Advanced (strategy guide)

Advanced edge templates

Template IVc

This is a two-piece template and is useful for squeezing edge connections and ladder escapes into relatively small regions. Also, many players are unaware of it. If the opponent intrudes on the template with any move other than those marked by '+', Red two-chains to template II by playing at one of the hexes marked '*'.

Solution to intrusion at b4

If Blue (the horizontal player) intrudes at b4, then Red responds with d3 — d3 is connected to the edge via template II and threatens a direct connection via d2. So d2 by Blue is forced. Red then two-chains to b3 threatening to connect either via a4 or c3.

Solution to intrusion at c2

If Blue intrudes at c2, then Red responds with b3; b3 is connected to the edge via template II and threatens a direct connectione via b2. So b2 by Blue is forced. Then Red plays at d2. Red threatens to extend d2 to template II at c3 and d3, and threatens to two-chain from d2 to the edge at c4. The only hex that is in the overlap of all these threats is c4 thus, Blue is forced to play at c4. Then Red plays at c3 completing the connection.

Template Va

The hexes marked '*' are not relevant to the template.

If Blue intrudes in the template at any hex besides the three marked '+', Red makes a move that reduces the situation to a closer template.

Solution to the intrusion at f4

The key move is the response d5. Yielding

Red is threatening to play at the marked hex which would complete the connection of the g2 piece to the bottom. Blue must block by a playing at some hex between Red's two pieces. Red then plays h3 forcing Blue to block at h4 yielding

Red has forced the most common pre-ladder formation. Red can get a second row ladder by squeezing through at g4 (Blue blocks at f6). Red's initial key d5 piece acts as a ladder escape which completes the connection. The final position is

Solution to the intrusion at d6

Red's best response is to two-chain to h3. To stop the threatened immediate connection, Blue must block at h4 or play the forcing move g3 in the two-chain. The first play is defeated by the forcing sequence f5, g3, f3, f4, e4 yielding

Red is now threatening to connect to the bottom in two non-overlapping ways, by playing e5 or by two-chaining to c5. Blue cannot stop both threats with a single move. The other play, g3 (after Red's h3) is defeated by the forcing sequence h2, h4, f5, g4, f3, f4, e4 yielding

Red is again threatening to connect to the bottom in the same two non-overlapping ways: by playing at e5 or two-chaining to c5. Blue cannot stop both threats with a single move.

Solution to the intrusion at f6

Red's best response is play e5 which is connected to the bottom and forms a loose connection with the g2 piece. To stop the immediate connection, Blue must play in the middle of the loose connection at one of the hexes marked "+" in the following diagram.

The move f3 is defeated by the forcing sequence g3, f4, g4 yielding

Red is threatening to connect to the bottom in two non-overlapping ways: by playing at f5 or two-chaining to h5. The alternative response, f4, is defeated by the forcing sequence e3, d6, e6, e4, d4 yielding

Red is threatening to connect to the bottom in two non-overlapping ways, by playing at d5 or c5.

Note that template Va occurs in a mirror-image form (in the mirror image form, the three hexes on the 5th row (from the bottom) are shifted over 1 hex to the G, H, and I columns). It may seem that this template is very strong because it reaches 5 rows into the board but it rarely occurs because of the huge size of the template; the template requires 31 empty hexes and 10 hexes along an edge — the entire edge on the 10x10 board!

Furthermore, the large perimeter makes it more vulnerable to encroaching adjacent plays and forcing moves. Additionally, template area surrounds the 5th row piece on both "shoulders" so that non-overlapping plays from the 5th row piece can occur in only two directions.

Template Vb

The hexes marked '*' are not relevant to the template.

If the horizontal player Blue intrudes in the template at any hex besides the three marked '+', Red makes a move that reduces the situation to a closer template.

Solution to the intrusion at f3

There are several solutions but the simplest is to respond with g3. Blue's only play to stop the immediate connection is f5. Then Red plays e4 yielding

The e4 piece is connected to the bottom via a 3rd row template and e4 is connected to the other group of red pieces through e3 and f4. Thus, the connection is complete.

