With each move a player’s primary goals are to improve his own position while degrading his opponent’s position. Naturally this requires an understanding of which features of a position are valuable and which are not. The novice understands that strong connections are valuable, and in fact we spent much of the last few chapters discussing what is and isn’t strongly connected. But strong connections can require many moves to make, and overcommitment to one area of the board comes at the expense of other areas. What is needed is an understanding of the value of features that fall short of a strong connection. Understanding these features allows the player to grasp the strategy of the early and middle game. Equally important, of course, is an understanding of the features of a weak position, as these are to be avoided in one’s own position (and encouraged in the opponent’s).
The other major aspect of Hex strategy that one needs to understand is the initiative. The player with the initiative is the one dictating play. By threatening valuable parts of your opponent’s position you can force responses and keep the initiative for yourself. Given the choice between a move that greatly improves your position while losing the initiative, and one which marginally improves your position while keeping it, the latter is often the superior choice (as long as you keep the initiative, you can always make that strong move later). Good players can make long sequences of moves that gradually improve their position by repeatedly exploiting weaknesses in their opponent’s position with forcing moves.
The initiative can change hands in a few ways. A player may simply exhaust all the threats they can make into the opponent’s position in pursuit of improvement to their own. Once this happens they’ll be forced to relent the initiative to their opponent. The move that hands away the initiative is often critical and must be chosen carefully. More dramatic is when a player without the initiative ignores a threat and plays elsewhere in order to regain the initiative. This can happen when the threat is not strong enough to be worth saving what is threatened—at least in relation to what is available to be gained by moving elsewhere. Often the group sacrificed retains some strength (it can often be used as a threat itself, or strengthened later). The last way the initiative may change hands is when a player sees an outright win—if you can just strongly connect your edges, it won’t matter what your opponent does on the next move.
The flow of the game typically consists of a player with the initiative making a series of threats to gain small advantages in their position until the initiative is lost, at which point the other player will return the favour. Between well matched players there can be parts of the game where the initiative changes hands with each move, as each player finds creative ways to avoid responding in the manner their opponent intends.
All of this is built on a foundation of tactical knowledge. Without understanding what is and what isn’t connected, effective strategic play is hopeless.
This chapter covers the two main strategic areas outlined above. The first is positional features, both good and bad. The second is the initiative: using it to improve your position through threats, and, when playing without it, responding to threats in the most effective way. Rather than dedicating one half of the chapter to positions and the other to the initiative, the two concepts will be interwoven throughout.
We’ll begin with the most salient positional feature, that being connections. We’ve already discussed connections extensively in the form of templates. Now we’ll go into more detail on the different kinds of connections.
Solid connections have no intrusion points. Chains of stones as in Diagram 108 are solidly connected. While solid connections are naturally a very strong positional feature, it can take many moves to make a solid connection, and these moves are often better spent elsewhere.
Diagram 108
Solid connection for Black.
Strong connections cannot be broken if the owner doesn’t allow it. Templates (interior and edge) are strong connections. Even though strong connections cannot be broken (unless allowed), intrusions can often be used as threats to gain territory. Diagram 109 shows a strong connection for Black.
Diagram 109
Strong connection for Black.
Lesser connections can be classified by how many moves it takes to make them strong. A strong connection is a 0-move connection by definition (since it is already strong). A 1-move connection takes one move to become strong. It is frequently the case that when one player has a 1-move connection in an area, the other player does as well. Diagram 110 shows an example of this. When a strong connection is intruded into, it becomes a 1-move connection (usually for both players), requiring a move to restore. After the intrusion, the player who originally possessed the strong connection is next to move, and therefore has the opportunity to restore the connection.
Diagram 110
1-move connection for both players. Either player could make a strong connection with one move.
Even weaker is the 2-move connection, requiring one move to turn into a 1-move connection and two moves to turn into a 0-move (strong) connection. Back in Diagram 109, whereas Black has a strong connection, White has a 2-move connection. This symmetry is typical: usually where one player has a strong connection the other has a 2-move connection through the same area. This is not universal, however. It’s possible for one player to have a strong connection through an area where the other player has a 3-move (or more) connection. An example is shown in Diagram 111. In this case the player with the strong connection can actually ignore threats! Try this out for yourself in Diagram 111. Notice how, even after White “intrudes” into the connection, Black still has two ways to connect and therefore doesn’t need to respond. Attempting to intrude into this connection is a wasted move for White. Such situations are rare, however, because it is usually a waste of moves to make a connection so strong in the first place.
Diagram 111
3-move connection for White and a strong connection for Black.
Technically, the status of any connection, be it 0-move, 1-move, etc., is precisely determined. However, in practise, there will always be positions that are too complex for a human to analyze. This depends on a player’s skill, of course, but everyone has a limit of what they can ascertain. In these situations a player must use their intuition to gauge the approximate strength of a connection. In this approximate sense, connections will range along a continuum from weak to strong. I may at times refer to connections in this approximate sense.
