (dense and diverse piece selection with a variable setup).

Exo-Hex and Iris

I briefly covered both these games before, but since then I’ve made some enlarged boards for myself, so I thought I’d share these here and urge you again to give them a shot. Both these games are from Craig Duncan, and they are unique connection games that are centred on scoring points rather than being the first to make a single connection. Both are rich and highly strategic, and well worth your time.

Why they’re great: Exo-Hex is essentially a distillation of Side Stitch into a simpler form, playable with a standard hexhex board with some extra stones around the edges. The more straightforward rules and minimalistic look are great for beginners who may not yet be ready to graduate to Side Stitch and its endless variety of possible playing surfaces. Exo-Hex is also much easier to construct with components you may already have around, so it’s more straightforward to pick up and play.

Iris, meanwhile, is part of the surprisingly small family of connection games with two-move turns. Simple restrictions on placement — you may either place two stones on same-coloured spaces on the edge of the board that are directly opposite each other, or two stones in the centre on non-adjacent spaces — means that the game moves quickly and has a huge number of possible moves per turn (a large branching factor), yet structures you will know from Hex and other one-move games still work. I’ve played Iris a lot against Ai Ai and I highly recommend it for any fan of connection games.

Lotus and Medusa

Lotus and Medusa are two under-appreciated territory games by Christian Freeling that are closely related — in fact Christian calls Lotus the ‘support act’ for Medusa. Both centre around the use of a mechanic from a game called Rosette. Over the years, numerous designers have tried to transport the game of Go to the hexagonal grid, only to find that the reduced connectivity of each point (from 4 adjacencies to 3) made it too hard for players to build stable groups of stones. Rosette addressed this by allowing groups of stones containing a rosette — a formation that occupies all six points of a single hexagon — to be immune from capture permanently. Lotus and Medusa adopt this clever tweak, while adding some fascinating additional touches.

Why they’re great: Lotus takes the cool-looking board from the rather disappointing game of Kensington, and turns it into the basis for a compelling territorial contest. Capture doesn’t just eliminate enemy stones, it flips them to your side, like in Othello, and occupying all six points of a hexagon keeps your groups alive forever, as in Rosette. Medusa takes this further by removing hexagons from the playable area of the board to further reduce its connectivity, and allowing players to either place or move a group of stones already on the board. Medusa also has the ‘Othellonian’ capture and rosettes of Lotus. Both games have the satisfying tension of a good Go-like game, but with very different play styles; Lotus is quick and deadly, while Medusa is a longer epic that allows groups to flow sinuously across the board. Both deserve more attention than they’ve received.

Nutty Shogi (and friends)

: A blank 13×13 Shogi board for all your Shogi needs.

: Starting position of Nutty Shogi.

: Nutty Shogi pieces flipped to show their promoted sides.

Nutty Shogi is here as a representative of the class of 13×13 Shogi variants. The only historic 13×13 Shogi variant is Heian Dai Shogi, which is a very early form of Dai Shogi that is unfortunately not very enjoyable to play. However, some modern Shogi variant fans have created some 13×13 variants that are worth your time, and given that 13×13 Shogi boards are not available anywhere, I decided to create one to print on a mat.

Why 13×13 Shogi is great: Nutty Shogi, designed by HG Muller, is a reduced version of Tenjiku Shogi, a 16×16 historic Shogi variant famous for its outrageously powerful pieces and extremely fast-paced and destructive play. Nutty Shogi condenses Tenjiku’s armies of 78 pieces per player, with 36 types of pieces, down to 50 pieces of 25 types — still much more than Chess or Shogi, but quite manageable. The selection of pieces is basically a Tenjiku Greatest Hits album, so the game retains the feel of Tenjiku in a more compact size. HG Muller also created two other worthwhile 13×13 variants: Cashew Shogi, a reduced form of Dai Dai Shogi; and Macademia Shogi, a reduced form of Maka Dai Dai Shogi. While you’re at it, do check out Mitsugumi Shogi, a condensed form of Suzumu Shogi, which is a modern variant of Tenjiku Shogi (still with me here?). All of these games pack a lot of action into that 13×13 area, so despite the large boards and starting arrays they are far from slow.

Odd-Y and Pex

Here we have two fascinating variants of the seminal connection games Hex and the Game of Y. Odd-Y extends the core concept of Y to boards with more than three sides, while Pex transports Hex to a grid of irregular pentagons.

Why they’re great: Odd-Y circumvents one of the shortcomings of Y, in my opinion, which is that the triangular Y board gives different areas of the board very different values, which means some parts of the playing area go largely unused. Odd-Y extends the goal of forming a Y — connecting three sides of the board — to boards of more sides, creating a more expansive feel. The new winning condition is a bit complicated to explain on larger boards, but Odd-Y with five sides — 5-Y — is beautifully simple: connect any three sides to win, so long as all three sides are not adjacent. This can then be translated to a six-sided hexagonal board by colouring the edges with five colours in a pattern like you see above (Craig Duncan came up with this idea). 5-Y feels very freeing — there are more winning connections available than in Y, creating more strategic complexity, and the entire board surface feels useful.

Pex was invented by connection game maestro David J Bush, world champion of TwixT and co-author of my post on that game. He transformed Hex by placing it on the irregular pentagonal grid you see above, keeping all the rules the same (not that there are many rules in Hex). The new grid forces significant changes in tactics, as cells now have different adjacencies, so standard Hex techniques won’t work. Pex is a challenging and interesting variant, definitely intriguing for experienced Hex players, but also simple enough for newcomers to pick up and enjoy within minutes.

Snodd (and Xodd/Yodd)

: 16×16 Snodd board.

: Snodd board with placed stones, showing the changed adjacencies of points.

Snodd is a variant of a pair of games by Luis Bolaños Mures called Xodd and Yodd. Xodd/Yodd are mind-bending games in which players are assigned a colour, yet may play stones of both colours; Xodd is played on a square grid, while Yodd is played on a hexagonal grid. On your turn, you may place two stones on the board, each of which may be either colour, and at the end of the game the player with the smallest number of groups on the board in their colour wins the game. There’s a catch, however: at the end of any player’s turn, the total number of groups of stones on the board must be odd! This single restriction is what makes the game so challenging and unique. When you start to play you’ll soon realise how this parity restriction allows you to catch your opponent out in all sorts of clever ways.

Why Snodd might be great: Snodd is my attempt to bridge the gap between Xodd and Yodd. Xodd is played on a square grid, where each square has four adjacencies (diagonal adjacencies don’t count), resulting in a tight, tactical game where groups are often split apart. Yodd is played on a hexagonal grid, where cells have six adjacencies, meaning groups stay connected more easily and the game feels more deliberate and strategic.

In Snodd I took the exact same rules and ported them to a snub-square tiling. When you play on the points of this pattern, each point has five adjacencies, placing it right between Xodd and Yodd’s geometries. In theory, this should make a version of the game with a nice balance between tactical fights and global strategies. Test games against myself have been promising, but more investigation is needed. Give it a try and let me know how you find it!

*Star and Superstar

*Star is another game I’ve covered before, but at the time I was a bit confused about the rules and had yet to try it. Boards are also hard to obtain, as they can only be ordered from America, and shipping from America now is ludicrously expensive, so I made two variations of the *Star board to print myself. Superstar is a predecessor of Starweb, a fantastic connection game from Christian Freeling; Christian says Superstar is no good now and fully superseded by Starweb, but he thinks lots of things are no good, so I wouldn’t take that to heart.

Why they’re great: *Star is the final iteration of Craige Schensted/Ea Ea’s set of connection games built around the goal of claiming edges and corner cells, then connecting groups of those cells together. *Star is a bit hard to understand at first, but once you get going, you’ll find a dynamic game of territory and connection, where both players writhe hectically around each other trying to weave their scoring groups together. The resulting play is complex and challenging, and games of *Star often exhibit subtle and sophisticated strategies. The *Star board also supports two excellent variants: Double Star, where players may place two stones per turn instead of one; and Star-Y, a pure connection game where players must connect three sides which are not all adjacent (just like 5-Y above).

Superstar’s relationship to Starweb is about more than the shape of the board — there’s a clear lineage here, where Christian was moving from Star/*Star toward what would eventually become Starweb. Despite Christian’s misgivings, I enjoy this game — it has a remarkable diversity, in that multiple types of formations are available for point-scoring: stars (a group touching at least 3 edge cells); superstars (groups connecting 3 or more sides, worth many points); and loops (worth more points for enclosing more cells, and many more points for enclosing enemy stones within). The feel in play is like a heady mix of Star and Havannah, where each player has incredible flexibility and must keep their wits about them to spot the myriad ways their opponent may be seeking to score. The mix of connection and surrounding elements gives it a bit of a territorial feel as well. For me it is a worthy entry in the Freeling canon, distinct enough from both Starweb and Havannah to have its own identity.

About the boards: The two *Star boards above are equivalent — on the blue one you will play your stones in the cells, and on the other you will play on the intersections. I made both since different players may find one or the other easier to parse visually, so I wanted to have both options available. The Superstar board is very similar to the Starweb board, with the notable difference that the light-shaded cells are not playable, but instead are there to indicate the point values of cells adjacent to them. The game would definitely be extendable to larger boards, but uncharacteristically I haven’t yet made one; I plan to write a full post on this game at some point (along with some other connect-key-cells games), so I will be sure to make a bigger board when that day comes.

Tamerlane Chess

Tamerlane Chess is a historic Chess variant from the 14th century; the game was allegedly invented by the Persian ruler Timur Lenk, but that may well be a myth. Tamerlane is a large-board variant of Shatranj, the Persian form of Chess and direct ancestor to the Royal Game we know today. This game takes the core of Shatranj and adds a bunch of unusual elements to the game, giving it a confusing and beguiling personality.

Why it’s great: Tamerlane’s board immediately stands out — not only is it large and oblong, forming a 10×11 grid, but there are two extra squares sticking off the sides. These squares are called citadels, and they serve a special purpose: if your King can reach the citadel on your opponent’s side of the board, you can secure a draw. These little boltholes of safety are just one of the quirks of Tamerlane:

Several unusual pieces are added to the base Shatranj army, including two pieces that leap like the Knight but in different patterns (the Camel and the Giraffe)
The Pawns — shown above as tiny versions of the other pieces — promote differently depending on what column they start from, and the ‘Pawn of Pawns’ (on A3 and K8) can promote three times to become an extra King
The Pawn of Kings promotes to a Prince, which also must be mated to win the game, so each player may have up to three Kings on the board

The result of all this craziness is a remarkably exciting game, with varied tactics thanks to the diverse pieces and unusual endgame strategies resulting from the promotion rules and citadels. Shatranj pieces are generally shorter-range than modern-day Chess pieces, and Tamerlane extends Shatranj with more leapers rather than long-range sliding pieces, so the feel is very different from Chess. Tamerlane may be 600 years old, but it feels modern and creative. I enjoy it a great deal, so I plan to do an article on this game once I finish writing about Courier Chess.

Trike and Tumbleweed

Unlike much of the rest of this list, these two games are extremely new — both Trike (designed by Alek Erickson) and Tumbleweed (designed by Mike Zapawa) were invented in 2020, and in fact are currently slugging it out to take the win in the yearly Best Combinatorial Game competition at BoardGameGeek. Both are very modern designs — they have extremely minimal rules, and are built to do one thing and do it well.

Why they’re great: Trike is an intriguing game in which players place pieces in their colour by moving a neutral pawn piece, then placing their stone underneath it. As the board fills up, the pawn has less freedom of movement, until eventually it can’t go anywhere; at that point, the player with the most stones of their colour adjacent to the neutral pawn wins the game. Trike is very tactically sharp and full of twists and turns, so despite its simplicity the play is complex and exciting. This game reminds me somewhat of Tintas, a brilliant game of moving a neutral pawn to claim a majority of pieces of seven colours. Trike has a quite different feel though and is inherently more flexible and scalable.

