I’m going to introduce the basic concepts of Hex by walking you through an example game. This game will be played on a 9×9 board, for simplicity.
The players open with e3c6e4 (Diagram 7, remember, you can click on the moves in the text to see them on the diagram). With move 3, Black plays adjacent to his opening stone, seeking to extend towards the southeast edge of the board.
White also wishes to extend her opening move towards the opposite (in this case, northeast) side of the board. But instead of an adjacent move, she plays e5. This is a stronger play than the adjacent move that Black made. Why? Because Black can’t actually separate from . If Black plays d5, White can respond at d6 and keep them connected. Alternatively, if Black plays at d6, White responds at d5. Another way to think of this is that White can connect to by way of one of two moves: either playing at (d5) or at (d6). Since Black has only one move himself, he cannot block both. This is called a double threat. Thanks to the double threat, we say these two stones are connected, even though they aren’t physically adjacent.
This particular pattern of stones is so common it has a name: the bridge. It’s also the simplest example of what’s called a template. Templates are patterns of connected stones, along with the empty hexes necessary to keep them connected. We will have much to say about templates later.
Example game, first 4 moves.
By using a bridge, White has moved across the board faster than Black has. In addition, she has also blocked Black’s progress towards the southeast edge. Hex cannot end in a draw or tie. If you can block your opponent’s connection, you yourself are connected. With this in mind, rather than trying to extend his group towards the opposite edge, Black decides instead to attempt to block White’s progress. Black considers his prospects of blocking from the northeast edge better than blocking from the southwest edge. So he plays f5 (Diagram 8). White sidesteps Black’s block with f4. Another attempt by Black is easily sidestepped as well: g4g3. What is Black to do?
Example game, moves 5 through 8.
The problem with adjacent blocking is that at each step, there are two empty hexes in front of the white stone, and Black cannot fill both hexes with a single move. Seeing that adjacent blocking is futile, Black steps back a row and blocks at i2 (Diagram 9). Now if White plays at h2 Black can respond at i1, and if she plays at h3 he can respond at i3. By moving back a row Black can meet both potential moves by White and his block is more effective.
Example game, moves 9 through 17.
White elects to play the h3i3 sequence. She follows up with h4, threatening to connect to the northeast edge at i4. Black is forced to respond there with i4. White continues to play forcing moves with h5i5h6i6. Sequences of forced moves along the edge like this occur frequently in Hex and are called ladders. In this situation we call White (who’s dictating play) the attacking player and Black the defending player.
White may be dictating play, but she can’t just push this ladder indefinitely. If she does so it will be Black who connects: h7i7h8i8h9i9. She’ll have to think of something else.
Rather than continuing to push the ladder, White steps forward a hex and plays at h8 (Diagram 10). The first thing to note about this move is that it’s connected to the northeast edge. Much as with the bridge, White has two options to connect to the edge. These are labeled and in the diagram. If Black attempts to block one of these, White plays the other: i7i8 (or i8i7). This stone , along with the two empty hexes and , is the simplest example of a type of template called an edge template: a stone (or group of stones) connected to an edge.
Example game, moves 18 through 22.
One way to view the h8 move is as blocking from the southeast edge. Another way to view it is as a forcing move: since is connected to the northeast edge, Black has no choice here but to block it from connecting to White’s main group, which he does with h7. White plays another forcing move with g8, which is connected to the northeast edge via . Black is forced to block from connecting back to the main group with g7.
At this point the situation is somewhat reversed. Black threatens to connect to the southeast edge by playing at f8 (recall the edge template we discussed on move 18), and White is forced to block him with f8. What we have here over the last five moves is another ladder, except one row back and with the roles reversed—Black is the attacker and White the defender. Ladders are labeled according to the row of the attacker, hence we call this a third-row ladder, while the ladder along the northeast edge was a second-row ladder.
As with the ladder along the northeast edge, Black cannot push this ladder all the way across the board or it will be White who connects. So he chooses to break here and plays e7 (Diagram 11). White responds with f7. Noticing how she threatens to connect back to her central group at g6, Black blocks her there with g6.
Example game, moves 23 through 25.
Now White plays f6 (Diagram 12). Here we have another double threat: White can connect to her central group through either or . In an act of desperation, Black tries to block her from the southwest edge with a7, but White easily steps around him with b5, using a bridge and our edge template to connect to the southwest edge. Now White has connected her southwest edge to her northeast edge with a series of double threats. Seeing the game is lost, Black resigns. There is nothing he can do to stop White (for example, a5a6b6c5d5d6e6g5i8i7 and White has connected solidly).
Example game, moves 26 through 28.
Having encountered the key concepts of this chapter through our example game, we will now explore each in greater depth.
Before continuing, let’s briefly review some common terms used to describe the hex board.