Naturally the presence of ladder escapes strengthens a player’s position. It’s easier to get a ladder than a direct connection to an edge, so having an escape makes connecting easier. Since ladders can usually start from anywhere on an edge to reach an escape, ladder escapes can exert influence at great distances, affecting the play over a wide area of the board. For this reason it’s very important to be mindful of the ladder escapes on all edges of the board as you consider your move.
In Diagram 112, White has a second-row ladder escape on the southwest edge due to . She has no ladder escapes along the northeast edge. Black, meanwhile, has second-, third- and fourth-row ladder escapes along the northwest and southeast edges thanks to and , respectively.
Diagram 112
In Diagram 113, gives White second-, third- and fourth-row ladder escapes on the southwest edge. gives her second- and third-row ladder escapes along the northeast edge. Neither of Black’s stones gives him a ladder escape.
Diagram 113
In the next chapter we will encounter more advanced ladder escapes. Of course you’ll want to keep these in mind as you evaluate board positions.
Territory is a fundamental concept of Hex strategy. For now, we’ll keep it simple: your territory consists of all the empty hexes adjacent to your stones on the board. Diagram 114 shows three examples with Black’s territory shaded. In general the more territory you have the better your position: as the game evolves territory provides tactical options which can help you complete your connection.
Since defense is the same as offence in Hex, every move can be seen as a block against your opponent. Jonathan Rydh (see Appendix D: Bibliography and other resources) categorizes moves as either strong or weak blocks.
A strong block upgrades a 1-move connection into a strong one. Or, viewed another way, downgrades the opposing player’s 1-move connection into a 2-move connection. Diagram 115 gives an example of a strong block.
Diagram 115
Strong block by Black, giving him a strong connection and White a 2-move connection.
A weak block reduces the opposing player’s strong connection to a 1-move connection—or, for the blocking player, makes a 1-move connection out of a 2-move connection. Intrusions into templates are weak blocks. Diagram 116 gives an example of a weak block.
At first, weak blocking may seem fruitless, since your opponent can respond by saving the connection with a strong block. But that makes it your turn again, and the stone you used to make the weak block gained some territory (if you placed it right). This is what it means to have the initiative: dictating play through moves that require responses, gradually improving your position. The player making strong blocks is usually the one without the initiative, responding to weak blocks by saving connections. A good weak block (and these are not always easy to find) should leave your opponent with little choice but to save their connection by responding with a strong block. If the threat isn’t strong enough, they may be able to play elsewhere, and you could lose the initiative.
Diagram 116
Weak block by White. Black saves the connection with a strong block.
Ultimately you’ll have to make a strong block somewhere in order to win the game, but if you’ve done a good job with your weak blocks you’ll have the stones you need to complete your connection.
I mentioned towards the end of 4.1 Types of connections that the exact status of a connection isn’t always clear, due to the limits of a player’s ability to analyze a position. In these situations connections must be judged on intuition as being somewhere along a spectrum of “strong” to “weak.” Naturally, the judgement of blocks may also be approximate in such situations.
Having discussed weak blocking in abstract terms, it’s time to look at some examples.
The simplest intrusion is into the bridge, as in Diagram 117. Bridges are very common and serve as natural points of intrusion. Notice here how Black adds the three shaded hexes to his territory. However White also gains a hex of territory in the exchange (). Notice that when intruding into a bridge you have two options. Other types connections can have more intrusion points. The best place to intrude will depend on the overall board position. In Diagram 118, Black intrudes into the west side of the bridge so as to orient the resulting territory towards the center of the board where it will be more useful for him. Had he invaded on the east side with j8i8, the resulting territory would have been nearly worthless. Since there’s no way to go around the east side of the – group (connected with Template M-4b), territory on this side of the group can’t help Black. Intruding into connections in the most advantageous way is an important skill which improves with practice.
Diagram 117
Intruding to gain territory.
Diagram 118
Black intrudes so as to get the most useful territory.
In Diagram 119, White’s e6 cannot be cut off from the southwest edge thanks to the potential ladder escape . Black can, however, gain territory by making several weak blocks until it is fully connected. As can be seen in the diagram, Black gains territory with each of his three moves and remains next to move, holding onto the initiative.
Diagram 119
Black makes multiple intrusions to gain territory.
In Diagram 120, White plays c7, connecting her group to the southwest edge via Template A-3. Black gains territory by intruding into the template with b9. Unwilling to let such a large group be cut off from the edge, White is forced to save the connection with a8, giving the initiative back to Black.
Diagram 120
Black intrudes into Template A-3 to gain territory.
In Diagram 121, Black threatens to cut off from the southwest edge. In particular, if he follows up by playing at , a fourth-row ladder will ensue, which will cascade through and ultimately be escaped by . White is forced to respond at herself with c10, and Black has the initiative back along with some extra territory.
Diagram 121
Black threatens to cut off .
Diagram 122 shows this game at a later stage. We can see how Black makes use of the territory gained earlier (the stone ) to connect his central group to the northwest edge. He uses to create a Trapezoid with d6.
Diagram 123 shows another game. Black can’t block from the southwest edge (for example, if c9c8a9 then White can escape the ladder with ). Instead, Black gains territory by threatening to cut off with c10, forcing the response from White.
