Theorems about templates

There are a number of theorems about templates, some of which can be useful in play. Some of these theorems concern how to construct new templates from existing ones. Others concern how to play when templates overlap. Others explain why templates tend to have particular shapes.

New templates from old

When we have theorems that allow us to construct new templates from known ones, there are fewer templates to memorize.

Corner clipping

We begin by observing that the following two positions are strategically equivalent:

A:

B:

Indeed, if Red plays first in the region, then x captures the entire triangle (x,y,z) in A, and w captures the corresponding triangle in B. Therefore, under optimal play, Red achieves exactly the same thing in A as in B. Similarly, if Blue plays first in the region, y captures the whole triangle in A and w kills the red stone and therefore captures the whole triangle in B. Therefore, under optimal play, Blue achieves the same thing in A as in B. It follows that A and B are equivalent.

Then we have the following theorem about edge templates:

Theorem 1 (corner clipping). Suppose an edge template has a corner of the form

A:

Here, the blue-shaded cells must not be part of the template, i.e., they must be outside of its carrier. Then the pattern where this corner has been replaced by

B:

is also an edge template. The converse is also true, i.e., if some template has a corner of shape B, the corresponding pattern with shape A is also a template.

Proof. Since the shaded hexes are outside the carrier, they may as well be blue stones. Then by the above observation, A and B are completely equivalent, so if some pattern containing A is connected, then so is the corresponding pattern containing B. Moreover, it is easy to see that if removing one cell from the carrier of A would yield a connected shape, then the same could be achieved by removing one cell or the red stone from B, and vice versa. Therefore, the template containing A is minimal if and only if the template containing B is minimal. □

Examples

Corner clipping shows that the ziggurat is equivalent to edge templates III2b and III2g, as well as a very compact 3-stone template:

⇔

⇔

⇔

Edge template IV2a:

⇔

Remark. For the clipped corner theorem to hold, the corner must be of this shape

and not merely that one:

In other words, the cell on the 3rd row should not be part of the carrier. However, if the corner is merely of the latter form, the clipped template is still valid. It may not be minimal. For example, consider edge template IV2b:

Clipping the right corner is no problem. If we clip the left corner, the resulting pattern is connected, but not minimal:

The three hexes marked "*" could be removed from the carrier while still remaining connected.

Corner bending

The idea of corner bending is similar to that of corner clipping. We again start with an observation about two positions. This time, we claim that B is at least at good for Red as A.

A:

B:

Indeed, if Red plays first in B, then y kills the red stone and therefore captures the whole region, which is certainly at least as good as anything that Red could achieve in A. On the other hand, if Blue plays first in A, then Blue occupies the whole region, which is certainly at least as bad for Red as anything Blue could achieve in B. Therefore, if any position containing A is winning for Red, then so is the corresponding position containing B.

We obtain the following theorem:

Theorem 2 (corner bending). Suppose an edge template has a corner of the form

A:

Here, the blue-shaded cell must not be part of the template, i.e., it must be outside of its carrier. Then the pattern where this corner has been replaced by

B:

is still connected. (It may fail to be an edge template only because it may fail to be minimal).

Proof. Since the shaded hexes are outside the carrier, they may as well be blue stones. Then by the above observation, B is at least as good for Red as A, so if some region containing A is connected for Red, then so is the same region containing B. □

Examples

The following is a ziggurat, followed by ziggurats with one or two bent corners.

Overlapping templates

When templates overlap, they are usually not both valid. However, there are some exceptions where templates can overlap and still be valid. It is useful to know them.

Edge template II in the overlap

Theorem 3. If the region in which two edge templates overlap is edge template II, then both templates remain valid.

Proof. The two empty hexes in edge template II are captured, and therefore they can be replaced by red stones without changing the strategic value.

Overlapping templates are only invalid if there are empty cells in the overlap. □

Example

+

=

Both templates remain valid, i.e., all three red stones can be connected to the edge simultaneously.

The ziggurat theorem

The following theorem is due to Eric Demer.

Theorem 4 (ziggurat theorem). Consider a ziggurat.

If the ziggurat overlaps another edge template in the cell x, and/or overlaps another edge template in the cell y, all templates (i.e., the ziggurat itself and its neighboring templates) remain valid. If Blue plays in the overlap at x or y, Red can restore all templates by playing at z. Moreover, this even remains true if the ziggurat has not been completed yet (i.e., if the template stone 1 has not yet been played).