Diagram 13 shows the major areas of the board. Edges are referred to by their compass directions: northwest, southeast, and so on. Corners can be referred to in the same way (north, east, south and west). Additionally, there are two kinds of corners on the hex board: the obtuse corners in the north and south, and the acute corners in the east and west. Obtuse and acute corners each have a distinct character and this affects the strategy of the game, as you will see.
Major board areas.
There are also two diagonals, shown in Diagram 14. The short diagonal stretches north to south between the two obtuse corners. The long diagonal stretches east to west between the two acute corners.
Hex is a connection game. One or more stones directly adjacent to one another form a chain. Diagram 15 shows three examples of chains (note that a single stone is trivially considered a chain). The object of the game is to build a chain which connects your two opposing sides of the board.
Examples of chains.
As we saw with the bridge in our example game, stones can be effectively connected even if they aren’t touching. Standard patterns of effectively connected stones such as the bridge are called templates. The key idea behind a template is that certain patterns of stones can never be separated, even if the other player has the first move (unless, of course, the player with the template allows them to be separated). We already encountered two templates in our example game. The first was the bridge, which I’ve reproduced schematically in Diagram 16 (by convention, all templates will be shown with black stones). It is important to understand that the template consists of not just the occupied hexes, but also of the empty hexes. In template diagrams such as Diagram 16, these are shaded grey. If the empty hexes are occupied by the opponent, the template is not valid!
If one of the template’s empty hexes is taken by the opposing player, this is termed an intrusion, and if the owner of the template does not respond on his next move, he risks losing the connection. There is always a response available within the template that saves it (if there wasn’t, it wouldn’t be considered a template). Since the bridge has only two empty hexes, it has only two possible points of intrusion. Diagram 17 shows both intrusions and Black’s responses to each. Templates are always minimal: no unnecessary empty hexes are included. Therefore every intrusion requires a response to save the template.
Black can always keep the two stones connected.
The bridge is an interior template. It can be played anywhere on the board. Of special significance, however, are edge templates: templates in which a stone or group of stones is connected to an edge. Diagram 18 shows a schematic of the edge template we encountered in our example game, Template A-2 (edge templates have a special naming scheme, we’ll go over this later). The up arrow () indicates that this stone needs to be connected to the other end of the board for Black to win. Diagram 19 shows the two intrusions White can make and Black’s responses.
By convention, edge templates will be shown for Black along the bottom edge. Naturally either player could play these along either of their two edges.
Template A-2 intrusions.
Using bridges and edge templates, a player can cross the board much more quickly. Compare the speed of Black and White’s progress in Diagram 20. By using bridges with Template A-2, Black moves across the board in much less time.
Note: when you see the icon next to a template name, hover your mouse over the template name to see the template.
A game of Hex never ends in a tie: the game is guaranteed to end eventually (because stones are never removed), and the only way for one player to completely block the other’s connection is to connect their own two sides. In Hex, offence and defense are the same thing: to successfully block your opponent is to win the game. Depending on the circumstances it can be better to think defensively or offensively. If you find yourself unable to think of a good move, the best thing to do is to look for a good block.
Given the situation in Diagram 20 above, what is White’s best hope to block Black from connecting to the southeast? It’s tempting to try and block by placing stones directly adjacent to Black’s f8 stone (termed an adjacent block), but as Diagram 21 shows, this approach isn’t that successful. As we saw in our example game, the opposing player can just keep moving around the block, essentially because there are two hexes ahead of him at any point, and the blocker can only fill one of those.
Attempting to block with adjacent moves.
Given this, maybe a better idea is to step one hex back and block at e10 (a near block). Black, however, can respond by bridging to the side, as in Diagram 22. If this continues he’ll reach the southeast edge eventually, so this doesn’t quite work either.
Attempting to block further away.
What we can do is combine these ideas. On one hand, noticing that Black can only bridge to the right, we could play f9e9d11 (Diagram 23, left), starting with the adjacent block and then the near block. Or we could play the blocks in the reverse order with e10g9g10f10e12 (Diagram 23, right), starting with the near block and then playing the adjacent block. As you can see, both blocks leave Black with no way to break through.
Combining simple blocks.
Another approach is to step back even further and play a far block at d11 or e11, as in Diagram 24. By blocking further back you can respond both to the direct advance and attempts to move to the side. This is sometimes referred to as the classic block.
The classic block.
Finally we consider the indirect block (Diagram 25). White first plays at f10. If Black attempts to bridge forward (e10), White can block him with d12 and he will be unable to bridge to the side.
The subject of blocking naturally leads us to the subject of ladders, since blocking often results in ladders. We encountered ladders in our example game. Now let’s consider them in greater depth.
White to block.