Diagram 123
In Diagram 124, White blocks from the northwest with l1. Black cannot block from the northeast edge because and can escape the second-row ladder that would result from m1. Black must instead start a second-row ladder along his northwest edge. But first Black gains some territory by threatening to cut off with l3m1. Then he starts the ladder with k2.
Diagram 124
Diagram 125 shows the remainder of the game after Black breaks the ladder. is connected to the bottom edge. As Diagram 126 shows, is also connected to Black’s large central group and therefore to the northwest edge. White resigns as a result. Notice how the territory gained in Diagram 124 allows to connect to the central group. A long line of stones that ends in a “hook” shape, such as – in Diagram 124, can be quite useful.
Diagram 125
Diagram 126
Situations such as these, where a series of blocks towards the obtuse corner lead into a half bottleneck, come up frequently. It’s a common rookie mistake to proceed straight to the ladder without first securing the territory as in Diagram 127a. You should almost always take the territory, as in Diagram 127b. The only exception is if you think your prospects are better blocking the second-row ladder along the southwest edge.
(a)
(b)
Diagram 127
It’s almost always better to take the territory in this situation, as in (b).
Sometimes a player will engage in a sequence of moves in order to gain territory. Diagram 128 shows an example. Black’s first move doesn’t gain anything of significance, but after White saves the connection of to the northeast edge, Black is able to gain territory with the two follow-up moves l8 and j7. Note how is connected to the southeast edge.
Have a look at Diagram 129, which looks superficially very similar to Diagram 117. Black intrudes and gains two hexes of territory. Or does he? This is an example where our simple definition of territory (the empty hexes adjacent to a player’s stones) doesn’t hold up. Black’s stone is completely useless because it can never take part in a connection which wouldn’t have otherwise been made. The two hexes of “territory” in Diagram 129 are touching, so the black stone could never connect two groups of stones which weren’t already connected on their own. So we have to modify our definition: territory consists of the empty hexes adjacent to any stones that could hypothetically take part in a connection. If a stone could never take part in a connection then it gives no territory. For this reason the black stone in Diagram 129 along with the two hexes of non-territory are often referred to as the useless triangle. With this new understanding of territory we can see that Black should never intrude at in Diagram 129 or similar situations. This not only gains no territory, it gives some to White. The only intrusion point Black should consider is , which would gain territory.
Diagram 129
Black gains territory?
In fact, since the stone can never affect Black, it can never affect White either. We can even mentally substitute the useless black stone for a white one and obtain an equivalent board position (Diagram 130). There is no situation in which Diagram 130a results in a different outcome from Diagram 130b. In fact, it is often useful to mentally make this substitution when analyzing board positions, so as to avoid being distracted by the useless black stone.
(a)
(b)
Diagram 130
These positions are equivalent.
Diagram 131 shows another example of a useless stone. Again playing at gains no territory because there is no potential connection which wouldn’t have gone through the black stone already present. Black can only gain territory here by playing at .
Diagram 131
Black gains no territory.
Sometimes, however, you will create a useless stone on the way to gaining territory. Diagram 132 shows one such situation. By intruding into the Span at the bottom, Black can later make two intrusions after White saves the connection. The initial stone ends up being useless but it allows for two useful stones to be placed. Attempting to intrude at one of those two points without the initial move would have allowed White to save the connection without offering up the second intrusion.
Diagram 132
Sometimes useless stones can help you get useful ones.
Having looked at using weak blocks to gain territory, we should now consider how to respond to such intrusions. If we opt to save the connection, then we are making a strong block. But there’s generally more than one way to save a connection. So how do we decide amongst them? There are two primary considerations: how much more territory are you offering, and how much can you expand your own territory by. We will consider these in the following two sections.
On the other hand, you may have the option of not responding at all and making a move elsewhere on the board. Done well, this can regain the initiative. We will consider moves like this in a later section.
Let’s start with the simpler of the two. If possible, you want to connect in a way that doesn’t offer your opponent an opportunity to create more territory, or offers as little, or as useless, territory as possible. For example, in Diagram 133, Black can bridge his two groups of stones by playing at any of the labeled hexes. By playing at , Black offers White no opportunities to create territory, because if White intrudes into the resulting bridge she ends up with a useless triangle. Unless Black wants the territory offered by playing at or , he should play here. Under no circumstances should Black play at or : these moves give White the opportunity to steal territory but gain none for Black.
Diagram 133
Black offers no territory.
In Diagram 134, White has moved towards her northeast edge. Black wants to block White from connecting via and leave her with just the third-row ladder heading southeast. The natural way to achieve this appears to be l3, to which White could respond by intruding into the bridge with k3l2, gaining a small measure of territory (the hex ).
Diagram 134
Black to respond.
Instead Black responds at m2 (Diagram 135), connecting by the double threat of playing at either or . Now there is no intrusion point by which White could steal territory (with connected to the edge, l3m3 would not gain any useful territory for White).
The difference may appear slight, but there’s no reason to ever offer even the slightest advantage to your opponent if you don’t have to. Minor advantages can add up.
With this concept in mind, have a look back at Diagram 119. Did White need to give up as much territory as she did? Can you find a better way that White could have played this situation?
Diagram 135
Black offers no territory.
Of course you may not always be able to offer no territory at all. In that case, you want to offer the lowest quality (least useful) territory you can. In Diagram 136a White responds to Black’s intrusion at b3 with b4. Black could follow up with c4c3, but there’s little value to the resulting territory (White still keeps her second-row ladder escape for example). On the other hand, had White responded at c4 (Diagram 136b), using Template A-3, Black could follow up with b6a5. This territory is more valuable. Notice for example how White is left without a second-row ladder escape.
The other consideration when responding to an intrusion is expanding your own territory. In particular, when comparing your various responses, you want to look for responses that strengthen your connection in the other direction. For example, say as Black you have a connection to the southeast edge, and your opponent threatens it. You want to save the connection in such a way as to strengthen your connection to the northwest edge.
In Diagram 137, White is threatening to cut off Black’s fourth-row stone (which was connected with Template A-4) from the bottom edge. Two possible responses which save Black’s connection are bridging directly towards the edge (Diagram 138a) or by playing an adjacent stone to form Template C-4 (Diagram 138b). While the first connection looks, in a sense, “stronger,” there are no bonus points for stronger connections. Your only goal is to connect somehow. Both responses retain the connection to the bottom edge, so they need to be evaluated on how well they help connect in the other direction. Since the second approach gives a stronger connection to the top edge, it is, absent other considerations, the better response. This kind of response is an example of minmaxing—retaining a minimal connection in one direction in order to maximize the connection in the other direction.
(a)
Diagram 137
White threatens to cut off Black’s fourth-row stone from the bottom edge.
(a)
(b)
Diagram 138
Possible responses to White’s threat.
In Diagram 139, Black plays b8, intruding into the Template A-3 connecting (and it’s associated group) to the southwest, taking some territory for himself. Rather than save the template directly by playing at (or similar), which would gain her no territory, White responds with the minmaxing move e2. This keeps her group connected with a double threat (she can still play at , as well as at with Template A-3) while extending her territory. The c4–b5 group, which could have potentially helped Black by escaping a second-row ladder, is now also blocked.
Diagram 139
White responds with a minmaxing move.
In Diagram 140, with c3, White threatens to cut off from the northwest edge (if Black fails to save the connection, she can play e4d4e3). White hopes Black reconnects with e4, after which she can play h6b10 and then use to escape the third-row ladder.
Diagram 140
Black responds with a minmaxing move.
Instead, Black responds with the minmaxing play c4. Not only does this save the connection (e.g. e4d3d2f2), it blocks White’s third-row ladder escape. She attempts another intrusion to recreate her escape with c5 (this would cascade a third-row ladder to a second-row ladder, which would be escaped by ) but Black again responds with a minmaxing move, b4. This blocks the cascading ladder while still maintaining the connection to the northwest edge (for example, e4d4b3e2e1f2).
In Diagram 141a White has intruded into the Template A-3 connecting to the northwest edge. Rather than saving the connection directly by playing at (or similar), Black plays b5 (Diagram 141b). If White attempts to block from the edge with e1 Black can respond with the ladder-escape fork b2, so Black has saved the connection. But in playing so far out Black has extended his territory, in addition to cutting off White’s second-row ladder escape .
(a)
(b)
Diagram 141
Black plays a minmaxing reply.
Diagram 142 shows what White intends to happen after her threat of i4. If Black responds to the threat by connecting directly to the northwest edge with j3, then White can play out the sequence shown, forming a Crescent with and , while counting on to help connect to the northeast edge.
Diagram 142
What White intends to happen.
Instead, Black responds with the minmaxing move l5 (Diagram 143). He can use this stone to later form a Crescent with ( can escape the second-row ladder) and after k7 White is unable to connect to the northeast.
Diagram 143
Black responds with a minmaxing move instead, foiling White.
Due to the threat of minmaxing, you have to be careful with your weak blocks. Make sure your opponent doesn’t have a strong, minmaxing response available before you intrude into their strong connections. For example, in Diagram 144 White’s attempt to invade Black’s Template A-3 allows him to respond so as to nullify her territory and expand his own.
(a)
(b)
Diagram 144
Bad intrusion by White gains no territory while giving Black the opportunity to make a minmaxing play.
Particularly inadvisable is intruding into the center of a strong connection. Intruding into the center often gains you little territory and gives your opponent an opportunity to respond with a minmaxing move. As a rule of thumb, the best intrusions are along the periphery of a connecting area.
One of the best responses to an intrusion can be an intrusion of your own in another area of the board, if you can find one. The primary goal of ignoring an intrusion in order to make your own is to regain the initiative. You have to be willing to sacrifice the threatened connection in order to do so. It’s important that the threatened connection is not vital for you—your opponent’s intrusion into your connection has reduced it to a 1-move connection, and by moving elsewhere you’re leaving them the opportunity to sever it entirely. On the other hand, if it isn’t vital, severing will require your opponent to play a second move in the same area, handing the initiative back to you again.
The early game usually provides more options to move on and ignore intrusions. In the early game players are less committed to the connections they have—the open board provides numerous opportunities for alternative connections. In the late game the situation is much less flexible and opportunities to ignore intrusions are rare.
Diagram 145 is a typical early game example. White’s h12 intrudes into the connection of to its respective edge (specifically, Template A-4). Rather than save the connection, Black decides to block from its respective edge with b10.
Diagram 145
Rather than save the connection, Black makes his own block.
In Diagram 146 we have another example. With i6, White threatens to sever two groups by following up at . Rather than save the connection, Black plays on the other side of the board with b10, attacking the connection between and the southwest edge.
What motivated Black to move on here? There are a few features of the position that likely factored into the decision. Black isn’t doing so well on the northeast edge. Although playing would connect the i4–i5 group to the j6–l5 group, this may not be particularly valuable. The i4–i5 group is clearly only weakly connected to the northwest edge, since White can block at i3, and the j6–l5 group is also only weakly connected to Black’s large group on the right which connects to the southeast (for example, if j5, then k6j7j8i8i9). When you’re losing in an area it’s a good idea to try to open up play in another area, so it makes sense for Black to leave this situation and take the first move in the relatively open area along the southwest edge. Also note that playing at gains virtually no usable territory for either player, since it is buried behind other stones. This gives Black even less reason to play here.
In fact, Black’s is too threatening for White to ignore. She will almost certainly have to respond in the area, or else Black could play a move like c8, which would leave White in a dire situation (in the actual game, White responded at c7). So Black isn’t giving up entirely on playing in the future. As the new fight in the lower corner evolves and interacts with the rest of the position, will be left up for grabs as a potential move for either player.
Diagram 146
Intruding elsewhere.
In Diagram 147, Black intrudes into the connection of to the southwest edge with c10. If Black follows up at then the connection will be severed. White could attempt to save the connection with d8, but after g8g9f9e11e10 (Template C-4) things don’t look so good. In other words, also threatens to separate from . Rather than attempt to save the connection here, White instead opens up play in a new area with i4. is connected to the northeast edge thanks to the ladder escapes that and provide, and it stands between and the northwest edge. may be useful later.
Let’s summarize what we’ve learned so far. Weak blocks, intruding into the opponent’s strong connections, can be used by a player to gain territory. Unless the player who’s been weak blocked can find a weak block of their own, they’ll have to respond to the weak block with a strong block to save the threatened connection. There are two primary considerations when strong blocking. The first is how much territory (intrusion points) you are offering by your reply. The second it to maximize your territory in the other direction (minmaxing). These two considerations are often at odds—minmaxing plays often leave more intrusion points, and plays that leave few intrusion points rarely give you much territory. The best strong block therefore requires consideration of the overall picture.
Another potential response to a weak block is to find another area of the board where you can make your own weak block. Say you have a group of stones in the center of the board, with one strong connection to an edge and another 2-move connection to that same edge. If your opponent plays a weak block into your strong connection (making it a 1-move connection), you may be able to play a weak block into the 2-move connection, upgrading it from 2-move to 1-move. Now you have two 1-move connections from your central group towards that edge. These two 1-move connections make a double threat (since either could be upgraded to a strong connection with a single move). Whether the strong (minmaxing) reply or weak block is better will depend on which does a better job of helping the central group connect to the other edge of the board.
When making a weak block, always consider your opponent’s best response. Make sure you get more out of it than he will! If the potential minmaxing reply is too strong then it may be better to not make the weak block at all. A good rule of thumb is that a weak block should be on the periphery of a connection’s area. Weak blocks played in the center of a connecting area gain little to no territory and permit strong minmaxing replies.
Generally, if an opponent responds to a weak block with a strong block to save the connection, you can assume he is unable to find a better response elsewhere on the board. In this situation you should attempt to plunder the area for all the territory you can, playing weak blocks until there is nothing more to get. Diagram 119 shows an example of repeatedly attacking an area to gain territory. If your opponent had a better reply in another area he would have responded with it after the first weak block.
In Diagram 148 White cannot stop the black stone with just adjacent blocks (Diagram 149a) nor with near blocks (Diagram 149b). She must either combine the two simple blocks, or play a far block. All of these choices yield ladders (Diagram 150). Since ladders are generally to the advantage of the attacking player (Black in this case), Black’s e9 stone is threatening. I call stones like this attacking.
Diagram 148
(a) White uses only adjacent blocks.
(b) White uses only near blocks.
Diagram 149
Diagram 150
White can hold Black to a ladder.
Diagram 151 shows attacking stones on the third, fourth and fifth rows, highlighting the area that needs to be free for a stone to be an attacking stone. The idea here is not that attacking stones are strongly connected to the edge (except on the third row), but rather that they tend to be more difficult to defend against. The further back the attacking stone lies the weaker it is. An attacking stone on the third row corresponds to Template A-3 and is therefore connected to the edge, while an attacking stone on the sixth row would be considerably weaker.
Diagram 151
Third-, fourth- and fifth-row attacking stones.
