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.
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.
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.
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.
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.
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.
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.
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 weak block.
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.
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 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.
Intruding to gain territory.
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.
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.
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.
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.
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 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.
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.
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.
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.
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 .
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.
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.
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 ).
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?
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.
White threatens to cut off Black’s fourth-row stone from the bottom edge.
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.
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.
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 .
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.
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.
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.
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 it’s respective edge with b10.
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.