Consider the situation depicted in Diagram 26. White wishes to block Black from reaching the southeast edge, so she plays d6f5f6e6d8 (see Diagram 27). Black can’t force a connection here, but he can force a series of moves to the left or right: for example e7e8f7f8g7g8. This is called a ladder. Notice how White’s responses are forced—after each Black move, he is one hex away from connecting to the edge, so White must respond there. Black’s moves are therefore called forcing moves and we can view Black as the attacking player and White the defending player. It’s often said that the attacking player has the initiative. Initiative is an important concept in Hex; the player with initiative has more options with each move and can dictate play into a favourable position for themselves. Ladders are just one example of forcing moves.
Ladders can form on any row, and are labeled according to the row of the attacking player, so the ladder in
Diagram 27 would be a second-row ladder.
A ladder forms.
The attacking player always has the choice to move elsewhere, while the defending player does not. Notice, however, that if Black keeps forcing the ladder to the left and right (as in Diagram 28) it will be White who ends up connecting her edges and winning—being on the attack is no guarantee of connecting. Even though the ladder on its own cannot connect to the edge, there are a wide range of tactical tricks available to the attacker to force a connection. Two of the most important are the ladder escape and the fork.
Suppose now there had been an additional black stone at , as in Diagram 29a. When the black ladder reaches this stone it is now connected by Template A-2, and so White is not able to prevent the connection. The stone is known as a ladder escape. An important early game strategy is placing stones that can be used as ladder escapes in the future.
A ladder escape.
However, once the ladder forms (or is about to form), it’s too late to place a ladder escape. Doing so takes up a move, allowing your opponent to simply block the ladder, as in Diagram 30.
Once the ladder has formed, it’s too late to place the escape.
Suppose instead that there was no white stone on d4. Now Black can use a ladder-escape fork to connect (Diagram 31). First he plays b7. Since this stone serves as a ladder escape, White responds by blocking the ladder with d7. But now Black plays c5, connecting to the edge with two bridges.
Black’s move is yet another example of a double threat. This move threatens to connect in two different ways: either by the ladder or by . Since White has only one move, she cannot block both. Always be on the lookout for opportunities to use double threats in your games.
A ladder-escape fork.
Another common way to create ladder escapes is to intrude into an opponent’s bridge. Consider the situation in Diagram 32. Black’s stone is connected to the northwest edge of the board through some bridges and he wishes to connect to the southeast. He can ladder to the left, but there is no ladder escape. So Black creates one by intruding into White’s bridge at c8 (Diagram 33), threatening to connect via and Template A-2. White saves the bridge with c7. Black now has his ladder escape. He plays the ladder and connects: f8e9e8d9d8.
Interior templates are like the bridge in Diagram 16: they are simply groups of stones which are connected to each other, anywhere on the board. In addition to the bridge, four more interior templates every hex player should be familiar with are shown in Diagram 36.
Due to symmetry, there are only really two possible intrusions into the Wheel. Diagram 37 shows the responses to these. In the response to the central intrusion, Black plays on any one of the exterior points. The lone black stone is connected to the chain of three stones by a double threat.
Responding to intrusions in the wheel.
Diagram 38 demonstrates responses to two of the four possible intrusions of the Crescent. In the left diagram, Black uses a bridge to keep the chain of three stones connected to the single stone. Can you work out the responses to the other two possible intrusions?
Responding to intrusions in the crescent.
In Diagram 39, we see some responses to intrusions into the Span. On the left we can see that Black can “double bridge” in response to an intrusion at the bottom. On the other hand, if White plays anywhere else (as in the right diagram) Black simply plays the bottom hex to connect the two sides of the Span.
Responding to intrusions in the span.
The solutions to intrusions into the Trapezoid are left as an exercise to the reader. Also note that my responses above are not necessarily unique! Some, but not all, of these intrusions can be responded to in more than one way. Can you find some alternate responses?
When you have many stones connected by chains and templates this is called a group. Diagram 40 shows a group of black stones consisting of (from left to right) a Trapezoid, a Span and a bridge. These eight stones are connected and White cannot separate them.
A group of stones.
The double threat concept can be applied to groups: if two groups of stones can be connected by one of two non-overlapping moves, then they are connected and can be considered one large group. In Diagram 41 we can see that the Wheel and Trapezoid templates can be connected by playing at either or , so all these stones together form a group.
We’ve already encountered the very basic Template A-2 (Diagram 42a). Now we’ll consider some more complex edge templates, such as Template A-3. Learning third- and fourth-row edge templates will strengthen your game considerably. Since edge templates can be so large, we’ll require new techniques to “solve” them.
(a) Template A-2
(b) Template A-3
A note on naming: There are too many edge templates to give them all individual names. Since many edge templates share similarities, I have placed them into lettered groups (A, B, C, etc.). Templates in each group share a common “theme.” Grouping them in this way tends to make them easier to remember. Within each group, the number indicates how many rows back the furthest connected stone lies—for example the stone in Template A-2 lies on the second row, while the stone in Template A-3 lies on the third row.
We’ve already solved Template A-2: if White intrudes at either of the two empty hexes, Black responds by playing the other one. What about Template A-3 (Diagram 42b)? It would be tedious to work out a response to eight possible intrusions, but we can take a shortcut. In Diagram 43 notice that, with one move, Black can reduce this template to the simpler Template A-2 (left), or alternatively, a bridge plus Template A-2 (right). This means Black has a response to every possible white intrusion: if White plays in any of the hexes involved in one reduction, Black simply plays the other one. This is another example of a double threat.
Template A-3 reductions.
Diagram 44 shows a succinct way to convey this information. This kind of diagram is called a pivot diagram. The small dots indicate Black’s potential moves. Connections between these potential moves and existing stones (or the board’s edge) are shown by lines. Bridges connections are shown by a pair of arcs.
Template A-3 pivot diagram.
From the pivot diagram we can see that Black has double threats on multiple levels. The two potential moves (the dots) are each connected to the edge through double threats. The leftmost potential move is connected to the stone at the top directly, while the rightmost potential move is connected to the stone through the double threat of the bridge. Since either of these two moves would connect the stone to the edge, together they constitute a double threat.
Looking back at our example game, we can see that this template was used a number of times! The stones on e3, e7 and c6 (Diagram 11) were all connected to their respective edges by Template A-3. It’s no surprise, then, that Black was unsuccessful in attempting to block White on move 27. She simply bridged to the side, employing one of our reductions.
Using reductions we can simplify the analysis of templates considerably. Diagram 45 shows another important template (Template B-3) and its reductions. These reductions consist of bridging to the left or the right and using Template A-2. Notice the unshaded hex within the template. This hex is not part of the template and could be occupied by White. Diagram 46 shows the pivot diagram for this template.
Template B-3 and reductions.
Template B-3 pivot diagram.
Diagram 47 shows Template B-4. Similar to Template B-3, the reductions of this template consist of bridging to either the left or the right, this time using Template A-3 (Diagram 48). Again, the unshaded hex in the center is not part of the template, and could be occupied by White.
Template B-4 reductions.
Diagram 49 shows the pivot diagram for this template.
Template B-4 pivot diagram.
Diagram 50 shows Template C-3 and Template E-3, edge templates with two stones. These templates can both be solved with two reductions. Can you find them?
And lastly, let’s look at an interior template. Diagram 51 shows the Parallelogram. This template can be reduced with one of two bridge moves, as can be seen in the pivot diagram.
Diagram 52 shows a winning position for White. I’ve shaded White’s connection in the diagram. Starting from the southwest edge, the connection consists of a Template A-3, a bridge, a Crescent, a Trapezoid, another Crescent, and finally connecting to the northeast edge with another Template A-3. Note that while Black has several groups of stones, none form a complete connection between his two sides of the board. Since Black is unable to break any of White’s connections, the game is effectively over. It would be customary for Black to resign in this position.
White has won the game.
Diagram 53 shows a winning position for Black. Black connects with Template A-3 to the northwest edge and Template B-4 to the southeast. Black also has two double threats (marked on the diagram): his northwest group connects to his central group through either or , and his central group connects to his southeast group through either or .
Consider the position in Diagram 54. Black’s stones and are connected to their respective edges via Template A-3, and they are connected together through a series of bridges. It would appear that Black has won the game.
White to move.
But there’s a problem! Black’s j9–k10 and k10–i11 bridges overlap at j10. If White plays here then she intrudes into both templates. Black can’t save them both with one move and so White breaks the connection (see Diagram 55).
This illustrates an important lesson: do not ignore the empty hexes which make up the templates forming your connection. All the templates given here are “minimal,” in that they are as small as they can be, so every intrusion requires a response to save the connection. If a hex did not require a response then it would not be included in the template. And so if a player can, with a single move, intrude into two templates then they can’t (in general) both be saved. In some circumstances you may be able to save the connection due to additional stones or space which may be present, but these are special cases. The rule of thumb is to not overlap the empty hexes in templates when forming connections.
Let’s combine what we’ve learned so far and look at a some examples. As an example from a real game, consider the situation in Diagram 56. White’s stone is connected to the northeast via Template B-4. It is also connected to the stones on and by a series of bridges. But how to reach the southwest edge?
White to move.
The key is the double threat of playing at c9 (Diagram 57), because a stone there would be connected to the main group by way of either or , and Black can’t block both. Since this stone is in a bottleneck, a ladder ensues, which White can escape with a fork at b12, winning the game.