Attacking stones are commonly played in the early game, often along the short diagonal from the third to sixth rows, with the fourth and fifth rows being the most popular. The short diagonal is popular because it places the attacking stone as close to the far edge as possible. However they don’t necessarily have to be on the short diagonal. For example in Diagram 152 Black plays off the short diagonal due to the presence of the white stone.
Diagram 152
Black must place the stone off the short diagonal due to .
Friendly and opposing stones in the area can affect the status of a potential attacking stone. In Diagram 153 the black stone on e9 is no longer attacking because the presence of means that White can easily block Black using just near blocks. In Diagram 154 the black stone on d9, which would not ordinarily be attacking just behind the short diagonal, cannot be stopped with adjacent blocks thanks to .
Diagram 153
White uses only near blocks.
Diagram 154
Black can play an attacking stone further west due to .
Somewhat counterintuitively, it’s often better to build out groups parallel to your own edge, rather than building directly towards your edge. The former builds a broad front (Diagram 155a), while the latter results in a narrow front (Diagram 155b). Broad fronts extend potential connections in both directions, and are much more difficult to block than narrow fronts. Note the overlap of this concept with minmaxing, where the goal is also to expand connections in both directions at the same time.
(a) Broad front.
(b) Narrow front.
Diagram 155
Of course Diagram 155 is merely illustrative—there aren’t even any white stones present! A more realistic example is given in Diagram 156. Here Black connects together his central group of stones into a broad front, extending potential connections in both directions, around White’s blocking stones.
Diagram 156
Black connects his central stones together to form a broad front.
In Diagram 157a, White has a ladder approaching the obtuse corner. Can she use to escape this ladder? She can’t, but as Diagram 157b shows, she can get a connected group extending through away from the edge. This is called a deflection. While this may not be connected back to the ladder, it’s strong in its own right, extending a connection from the edge far out into the field of play.
(a)
(b)
Diagram 157
The key aspect of deflecting is that, while a player may be unable to force a connection between a group and an edge (or some other group), they are able to get a connection from that edge extending in another direction. Deflections are an important strategic concept. In Diagram 158, White manages to successfully block (already connected to the northwest) from the southeast edge but loses anyway as Black deflects and connects to the northwest by a different route (note that is connected to the southeast because White can at best hold Black to a second-row ladder which is escaped by ). This is very much a case of winning the battle while losing the war!
Diagram 158
Deflections vary considerably, depending on the overall arrangement of the board and the area into which a player can deflect. Naturally the strength of a potential deflection can change as the game progresses. Be mindful of possible deflections and how play might develop as a result of them. You might see an opportunity to block your opponent’s, or strengthen your own.
Let’s look at a few more examples. In Diagram 159a, Black has a third-row ladder approaching . As we saw in the last chapter, White can prevent from escaping the ladder. But Black can still deflect around the white stone as Diagram 159b shows. Note that is connected to the edge, because if j11, then l10 is connected by Template D-4.
(a)
(b)
Diagram 159
In Diagram 160 White wants to block (which is connected to the northwest edge) from connecting to and on to the southeast edge. Black first gains some territory with h7 and h8. Then he uses a sequence of forcing moves, each threatening to connect back to the group, to deflect from to the opposing edge, winning the game.
Diagram 160
In Diagram 161, is connected to the southeast. White must block it from the northwest. She succeeds in doing so but Black is able to deflect to . From here he threatens either e8 or g6h5h6i5h7. Either leaves him in a strong position.
Diagram 161
In Diagram 162, is connected to , which in turn is connected to the southeast edge. Clearly, White must block it from the northwest edge. Mindful of the threat that poses in the corner, she plays i2 (Diagram 163). Black replies with h2h3f4. is connected to the northwest edge; if g3 then f3e3g2. White is forced to block with g4. Black then works his way down the board with f5g5e7, connecting to with a Trapezoid on the last move. Although this is technically also connected to , we still consider this a deflection since Black’s connection does not run through .
Normally with every move of the game you want to make the best move possible. The first move is different. Due to the swap rule, if you make a move that is too good your opponent will likely swap it, and you will be forced to play against it. On the other hand, if your opening move is too weak your opponent will likely not swap, and you’ll be stuck with it. So your goal with the first move is actually to play the most average move possible. This section is concerned with finding these average opening moves. In doing so we will also help shed some light on the choice facing the second player—whether or not to swap the move of the first player.
Without the swap rule, every opening move is either a win or a loss for Black. When I say that an opening move is a “win for Black,” what I mean is that Black wins with that opening so long as he doesn’t make any mistakes in the subsequent moves (again, this is without the swap rule). On the other hand, if an opening move is a “loss for Black,” it means that White will win so long as she doesn’t make any mistakes.
