< Hexagain

Proof of the Fifth-Row Edge Template

We can see that the template is one stone away from reducing to one of many possible smaller templates. We only need to consider the points at which every one of these templates (and the paths required to reach them) overlap. But that’s hard and boring to do mechanically, so let’s do it a little informally.

The two triple templates cover the most area, so we’ll start with all the points in the trapezoid at which they intersect. Then we’ll subtract each of the points spanned by a suspension bridge.

And an intrusion into either of the rightmost points has an obvious response:

Which leaves us with three interesting weak points. We won’t be able to respond to an intrusion at any of these points by reducing our path to a smaller template. Instead, we’ll answer with forcing moves that will eventually conspire to get us a connection.

Let’s go through these one at a time, starting with the topmost.

This play creates a strong threat that white must respond to and, as you might have guessed, sets up a ladder escape that will be useful later on. It doesn’t matter where white blocks, so we’ll continue as if every point in the path were occupied.

There are two points where these paths overlap, and white is forced to play in one of them. It doesn’t matter which one: a ladder will form either way, and the escape is ready.

So an intrusion at the top point doesn’t work out for white. What about the left?

White has a choice of how to respond here, but either intrusion has the same response, so we’ll consider them together:

Hang on a minute. What if instead of one of those “forced” responses, white had intruded on the initial bridge? Since white eventually has to intrude on it as a forced move, and by that point black gets to ignore the intrusion, what if white comes out swinging?

Well, it turns out it didn’t help. White is still forced to defend here, just at a different point. Black takes back the initiative, and the rest of the play proceeds identically.

Alright. That leaves us with one final intrusion point, and we’ll be done proving that this template is valid.

Once again, white has a choice of how to respond. They can take the high road:

Or the more interesting, but no more effective low road:

And that’s it: we’ve shown that no intrusion into the template allows white to stop the connection to the edge. QED, as they say.

Don’t memorize these responses! The important takeaway here isn’t how to defend the fifth-row edge template, but the technique of playing forcing moves that will help you connect a few turns away. You should try to re-create this proof yourself, and work out the same analysis you saw here. It’s a fun exercise.