Proof. If Blue plays at x or y (or both), clearly Red playing at z defends the ziggurat. What we must show is that it defends the neighboring templates as well. But if Red plays at z, then the two hexes just below z are captured.

Then the neighboring templates are valid by Theorem 2 above (the corner bending theorem).

The other thing we must show is that if Blue starts by playing in the ziggurat anywhere other than at x or y, then Red can always reconnect in a way that either captures or no longer needs x and y. Indeed, if Blue plays anywhere on the right, Red can play like this:

which captures x and no longer needs y. And if Blue plays on the left, Red responds like this, which captures y and no longer needs x:

□

Example

The classic application of the ziggurat theorem is edge template IV2c:

Red is threatening to play at a or b, getting a ziggurat each way. Blue's only hope is to play in the overlap. Alas, by the ziggurat theorem, this does not work. Red knows that she should play at 2 (the symmetric move would of course also have worked):

Now Blue is sure to lose:

There are other ways for Red to connect here; for one, Red's moves 2, 4, 6 could have been played in a different order. But by using the ziggurat theorem, Red can easily know what to do without having to think hard, and can concentrate on other trickier areas of the board.

Generalizations. There are many other edge templates (besides the ziggurat) for which a version of the ziggurat theorem holds, but it is not known whether it holds for all templates of the appropriate shape. Perhaps a list of such templates could be added here at some point.

The shape of templates

You may have noticed that many edge templates resemble each other: their boundaries tend to follow a relatively small number of possible shapes. This is not a coincidence. Some of it is due to the following theorems.

Observation: the two empty cells in the following pattern are captured by Blue.

Indeed, if Red plays in one of these cells, Blue can play in the other, killing Red's stone. From this, we immediately get the following theorem:

Theorem 5. There is no edge template with an empty corner of height 2, i.e., with a corner of this shape:

Here, as in previous theorems, the blue-shaded cells indicate the outside of the template, i.e., they are not part of the carrier.

Proof. Since the shaded hexes are outside the carrier, they may as well be blue stones. But then the two empty cells are captured by the previous observation, which means they can also be filled in with blue stones, contradicting the minimality of the template. □

We note that the theorem only says that the corner of an edge template cannot be of height 2 if that corner is empty. When there are stones in the corner, height 2 is possible, as in the following examples:

Observation: the following two regions are equivalent.

This is much harder to prove and has been verified by computer using combinatorial game theory. We do not give a proof here. However, it yields the following theorem:

Theorem 6. There is no edge template with an empty corner of height 4, or more specifically, with a corner of this shape:

Here, as in previous theorems, the blue-shaded cells indicate the outside of the template, i.e., they are not part of the carrier.

Proof. Since the shaded hexes are outside the carrier, they may as well be blue stones. Then by the previous observation, the topmost empty cell might as well be blue, contradicting the minimality of the template. □

Similarly to the remark after Theorem 5, we emphasize that the theorem only holds when the entire 10-cell corner is empty. If there are stones in the corner, height 4 is possible, as in edge templates IV2a, IV2d, IV2i, etc.

Observation: The following two regions are equivalent:

To see why, first assume that Blue plays first in the region. Then Blue x captures the entire region, so the stone at y no longer matters. Next, assume that Red plays first in the region. If Red plays anywhere other than x, then Blue x kills the red stone. If Red plays at x, then y is dead, so the stone at y no longer matters. As a consequence, we get the following theorem:

Theorem 7. There is no edge template with a piece of boundary of this shape:

Here, as in previous theorems, the blue-shaded cells indicate the outside of the template, i.e., they are not part of the carrier.

Proof. Since the shaded hexes are outside the carrier, they may as well be blue stones. Then by the previous observation, y might as well be blue, contradicting the minimality of the template. □

Once again, we remark that Theorem 7 presupposes that there are no stones in the relevant portion of the template. For example, the ziggurat is an edge template that ends in two columns of height 3, but this does not contradict Theorem 7 due to the presence of a red stone in one of these columns.

Theorem 8. There is no edge template with a corner of this shape:

As always, the blue-shaded cells indicate the outside of the template, i.e., they are not part of the carrier.

Proof. This is very similar to Theorem 7. By the previous observation, y might as well be blue, contradicting the minimality of the template. □

Theorems 5–8 explain why the rightmost few columns of edge templates frequently have one of these shapes

and never one of these, unless the template contains stones in those regions: