Difference between revisions of "Advanced (strategy guide)"
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| − | Adapted from: [http://www.gcrhoads.byethost4.com/GamesPuzzles/Advanced.html Glenn C. Rhoads's Advanced Strategy Guide]. | + | ''Adapted from: [http://www.gcrhoads.byethost4.com/GamesPuzzles/Advanced.html Glenn C. Rhoads's Advanced Strategy Guide].'' |
== Advanced edge templates == | == Advanced edge templates == | ||
Revision as of 00:57, 9 February 2022
Adapted from: Glenn C. Rhoads's Advanced Strategy Guide.
Contents
Advanced edge templates
Template IV-2-a
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 Blue intrudes anywhere but a or b, Red can just bridge to the edge. So the only possible intrusions are at a or b.
Solution to intrusion at a
If Blue intrudes at 1, Red responds with 2, which forces Blue to respond at 3 (in all other cases, Red will connect to the edge easily). Now Red's 4 connects to the edge by double threat at the hexes marked "*". Alternatively, Red can also play 4 before 2.
Solution to intrusion at b
If Blue intrudes at 1, Red responds with 2, which forces Blue to respond at 3. (Blue could also first intrude on Red's bridge, but this would not change anything). Now Red connects at 4 using a trapezoid. Alternatively, Red can also play 4 before 2.
Template V1a
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.
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 V2a
The defense of this template works similar to template IV2a above. If Blue intrudes in the template at any hex besides a or b, Red makes a move that reduces the situation to a closer template.
Solution to the intrusion at a
There are several solutions but the simplest is the following:
4 is connected to the bottom via a ziggurat and to the rest of Red's group by double threat at "*". Thus, the connection is complete.
Solution to the intrusion at b
Red's best response is as follows. This piece is connected to the bottom via a 3rd row template and hence Blue must block at g3. Red then plays e4 yielding
Since 2 threatens to connect to the edge, Blue must defend on the left. Then Red connects with 4, via a trapezoid and ziggurat. Unlike template V-1a, this template is not a rare occurrence. Still, 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 the strongest possible connection is not necessarily 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.)
Example 1
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.
Example 2
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 ladder escape fork 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 ladder escape fork at h9 wins the game.
Example 3
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.
4 at h4 would also be possible. Now I have a forced winning ladder down row 2, 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, c3, c2, b3, b2, a3 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.
Example 4
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.
See Glenn_C._Rhoads_vs._unknown
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 are 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 ladder escape fork 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 ladder escape fork 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 which makes use of what I call "Tom's Move" (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!! (f8 is "Tom's Move").
This brilliant move is the only way to win (it is, essentially, Tom's move). 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 ladder escape fork; 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 escape
(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 exccape. 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 escape by playing g7. Of course, the existence of other pieces in the area can change the possibilities.
For whom who understand The parallel ladder escape !
This trick is useful only for ladder 2nd and 4th!
Consider the following position with Red to play and win. Red's has winning position even with a blue stones in h5 from the beginning.
Note from original author (Glenn C. Rhoads): I realized that my example had this improved solution which ruins the example but I never got around to writing a correct example. The move 1.f8 is essentially what I call "Tom's Move" in the last example of the section "Reconnecting template IIIa after an intrusion." The parallel ladder escape is when you can push the escape up by using the threats to connect to each branch of the parallel ladder. You might think Tom's move does away with the need for the parallel ladder escape but this is not the case. If you take the above position and shift every piece one hex to the right, then Tom's move no longer works because after blue's intrusion, there is not enough room to reconnect this piece to the bottom by playing the "away" connection (see Blue's third option in the provided solution). Yet there is still enough room to potentially ladder escape up the third row from the right edge. The example has to be modified to make the parallel ladder escape work and be the only way to win. Also, there are cases where the ladder starts at the far left and you must push the parallel ladders to the right in order to avoid some centrally located pieces. After pushing past them, Tom's move can then work. Both ideas are useful but Tom's move seems to work more often than the parallel ladder escape.
Also, I did have one game where I managed to pull off a parallel ladder escape from rows 3 and 5! I wish I had saved the moves of the game. The position was much too complex for me to ever remember it.
As an aside, I recall the conventional wisdom that every chess book ever written has mistakes in it (not literally true but a chess book without an error is a rarity indeed; even most beginner chess books have mistakes in them). Perhaps a similar thing could be said about Hex. Both games are at times very difficult and complex where it is all too easy to make a mistake. Of course I had no such intention of providing supporting evidence by my mistaken example. There is another mistake in my advanced guide. In the fourth example in the mini-maxing section, my first mini-maxing play can be defeated! In the subsequent game, my opponent plays an excellent move which I believe would have given him a winning position had it been played in the opening (with this hint, can you find the move?). I didn't bother trying to come up with another example out of sheer laziness. I wanted an opening example where mini-maxing would upon a close examination of the subsequent game, make a more or less direct difference in the outcome of game. Most examples of mini-maxing in the opening help in a less direct extremely complex way. I was reluctant to spend the time to find a good example and to provide a detailed analysis of the subsequent game when it seemed unlikely that anybody would notice the problem!