Solution to the intrusion at e5

Red's best response is g4. This piece is connected to the bottom via a 3rd row template and hence Blue must block at g3. Red then plays e4 yielding

The e4 piece threatens to connect to the bottom in two non-overlapping ways, to d5 and to g4 (through f4). Hence the connection is unstoppable.

Unlike template Va, this template is not a rare occurrence and many hex players are not familiar with it.

Advanced templates as ladder escapes

Templates IVc and Vb are valid escapes for row 2, row 3, and row 4 ladders. Template Va is not a valid ladder escape.

Exception: Template Vb is not valid for 3rd and 4th row ladders coming from the right side in the above diagram if the Horizontal player has a piece at h3. For the horizontal player to defeat the 3rd row ladder in this case, connecting to h3 must provide a strong threat that the vertical player needs to respond to.

Note: The unique way to win with template Vb and a 2nd row ladder is as follows. As soon as your head ladder piece intrudes on the template, your very next move must be to two-chain up to the 3rd row (this is true no matter which side of the template you are entering from). Then you break off the ladder (this piece will be connected to the edge via a smaller edge template).

The minimax principle

(See also the page Minimax)

Suppose you have multiple ways of establishing/maintaining a connection to an edge. A move that maintains as strong a connection as possible is not preferable to other connection moves because you only need to get some connection; you don't win extra points by connecting more strongly.

In fact it is generally better to play a move that maintains as weak a connection as possible; the reason being that such a piece may help you extend the connection towards the opposite edge. This principle is sometimes called "mini-maxing."

The idea behind the term is that you are playing a move that maintains a minimal connectivity in one direction while building up (i.e. maximizing) your strength in the other direction. I'll illustrate this with a couple of positions from my games. (Note that this principle applies equally well when establishing/maintaining a connection to a group of pieces.)

My opponent, Blue played the minimax move f4. This move maintains a minimal strength connection to the left while building up strength to the right; in fact the f4-f5 group is almost connected to the right edge via template Vb. I responded with my own minimax move d5 (d6 is the other minimax option) yielding the following position.

d5 maintains a minimal strength connection to the bottom while maximizing my strength to the top. (d6 would have maintained a minimal strength connection to the top while maximizing my strength to the bottom.) A move that is even stronger towards the top, such as d4, would be a mistake. My opponent could then block at the bottom with c7, which is connected to the left edge via a 3rd row template and which threatens to link up with the central group. If I try to stop the connection to the central group with e6, my opponent responds with d5 yielding the following position.

d5 is connected to the central group via a 2-chain and the combined threats c5 and d6 guarantee a connection to the left edge (a7 is defeated by c5, b5, b6, a6, b7, a8, b9). I would be in dire straits as the central pair f4-f5 is almost connected to the right edge.

Now back to the game; after my minimax move d5, I can safely meet c7 with e6. Yielding

In fact, the c7, e6 sequence occurred in the actual game. I eventually won after a close hard fought battle.

In this position, I was the vertical player and was expecting f6 to which h4 would give me an excellent position (with best play, this position would in fact be winning though this is not obvious). Instead my opponent played the excellent minimax move f4. This move fights in both directions and is in fact a killer move. I can't block the f4-f5 pair from the right due to the forking ladder escape at h9. Thus, I must meekly submit to the forcing sequence f6, e7, e6, d5 yielding

The game is over. The f4-f5 pair is connected to d5 which in turn threatens to connect to left in two non-overlapping ways, c5 (a 3rd row template) and d6, hence the pair is connected to the left. If I try to block at the right, the best I can do is yield a ladder (e.g. h4, h3, j2, i3 and H has a second row ladder) and then the forking ladder escape at h9 wins the game.

In the next example, I am the horizontal player and it is my move.

Most hex players would probably connect to the left side with a7 (or b6 or b7). Despite its apparent necessity, this move actually loses (against best play). Instead I played the winning minimax move d3! By adding a second non-overlapping connection threat to the left, my group of pieces maintains a connection to the left. And despite its modest appearance, d3 also helps out on the right and in fact guarantees a winning connection from f5 to the right by defeating one of the main potential blocking plays.

E.g. suppose V tries to block the f5 piece from the right as follows. g5, g4, i3 (or h4), i2 and now I have a forced winning ladder down row 2 — h3, h2, g3, g2, f3, f2, e3, e2 completing the win.

This line clearly shows the usefullness of d3. If I hadn't played d3 (playing a7 instead, for instance), the vertical player could continue d3, d2, a4! and eventually winning with best play (considerable deep analysis is needed to show this).

Minimax moves are not always "parallel" moves. The principle of maintaining a minimal amount of connectivity in one direction while maximizing your strength in the opposite direction is more general than that. The final example from a game of mine illustrates non-parallel mini-max moves. I was the vertical player and opened with 1. a3 and my opponent responded with 1... e4 yielding the following position.

I played the minimax move 2. f5 yielding

By connecting as far away as possible from the top, I increase my strength towards the bottom. (i.e. I am maintaining a minimal strength connection to the top while maximizing my strength towards the bottom). Before playing such a move, I have to verify that my opponent can't stop me from reaching the top. I could meet the attempted block with 2...g4 or 2...h2 by getting a third row ladder (2...g4 3.f4 g2 4.f3, etc. or 2...h2 3.g3 g2 4.f3, etc.), laddering down to e3, and then playing b4 (how to play a third row to a3 is described in a later section). I would be happy with such a line. My opponent however played the excellent e3. This move takes away the ladder, hence forcing me to reconnect to the top, while at the same time increasing his strength to the left.

Here I played the minimax move g4. g4 has the potential to help block my opponent from going across the bottom of the board (e.g. Blue e7, Red f7, Blue f6, Red h5 and now g4 is helping out) or equivalently helps me to connect downwards on the right. I.e. g4 maintains a minimal strength connection towards the top while maximizing my strength towards the bottom. Note that a stronger move towards the top such as g3 does not have the same potential to help out towards the bottom. This potential may seem remote but in fact I would not have won the game without it! The rest of the game does not illustrate minimaxing but it is instructive nevertheless. The most important variation is as follows (there were two mistakes in the actual game which took the game out of the path it should have followed into a shorter less instructive branch).

3.	g4	h3	Again forcing me to reconnect to the top and hence, getting a free hex that could potentially help him connect to the right (h3 provides an escape for ladders up row I).
4.	g3	f8	As good of a block towards the bottom that there is.
5.	e8		Necessary.
5.	...	e9	Essential. Note that this move stops any ladders on row 9 coming from the right and using e8 as an escape.
6.	g7!		Excellent. Reconnecting e8 to the bottom with the d8 or d9 is defeated by e7. I can't stop e7 from connecting to the left because e4-e3 provides enough help (e.g. 6.d8 e7 7.c7 d6 8.b6 c6 9.c7 b5 10.c5 c4) nor could I stop e7 from connecting to the right — e7 is aided by f8, e9, and h3 (e.g. 6.d8 e7 7.f7 f6 8.h5 g6 9.h6 g7 10.h7 h8 11.g8 f10 12.g9 g10 13.i9 h9 14.j7 i7 and the h3 piece provides the ladder escape). 6.g7! may look strange and unconventional but it maintains a very slim advantage.

6.	...	f7	The toughest move to meet. Note that 6.... g8 loses quickly to 7.d8 e7 8.f7 f6 9.h5
7.	e7		I can't stop f7-f8-e9 from connecting to the right, so I block to the left in the only satisfactory way. I can't allow my opponent to connect from his f7 towards e3-e4.
7.	...	d9	An essential block.
8.	c8		c9 is no good. I can't allow my opponent to come up row D towards e3-e4. I need to force him as far away from e3-e4 as possible.
8.	...	b10
9.	a10	b9	I've forced my opponent into a second row ladder which is not sufficient because my a3 piece is just barely inside the Vb template.
10.	a9	b8
11.	a8	f2!	A nice idea. e4-e3-f2 is just strong enough for the second row ladder to work and at the same time, it threatens to cut off my main group of pieces from the top. This looks like it wins but there is a way out!

