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Hex:
A Strategy Guide

Matthew Seymour


Chapter 6

Equivalences

This is a rather brief chapter. We will look at a few assorted topics concerning patterns that can be considered equivalent in some way. Noticing the similarities between different patterns simplifies the analysis of board positions and makes you a stronger player as a result.

6.1 Edge templates as interior templates

It can be helpful to view the players’ respective edges as lines of stones of their own colour (Diagram 228).

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Diagram 228

Diagram 229 compares several edge templates to their interior equivalents. In each comparison, the edge has been replaced by a line of stones. We always use the minimal number of stones to replace the edge. Where there are two hexes of the template contacting the edge, we only need a single stone to replace it, as when we compare Template A-2 to the bridge. Where there are three hexes of the template in contact with the edge, we can replace the edge with a pair of stones, as with the transformation of Template E-3 into a Span, or of Template C-3 into a Parallelogram. And in the final comparison, with the four hexes of Template A-3 in contact with the edge, we replace the edge with a line of three stones.

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Template A-2

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Bridge


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Template E-3

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Span


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Template C-3

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Parallelogram


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Template A-3

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Interior version of Template A-3.

Diagram 229

6.2 Extensions of some interior templates

As Diagram 230 shows, the Trapezoid can be “extended” indefinitely and still remain a valid template. The straight row of stones along the bottom operate as an edge, with the stone on the right serving as a ladder escape, as Diagram 231 shows.

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Diagram 230

Extensions of the Trapezoid.

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Diagram 231

Trapezoid extension as a ladder escape.

Diagram 232 shows how the Crescent can be extended as well. In the same manner as above, we can view this as a ladder escape along an edge. Diagram 233 shows the equivalence. Have a look back at Diagram 125 for a practical application of extending the Crescent.

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Diagram 232

Extensions of the Crescent.

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Diagram 233

Trapezoid extension as a ladder escape.

6.3 Dead hexes

Dead hexes are hexes that have been rendered irrelevant to the game’s outcome. Any possible connection for either player that might go through a dead hex would have been a connection anyways. A dead hex is still considered dead whether it is occupied (by either player) or not. When a hex is dead, the outcome of the game is identical whether it is occupied by Black, White, or nobody—so we are free to imagine it being occupied by whoever we like (Diagram 235).

If a dead hex is empty, never play in it. It’s a waste of a move.

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Diagram 234

Dead hex patterns. In each pattern, the marked hex is dead and can be occupied by either player without changing the outcome of the game. See D.4 Dead and captured cells for references.

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Diagram 235

All three patterns are equivalent.

6.4 Captured hexes

Have a look at Diagram 236. Each pattern has two marked marked hexes. If White plays at one hex, Black responding at the other renders the initial hex dead (try this yourself and compare to Diagram 234 above—remember that a hex can still be dead if it’s occupied). We say the marked hexes are captured by Black. Since the best outcome for White in these patterns is to occupy a dead hex, these patterns are equivalent when the marked hexes are occupied by Black.

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(a)

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(b)

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(c)

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(d)

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(e)

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(f)

Diagram 236

Captured hex patterns. In each pattern, the marked hexes can be occupied by Black without changing the outcome of the game. See D.4 Dead and captured cells for references.

It’s almost always a waste of a move for either player to play on a captured hex. There is one exception: if the game is settled—by which I mean one player has a strong connection between their edges—but one player refuses to concede, it may be necessary for the player with the winning position to fill in the remaining hexes in order to complete the win. Doing so may involve playing in their own captured hexes.

Pattern (a) from Diagram 236 is interesting when we replace the three stones with a black edge (Diagram 237). Playing a stone on the second row from the edge captures both hexes adjacent to that edge. This means that it’s almost always better to play one stone back from your edge than directly adjacent to it! There are only a few exceptions to this principle. The first is if you don’t have two open hexes along the edge where you want to play a stone (this is quite rare). The second is on the first move of the game, where the swap rule implies you don’t always want to make the strongest play possible (in fact, a13 is a common opening on the 13×13 board). And the third is the aforementioned case where you need to fill in the remaining hexes of your connection to finish a game.

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Diagram 237

Playing on the second row from your edge captures both stones on the first row.

We can apply the capturing rules repeatedly, filling in captured hexes and then looking for new patterns of captured stones. Diagram 238 shows an example. Keeping in mind that the edges of the board are equivalent to a rows of stones owned by their respective players, we notice that a black stone on c2 captures the hexes c1 and d1 (Diagram 238a). Filling these captured hexes with black stones (Diagram 238b), we notice that a1 and b1 are captured due to the pattern from Diagram 236f. Filling these in with black stones (Diagram 238c), we notice the same pattern yet again, this time capturing a2 and b2. Filling those captured stones in leaves us with (Diagram 238d). This leaves us with the conclusion that a single black stone on c2 captures the entire corner for Black (presuming that none of these hexes are occupied by White). Therefore a2 and b2 are inferior moves for Black compared to c2 when the area is empty.

a b c d e 1 2 3 Reset board to initial state Next move Previous move Last move First move

(a)

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(b)

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(c)

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(d)

Diagram 238

c2 captures the entire corner for Black.

6.5 One-sided capture

There are also patterns where only one intrusion point can be rendered dead in response. In Diagram 239, if White plays at A , Black can kill her stone by playing at B . The converse is not true: if White plays at B , Black cannot kill her stone. Since it’s possible for White to gain some territory playing at B , we cannot mark A as belonging to Black in the same sense as we did with the truly captured hexes in the previous section. Nonetheless, we can still consider the hex A to be captured as far as the play on that side of the pattern is concerned.

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Diagram 239

Diagram 240 shows an application of this idea: Black is able to escape his fourth-row ladder because he effectively has Template C-4 thanks to the captured hex.

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Diagram 240

Black can escape the ladder with Template C-4, taking advantage of one-sided capture.

Another instructive example is Template D-4 (Diagram 241). B and C are captured due to the pattern in Diagram 237. But A is also captured on the left side, as White cannot prevent a stone played here from becoming dead. In Diagram 241b all four hexes on the bottom row are captured by Black. This means and are collectively dead, since they cannot participate in any connections. As Diagram 241c shows, although is dead, White can produce usable territory on the right side, so we can’t fill A with a black stone to get an equivalent pattern. But the left-sided capture of A combined with the captured B gives Black an effective line of four stones, to which he could for example attach Template A-3, as can be seen in Diagram 242.

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(a)

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(b)

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(c)

Diagram 241

White cannot make a stone at A in Template D-4 live, although she may get territory on the other side of the template.

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Diagram 242

Black can take advantage of the effective line of stones on the left edge of Template D-4 to attach a template such as Template A-3. A is attached to the bottom through the two templates.