As an example, let’s look at the 2×2 board (Diagram 164). Although there are four empty hexes, due to the symmetry of the board the first player really only has two openings: a1 (equivalent to b2) or a2 (equivalent to b1). Without the swap rule, Black can win by opening with a2 (Diagram 165a). This move is connected to the southeast edge of the board, and is connected to the northwest via Template A-2. As long as he responds correctly to White’s reply, Black will win the game on his second move. Hence we can call a2 a winning opening for Black on the 2×2 board. Now suppose instead Black had opened with a1 (Diagram 165b). White could then respond with a2, connecting her two sides of the board. Therefore we can say that a2 is a losing opening for Black on the 2×2 board. Since these are the only two openings, we have now “solved” the 2×2 board.
Diagram 164
The 2×2 board.
(a) Win for Black.
(b) Loss for Black.
Diagram 165
The two possible openings on the 2×2 board.
Larger boards can be solved in this way, either by hand or with a computer, to categorize every opening as either winning or losing (remember, there are no ties or games that don’t end in Hex, so all openings must fall into one of these categories). As of the time of writing, computers have solved all sizes up to the 9×9 board (Diagram 166). Although the 13×13 board hasn’t been solved (and probably won’t be for a long time), we can look at the trends in these boards to get some insight into the larger game boards.
Diagram 166
Winning opening moves for Black on boards up to 9×9. All other moves are losing for Black. See D.3 Solved openings for references on these solutions.
These diagrams tell us who wins with perfect play. But all the winning openings aren’t created equal (nor are all the losing openings). For example, on the 9×9 board we can see that Black can win after opening either at e5 (the center) or at a6 (along the edge). But humans don’t play flawlessly. Between two human players, Black would likely have a much more difficult time with the edge opening. With the edge opening his margin for error is much smaller than with the center opening. We can reasonably guess that the fairest openings—those which give a pretty much equal chance at winning to either player—are likely those along the boundary separating Black’s wins from Black’s losses. This boundary is shown in Diagram 167 for the 8×8 and 9×9 boards, and the hexes adjacent to it are shaded. These shaded hexes are good candidates for opening moves. The unshaded hexes in the center should be swapped because they are too strong, while the unshaded hexes near the edges are too weak and should not be swapped.
Diagram 167
Borderline openings. These are good candidates for opening moves under the swap rule.
Of course we aren’t playing 9×9 hex, which is too simple a game to be interesting. 13×13 is too large to be solved by computers anytime soon. Nonetheless we can look at the general patterns from the smaller boards as well as the games of stronger players to determine what moves can serve as suitable openings.
The entire short diagonal is winning on every board in Diagram 166. It also appears that the third-to-last hex on the a-file loses on the larger boards (for example, a7 on the 9×9 board). The region of potential openings outlined in Diagram 167 is confined to the edge of the board, hugging the white edges more closely than the black edges. In general, most of the a-file serves as a good opener, although a1 is too weak. These trends seem to hold on the 13×13 board, where openings on the a-file are by far the most popular amongst strong players. Interestingly however, these same players seem to ignore opening along Black’s edge, except near the corners. Diagram 168 shows the most common openings plays on the 13×13 board. As you can see, the majority are played along the edge on the a-file (note that for every opening move, there’s an equivalent move on the other side of the board; for simplicity I’ll only talk about the moves on the west side of the board). I’ve also indicated here my best guess as to the border that separates the winning openings from the losing ones. I must stress that this is only a guess, although it is somewhat informed from data on opening move winning percentages.
Although the openings indicated in Diagram 168 are the most common, I would advise against playing a4 or a5, as these openings are too strong. If your opponent plays these openings, I would recommend swapping them. Of the “unpopular” a-file openings, a1, a2 and a11 are too weak, while a12 is too strong.
Diagram 168
Most popular openings on the 13×13 board. I’ve also indicated my best guess as to the border that separates the winning from losing openings, with the winning openings on the interior and losing openings on the exterior (note that a11/m3 are on the losing side). This border is highly speculative; this board size has not been solved and other players may have differing opinions.
What makes the a-file good for openings? One way is to think in terms of strengths and weaknesses, keeping in mind that fair moves will have equal proportions of both. The weakness of these moves is their distance from the central area of the board, which is where most of the action occurs. Any influence they have will be at a distance. And they leave the far side of the board almost completely unaffected. Whether the move is swapped or not, the next player will have a wide open area in which to make a strong move.
The strength of these moves can be understood in the context of the 4.12 Attacking stones section above. A stone on the opponent’s edge can force them to play closer to that edge if they need a threatening stone of their own. And that means they must play further from the center. To make this concrete, have a look at Diagram 169, which shows three examples of white attacking stones. On the open edge, White can play attacking stones further back and closer to the center of the board (I should reiterate here that attacking stones are not guaranteed connections), while on the southwest edge, the stone forces play closer to that edge and further from the center.
Diagram 169
Three white attacking stones. Notice how the presence of the black stone forces White to play much closer to that edge, and therefore further from the center.
You could also think in terms of templates and ladders. “Fitting” an edge template into an edge with an opponent’s stone present is more difficult than on an open edge. And a ladder on your edge will be blocked if it hits an opposing stone. No matter how you approach it, an opposing stone along a player’s edge forces them to play closer to threaten or make connections to it. And that’s the strength of the a-file openings, especially the central ones (a4 through a10). A key consideration in the early game after such an opening is that White will need to strengthen her edge, and Black must attempt to exploit that (these roles are reversed if the move is swapped).
The more peripheral a-file openings, as well as the ones off the a-file, will have other effects. While they attack White’s edge less, they strengthen Black’s. a3 and a13 can be used as a second-row ladder escape, while c2 and c12 can serve as both second- and third-row ladder escapes.
It’s interesting to note that very few players open along the black edge. I’m not entirely sure why that is. An opening right on the edge such as g1 would be very weak, while an opening further out (say on g5) would be too strong. This would seem to imply that somewhere in between these two points there would be a fair opening. Perhaps the line that separates weak and strong openings is very “sharp” along the black edge. By this I mean that as you move out from the black edge you at some point abruptly jump from a very weak opening to a very strong opening—with no fair opening in between. Or perhaps people simply open on the a-file out of habit, and there are in fact very good openings along the black edge just waiting to be discovered! If you’re feeling adventurous give it a shot. My best guess would be that moves on the third row might be good candidates, but I want to stress that’s just a guess.
a4swapi5g9j9j7e9g6 (Diagram 170): Black opens with a4 which is quite strong. White swaps. Black’s next three moves are in the north, east and south corners. White’s , and are probably best seen as blocking moves. At this point Black has a very strong southeast edge. Were Black to make a move like h7, they could threaten to connect to either of or . But such move would do little to strengthen their connection towards the northwest edge.
e6: Now Black has multiple options open: either play at to connect to ; or play at to connect to one of or . And has done much more to strengthen the connection to the northwest edge than playing at would have. Viewed another way, this move is a weak block of from the southwest edge in order to gain territory.
g3g5: White attempts to block . Black wants to both strengthen the connection of to the northwest and to . If Black can turn this into a single group connected to the northwest then there will be two paths towards the southeast and the game will nearly be won.
j2f3f4d5e4d4e3c3: White tries to block Black from the northwest to no avail. From Black can either play or ladder to the right and play a fork at .
Diagram 170
e7h7 (Diagram 171): White blocks one of Black’s connections to the southeast so Black responds with the other one. can connect to either or .
h5j3k2i3i2h3h2b2: White fails to block Black from the northwest edge.
f8i8i9j8i11h10g11h11g13j12: White makes a final attempt to block Black from the southeast edge, but Black is able to squeeze through and use a ladder-escape fork. White resigns.
k10k12l11: Black moves to take the east corner. is connected to the edge with Template A-4 and can escape ladders on the second and third rows. White intrudes to block this escape and gain their own second-row escape. Black responds to save their own ladder escapes (can you see how?) while nullifying White’s escape. This sort of fighting in the corner with multiple invasions and responses occurs commonly in the opening, and will be covered in greater depth in Chapter 7: Advanced strategy.
e2: Here we have an example of “moving on.” Rather than respond directly to Black, White decides to strengthen their position in the west corner. This move also decreases Black’s space along the northwest edge.
i5j4i4: Needing more influence along this edge, Black moves into the north corner. White intrudes to gain a foothold in this corner, forcing a response.
l12j12: White returns to the situation in the east corner. This move re-establishes their second-row ladder escape while blocking Black’s escapes. Black’s response saves their own ladder escapes, but doesn’t block White’s, leaving White with the initiative.
Diagram 172
j3i3 (Diagram 173): White intrudes again to gain more territory along the northeast edge. White now has Template C-4 here.
g9: White steps far back to block Black’s from the southeast.
f9: Black responds by blocking from the southwest with an adjacent block.
g8f8g7f7g6f6g5f5g4f4: A ladder ensues. Black avoids yielding due to the strength of the – pair.
g2: White breaks here, knowing that can’t escape a third-row ladder. Before we look at Black’s next move, let’s consider some sequences. If Black attempts to play out the third-row ladder it could go something like this: g3h2h3j1j2k1l2k2k4j5k5j7j6i6i7i8. Clearly this isn’t good for Black. On the other hand, if Black allows White to connect to , say with h3g3, then it will be difficult to block White’s large central group from the northeast edge: j6i6j5j7i7i8.
j7: Black attempts to meet both of these threats, by playing in an area where both sequences overlap.
Diagram 173
h2 (Diagram 174): White connects to . Note the choice to connect with a bridge rather than directly with g3. This gives White better territory (you could also think of this as invading the Template A-3 formed by ). Neither intrusion point in this bridge can gain Black usable territory, so this extra territory is free. This is smart hex. Always consider the different approaches to connecting groups and the tradeoffs involved.
i6: With the central group now connected to , Black tries to block it from the northeast edge.
j1l2j2k4j6i7j5l5k6l6k7l7: White pushes through to get a third-row ladder. Note how important is to this sequence. Had White played g3 this would not have worked as it did.
k9: This stone can at best be held to a second-row ladder which would be escaped by the group: m8l9m9l10m10k11. Therefore, it’s connected to the edge.
k8j9j8h10: White uses a sequence of forcing moves to connect back to the central group.
f2f3d4e3d2d3b4c2: Black tries to block from the southwest edge but White uses and to push through and connect. Black resigns.