12.	i2!		A subtle improvement over simply reconnecting with g2 or h2 (both of which are losing). This maintains the connection and i2 interferes with the usefulness of h3. This may seem insignificant but it makes the difference between winning and losing! (In hex, the difference between winning and losing is often very slight)
13.	...	c6	Continuing with his plan to connect to the left which now works due to f2.
14.	h8!		The unique move that stops my opponent from connecting on the right. Note that 14.g8 is not satisfactory. 14.g8 g6 15.i5 h9. h9 is a forking ladder escape that guarantees a connection to the right.
14.	...	g6
15.	i5	h7	Note that h7 is threatens to connect to the horizontal player's main group in two distinct ways, through g8 and through h6, and thus is connected to his main group.
16.	g8		Not necessary but it doesn't hurt as the reply is forced.
16.	...	h6
17.	j6	i9
18.	i8	g10
19.	f10	h4
20.	j3		I now have an unbreakable chain on the right side.

Special situations, tricks, etc.

Reconnecting edge template IIIa after an intrusion

In this diagram, suppose you are Red and Blue has just played d3 intruding upon the third row template connecting your e2 to the bottom. Most hex players would reconnect with e3 without giving it much if any thought, yet there three distinct ways to reconnect and there is often a reason for preferring one over the other.

A second way for Red to reconnect is to play f2 — the hex f2 and the empty hexes g2,e3,f3,g3,d4,e4,f4, and g4 form edge template IIIa; hence f2 has an unbreakable connection to the bottom and f2 is connected to e2.

The potential advantage of reconnecting with f2 over e3 is that it is easier to connect other pieces to the the group e2-f2 than to the group e2-e3 (e.g. h1 is a two-chain away from f2 but is not a two-chain away from either e2 nor e3). The extra connection possibilities can make a critical difference. For example, consider the following position with Red to move.

Red can win by laddering 1. d7 d8 2. e7. Suppose instead Red plays 1.h5 intruding on the g6 edge template. If Blue reconnects with h6, then Red would have nothing else to do except play the winning line. So Blue reconnects with g7 making the win tougher. (Red could still win by d7, d8, e7, e9, f8, f9, h8! — a forking ladder escape which decides the issue).

Now suppose that Red again intrudes on the edge template with 2. h6. Now the game continues 2...g8 (again reconnecting by playing parallel to the edge) 3. h7 (persistent) h8, 4. d7 d8, 5. e7 e9! and now Blue has an unbreakable winning chain at the bottom. By reconnecting with the parallel moves instead of the direct reconnection, Blue's group had a new way to connect to the left and this extra possibility turned a defeat into a win.

So is it always better to reconnect with the parallel move? No!! Sometimes the parallel reconnection can lose the game while the simple direct connection wins! The potential weakness of the parallel reconnection is that your opponent might then be able to use a double threat to defeat the edge connection. For example, consider the following position with Red to move.

With best play Blue wins, so Red tries 1. h4. If Blue responds with the direct reconnection h5, then the win is assured and Red may as well resign. Suppose instead that Blue reconnects with 1... g6. Then Red can respond with 2.h7! — this forking ladder escape is a killer. Red now has two disjoint winning threats, laddering from d7 to h7 and play i5 (This double two-chain cutoff threat occurs in situations besides cutting off third row edge templates. It is well worth being familiar with this idea.). Blue cannot stop them both so Red wins.

But this doesn't exhaust the reconnection possibilities. There is a third way to reconnect; a way that most players don't seem to discover.

Again starting at the initial position in this section, Red's e2 piece is connected to the bottom via edge template IIIa and Blue intrudes upon it with d3. In addition to e3 and f2, Red can reconnect with the surprising f1!

Red is threatening to connect e2-f1 to the bottom with e3. If Blue tries to block this with e3, then Red can reconnect by playing g2. g2 is connected to the bottom via template IIIa (Blue's e9 piece is just outside of this template) and h3 connects to f1 via a two-chain.

But what if Blue blocks with e4 instead of e3? (note the e4 is within the g2 piece's edge template). Then Red can still reconnect by playing as follows. 1. e3 d4 (forced) 2. g3 f3 (forced again) 3.g2 ending up with the following position.

How does this method compare to the previous two? Compared to the parallel reconnection, it is quite a bit more susceptible to forking plays and plays that encroach upon the increased area that is needed to reconnect, but by playing away from the edge, you have even more potential to connect the edge group towards the opposite edge. Sometimes the extra connection possibilities generated by playing away from the edge is exactly what is needed.

For example consider the beautiful solution to the following position (I wish I could take the credit for its discovery but the original over the board play was found by Tom239 on _Playsite_ (he was at the orange level at the time!). The position below is a slight modification of one constructed by Kevin O'Gorman, the maintainer of the Ohex data base).

It is Red's move. To win, Red must connect his a9 piece to bottom. To do this, Red must make some ladder escape that additionally must somehow use the d7 piece to threaten another way to connect to the ladder. This looks impossible but yet there is a way. Red can win by starting with 1.b9 b10 2.c9 c10 3.f8!!

This brilliant move is the only way to win. 3.g7 is defeated only by 3...d9 and 3.d9 d10 4.g7 is defeated only by 4...f8 (it takes a lot of analysis to demonstrate these claims). Blue's only good attempt is to intrude on the edge template with 3... e9. But Red can defeat this by reconnecting with 4.g7! (this is what Red had in mind when playing 3.f8!!)

Now f8-g7 has an unbreakable connection to the bottom and Red threatens two distinct ways of connecting this group back to the group containing c9; Red threatens f6, double two-chaining between d7 and g7, and Red threatens e8 two-chaining to c9. Blue's only possible defense is the forcing move 4...d8. This interferes with the immediate connection threat between c9 and f8, and it prepares to meet the f6 threat with c8 cutting off d7 from c9. But this move is still not sufficient because after 4...d8, Red can win with 5.d9 d10 (forced) 6.e8.

In practice, you can think of the parallel reconnection as your "standard" response (more often than not, it is the correct choice). But if the potential threat to cut off the parallel play from the edge is serious, then go with the direct reconnection. The "away" reconnection entails a substantially increased risk of being cut off from the edge but if you can see that it will be safe or if you need the stronger connection possibilities towards the opposite edge, then go with the "away" connection.

Third row ladder to a3 and its symmetric analogues

(See also the page a3 escape trick)

The following position is from one of my games.

I am Blue and it is my move. Red's e6-f6(-f4-g4) group is connected to bottom via template Vb. Red's i2 piece is connected to the top via edge template II. In order to stop these two groups from connecting to each and completing a win, I must start laddering down column H. So I ladder down to h6 forcing Red to follow down column I to i6 yielding the following position.

My h10 piece is not a valid ladder escape. If I ladder all the way down to h10, then Red follows down to i8 and his response to h9 is not i9 but j9!

Red has a winning chain on the right side. You might think I could win by instead laddering down one more hex, and then double two-chain to the h10 piece yielding the following position.

This may appear to settle the matter in my favor but in actuality, Red has a winning position! Red can win by 1. h8 (h9 also works but h8 is slightly more deceptive). If I respond by saving the link, i.e. by 1...g8, then Red wins by playing 2.h9 g10 (forced) 3. j9.

Red has an unbreakable winning chain down the right. Instead it is better for me to respond to Red's 1.h8 with 1...h9. My g9-h9-h10 group is now solidly connected to the right but Red can continue 2.g8 and I cannot stop g8 from connecting to the bottom because of the help provided by Red's e6-f6 pieces (work it out!)

In the initial position I cannot afford to ladder down any farther than g6. If I ladder down one more hex, I lose against best play no matter what. If there are no other pieces in the area, as is the case here, then the strongest way to play it is to ladder down one hex short of the hex that could double two-chain to the "almost-escape" piece, and then two chain up from the almost-escape piece which in our present case yields the following position.

Red has three tries to stop the connection between the h6 and g9 pieces.

• g8 is defeated by continuing the ladder down (try it!).
• h7 and h8 are best met by f8 (double two-chaining in the same direction).
• Meeting the play h8 with g8 (connecting up to h6) doesn't work for the same reason that laddering down to h7 and double two-chaining to h10 doesn't work (work it out and you should see what I mean).

Also, note that Red's attempt h9 is of no consequence. Against h9 you should save the link with g10 and then again meet either h7 or h8 with f8.