Tumbleweed is a game of territory based on a line-of-sight mechanic — on each turn you may place a stack of pieces of your colour in one cell on the board, with the height of that stack determined by the number of your pieces within unobstructed line-of-sight of that cell. You may capture and remove an enemy stack in that cell if your stack would be larger, or you can reinforce your own stack in the same way. At the end of the game, the player who holds the majority of the board wins. Tumbleweed is gaining a lot of attention since its creation, because the simple line-of-sight stack placement idea immediately creates interesting tactical situations and strategic dilemmas. Apparently the community of players is settling on hexhex-8 boards, but I prefer to play on the original hexhex-11 board. Playing in real life is a bit challenging, mainly because you need a huge number of counters to potentially stack them six deep on numerous cells, but playing online or via Ai Ai is straightforward and very enjoyable. My board above plays on the intersections rather than in the cells, which just intuitively makes more sense to me given the line-of-sight mechanic.

Volo

Volo is an innovative game of unification by Dieter Stein. The game was inspired by the flocking of birds, as illustrated in the famous Boids paper by Craig Reynolds (read more about the game and its influences in this paper). The Boids simulation was also seriously influential on me when I was young and first discovered the scientific field called Artificial Life, so I feel a certain kinship with this game. Volo’s rules are fairly simple, but the mechanics are evocative of the theme: the board starts empty, and as you gradually place birds you will need to fly whole flocks of them around the board at once in an attempt to join them together into one giant flock. Being able to move an entire line of pieces at once is fairly unusual in abstract games, so it feels quite satisfying. The first player to create one unified flock including all their birds is the winner.

Why it’s great: Volo is a creative game, and its inspiration comes through beautifully in its clever rules. You will feel like you’re navigating your flocks through treacherous skies, trying to bring your birds together to safety. Volo is also a fine example of the unification genre, which is surprisingly small; the most famous examples are probably Lines of Action, which is a brilliant game with an oddball movement mechanic, and Ayu, a compelling game playable on a Go board where every move is an approach move. The unification genre is small but mighty, and Volo may just be my favourite of the lot; the ability to move lots of pieces in a single turn gives it a sense of freedom and allows for some highly creative moves.

About the boards: The standard Volo board is a hexhex-7 board with corners and the center point removed. In the spirit of experimentation I’ve been playing with larger boards, so you can see above I’ve constructed hexhex-9 and hexhex-11 boards for more epic Volo games. On all the Volo boards you place your birds on the intersections, rather than within the triangular spaces.

YvY

YvY is another forgotten connect-the-key-cells game from Christian Freeling, developed as a vision of a simplified Superstar, then refined into its final form in collaboration with David J Bush. In YvY, players take turns placing one stone of their colour onto the oddly-shaped hexagonal grid, and attempt to occupy and join together the green ‘sprouts’ sticking off the side of the board. At the end of the game, each player scores points equal to the number of sprouts they occupy, minus twice their total number of ‘live’ groups (live groups being those occupying at least one sprout). So, as with Star and *Star, the scoring system forces you to try to connect your occupied sprouts with as few groups as possible. Intriguingly, YvY also offers a ‘sudden-death’ victory condition: if either player forms a contiguous loop of stones of any size, they win immediately!

Why it’s great: I’m a sucker for a connection game with multiple objectives, and YvY fits squarely into that category. The need to connect groups across the board to score well gives the game a territorial feel, while the loop-formation win condition adds some tactical sharpness on top. In play the game bears a certain resemblance to Havannah, and the need to score points via multiple connections encourages board-spanning play with great subtlety. Christian views this game as obsolete, but I see it as another intriguing take on the connect-the-key-cells genre, alongside Star, *Star, Superstar, Starweb and Side Stitch. For my money this category of games offers a lot of depth and intrigue, so I recommend trying several of them and seeing which one best fits your style of play.

About the boards: As per usual, I made a few different sizes of boards for this game, to allow potential players to choose a game length that suits them. The YvY board is oddly shaped, with three of the sides being two hexes longer than the other three; as a consequence of this shape and the need to place sprouts evenly around the outside edges, the boards all have even-length sides. As is typical with games like this, the larger boards produce longer games of greater strategic complexity; the size-12 board above has 330 interior cells and 33 sprouts for a total of 363 cells, almost exactly the same as a Go board’s 361 points. The size-12 board is thus suited for intense strategic contests; the size-8 board is great for beginners and more casual games, while size-10 offers a nice balance between depth and brevity. If you’re feeling particularly adventurous, have a go on the size-14 board, with a whopping 468 interior cells and 39 sprouts.

New boards for old favourites

Side Stitch

I’ve talked about Side Stitch before, of course, but in the last few months I’ve gone back and tidied up the boards I made previously, and added two new ones — the hexhex-11 with 15 colour-sides, and the 14×14 Hex board with 13 colour-sides. Side Stitch is a favourite of mine not just for the actual game, which is great, but also the aesthetic — making boards for this game is really fun.

Why it’s great: Side Stitch is a member of a class of connection games that I really enjoy — connective scoring games, where different types of connections have different values. These games spice up the connection-game formula by allowing for a wide variety of winning connections, and the need to stretch across the board to connect key areas and score points gives them a dynamic flavour. Side Stitch is even more dynamic than most, since players connect colours along the edges of the board which need not match up with the actual board’s sides, so there are a tonne of interesting board setups you can try. I just wish Side Stitch was playable on more game servers, so that more people would get acquainted with this excellent game.

About the boards: All of the boards above were based on designs originally uploaded to BoardGameGeek by the inventor of the game, Craig Duncan; I have simply replicated them in Illustrator and made them as clean and sharp as I can. The ‘standard’ Side Stitch board is the hexhex-8 with 7 colour-sides (top middle in the above array). The hexhex-7/9-colour board is great for quick games. My personal favourites are the hexhex-10 with 9 colour-sides and the hexhex-11 with 15 colour-sides; note that I have two variants of the 11/15 board available, one with some repeated colours and another with all unique colours. To my shame I have not tried the 14×14 Hex board version yet!

Star

Star is a classic game of connecting edge cells by Craige Schensted/Ea Ea, which I’ve covered before on this blog, so I won’t spend too long explaining it. These boards are slight updates of previous ones that I have made, with slightly cleaned-up cell placement and updated fonts.

Why it’s great: Star is an unfortunately overlooked game, I think partially because the published version in Games Magazine years ago was on a too-small board that didn’t adequately showcase its marvellous depths, and also because it was followed by *Star, which seemed to overshadow it. I think Star deserves more recognition than it gets, as it an accessible game only slightly more complex than something like Hex or Y, but the introduction of scoring and a group penalty takes it into a more territorial, strategic realm. On larger boards like those you see above, Star becomes a deeply challenging contest, and often a game will see much of the board filled with complex, winding connections. I highly recommend it both on its own merits as a beautiful game, and as a first foray into the connect-the-key-cells genre.

About the boards: My boards adopt the standard uneven hexagonal grid used by the original game, and simply extend that to larger sizes. I should note that the designer felt the corner cells, which on these boards would be worth three points due to being adjacent to three exterior edge cells, should be adjusted to only score two points; I don’t have particularly strong feelings about this, but in the future I do intend to make versions of these boards with corners altered in that way. Of course you can use these boards and simply adjust the scores accordingly when you play, but certainly having the scores clearly visible from the board geometry would be better. The largest board above, Star-12, contains 363 cells, similar to the Go board’s 361 points. Given that Star games often use most of the board, Star-12 is probably the largest size most players would be willing to use, and above that size the game is perhaps a bit too much of a marathon.

Poly-Y

Poly-Y is the ancestor to Star and *Star, and marks the first attempt by designer Craig Schensted/Ea Ea to impart a connection game with a bit of territorial flavour. In Poly-Y, players strive to control more corners of the board than their opponent; in order to claim a corner, a player must form a Y-shaped connection, connecting the two sides adjacent to the corner with another non-adjacent side.

Why it’s great: Poly-Y takes the connection goal of the Game of Y and adds a territorial element, using that connection as a way to claim parts of the board and score points. The addition of the point-scoring element gives the game an appealing strategic flavour, while adding minimal rules complexity. The importance of corners in this game means that oddly-shaped boards with larger numbers of corners are particularly well-suited for Poly-Y play, which adds a certain quirky visual appeal. If you want the depth of something like Star or *Star with simpler score calculations, Poly-Y is a great option.

About the boards: Out of the three boards presented above, only the middle one is for playing stones within the cells; on the other two, you should place your stones on the intersections. Making these boards was a bit of a challenge due to the odd geometry, but the final result is quite visually pleasing. All three boards are nine-sided, which seems to be the most-recommended shape by the designer, so they will play similarly; just pick the one that most suits your aesthetics.

Game of Y (Kadon-shaped)

As I mentioned in the Game of Y/Poly-Y/Star/*Star article, the published version of the Game of Y uses a board of 91 points with a distorted triangular shape, designed to balance out the in-game value of the centre, edge and corner points. However, the board published by Kadon is simply too small, meaning that every opening move by the first player should be swapped. A better option is to use the same board geometry but substantially larger, and that is what I have attempted with this board.

Why it’s great: Y is the most elemental connection game, even more fundamental than Hex — in Hex the two players have asymmetric goals, and are attempting to connect different sides of the board, while in Y both players have precisely the same goal. The need to connect all three sides of the triangular board can produce some interesting tactics, and it has a bit of a different flavour from Hex as a result. For people new to connection games, or to abstract strategy games in general, Y is right up there with Hex as an instantly accessible gateway to the genre.

About the board: The board above is 17 points long on each side, meaning that games will be substantially longer and more balanced than on the 91-cell Kadon board. Besides being visually appealing, this board geometry helps balance the values of board cells. The downside is that I haven’t yet found a straightforward way to extend this board in Illustrator without reconstructing large portions of it, so for now this is the only large board of this type that I’ve made.

So, that was a whirlwind tour of some of the games I made boards for over the past 12 months or so. Over the coming months I’ll try to cover a few of these gems in more detail, but at least for now I hope this will give you some ideas if you’re looking to try out a new game.

Next up: more Courier Chess!

Tagged *Star, Catchup, Chess, Connection Games, Exo-Hex, Game of Y, Games, Hex, Iris, Lotus, Nutty Shogi, Odd-Y, Pex, Poly-Y, Shogi, Star, Superstar, Tamerlane Chess, Trike, Tumbleweed, Volo, Xodd, Yodd, YvY

Feb 21 2020

8 Comments

Games

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.

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:

The player to move may add one stone of their colour AND one neutral stone to any empty hexes on the board, OR
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.

Tagged Abstract Games, Connection Games, Games, Hex

Dr Eric Silverman

Science, Philosophy and Miscellaneous Musings

Tag Archives: Hex

A Beginner’s Guide to Hex, Part II: Sample Games

Commented Hex Game: LG #2195354, kspttw vs Arek Kulczycki

Y Sample Game: PCM vs Matthew Seymour

A Beginner’s Guide to Hex

Basic Tactics

The edges

Interior templates: connecting chains

Defending at a distance

Ladders

Strategic Considerations

Openings

Playing in the corners

Influence

Territory

Taking the initiative

Tenuki

The Joy of Hex

Next moves

Extra nerd stuff

Quick picks: interesting abstract games in brief

Catchup

Chess: Supersized

Exo-Hex and Iris

Lotus and Medusa

Nutty Shogi (and friends)

Odd-Y and Pex

Snodd (and Xodd/Yodd)

*Star and Superstar

Tamerlane Chess

Trike and Tumbleweed

Volo

YvY

New boards for old favourites

Side Stitch

Star

Poly-Y

Game of Y (Kadon-shaped)

Connection Games I: Hex

Hex

The current state of play

Playing Hex

Hex Variants

What next?