In the actual game my opponent played h7 and I responded with f8. f8 threatens to connect with with h6 through g7. So my opponent played g7 to which I responded with f7. Again this threatens a winning connection from f7 to h6 through g6. So my opponent played g6 and I responded with c9 with a winning position.

Further play no longer concerns the topic under discussion but the remaining moves were d9, e7, d7, d8, b9, c8, a8, b8, a9, b7, a7, d6, resigns. My opponent doesn't need to see g8, f9, h9, g10, j9, i8

The key play of two-chaining up from the escape piece is also useful in another common type of third row ladder position. For example, consider the following position with the vertical player to move.

Red has a chain running from the bottom at c9 up to d4. The only way Red can win is to connect this group to the top. Red can ladder d3, c3, b3 but as we saw earlier, the a3 piece is not a valid ladder escape. But Red can still win by two-chaining from a3 to b4.

This threatens a winning connection to c5 through b5. If Blue blocks this with b5, then Red plays the ladder because now the pair a3-b4 are a valid ladder escape. If instead Blue blocks off the ladder with say c3, then Red wins with the line b5, b3, a4 (forced), b1, d2!

d2 is a forking ladder escape; it threatens d3 and it provides an escape to the 2nd row ladder starting with b2. Blue cannot stop both winning threats with a single move, thus Red wins.

a3/j8 is a common opening move. If you frequently play it or play against somebody who does, then you will run into these 3rd row ladder situations and hence, it will be beneficial to learn how to play them.

The parallel ladder trick

(See also the page Parallel ladder)

Consider the following position with Red to play.

All of Red's pieces form a connected group. This group is connected to the top. At the bottom, Red has a second row ladder with no possible escape on the left. The potential escapes on the right are inadequate. For example, suppose Red ladders to f9. Then tries to escape with 5.h9 g9 6.h8 g8 7.h7 f7.

Now Red's only reasonable try is 8.g7 f8. Now 9.g6 loses to 9...f5 and 9.h5 loses to the forcing sequence 9...g6 10.h6 h4 11.g5 f5. All the other escape attempts likewise fail. Is Red done for?

No! Red can create a sufficient escape by making use of a parallel ladder. In the original position Red plays 1.e7. How can Blue stop Red from connecting to the bottom? d9 lets Red two-chain from e7 to f8 connecting to the bottom; e9 and e10 allow d9 which is connected to the bottom and threatens to connect to Red's big group through c9 and e8; d10 loses to e8, f9 (forced), c10; hence, Blue is forced to play the parallel ladder move 1...e8. It is simplest for Red to repeat this and ladder to f7 forcing the 2...f8 response.

Now Red now goes back to the second row ladder and tries to escape. What have we gained by preceding this with the parallel ladder moves? When trying to escape, the threat to connect to d7-e7-f7 is stronger than the previous weak threat to connect to d7. This extra threat will let us push our escape chain farther up the board and in this case, just far enough to win the game.

Red's winning sequence is long but rather simple because every one of Blue's replies is forced. As before, Red ladders to f9 and escapes with 7. h9. Play continues 7...g9 8.h8 g8 9.h7 g7 10.h6 g6 11.h5. Red is threatening to play g5 with the double winning threats f5 and f6. But if Blue blocks this, say with 11...g5, then Red continues 12.i3 i2 13.h3 and 14.g3 completes the win.

I have managed to pull this trick off from one row farther back; i.e. with ladders on row 3 and 5 but this occurs far less frequently and you have to examine some additional defensive possibilities. Consider the following position.

Red has just played e6 trying the parallel ladder trick. With the closer ladder on row 2, we saw that Blue was forced to respond with the parallel ladder play e7. But here Blue has two additional possibilities e8 and c9 (the only other possibility where Red doesn't have a way to force his group to connect to the bottom is c10. But Red can respond with f8 and now Blue has nothing better than e7, g6).

e8 yields a second row ladder after d8, e7, c8, c10, d9. The play c9 also leads to a second row ladder after the likely f7, f8, e8 (d9 is met by e7) d10. In the latter case, Red could again try the parallel ladder trick by playing g7. Of course, the existence of other pieces in the area can change the possibilities.