LifeWiki talk:Did you know/67

1515 From: Richard Schroeppel
Date: Tue Aug 7, 2001 2:04pm
Subject: Coolout Conjecture counterexample


An outstanding question is the "Coolout Conjecture":

Given a partial Life pattern that's internally consistent with being part of a still life (stable pattern), is there always a way to add a stabilizing boundary? Is there an upper bound to the required boundary size, perhaps 3 cells thick? [Variations for stabilizing/completing partial oscillators are also proposed.]

Counterexample:

xx..xx
x.xx.x

The pattern is internally self-consistent with stability: Each cell has a number of live neighbors that, with possible boundary help, makes it stable.

But there's no way to stabilize the top edge: To preserve the xs adjacent to the corner cells, the row above the top edge must have six consecutive OFF cells. But this prevents stabilizing the two OFF cells in the middle of the top edge, each of which needs one or more
ON neighbors in the stabilizing row.

This example shows that internal consistency is not enough for stabilizability; some additional hypothesis is required. The obvious extra hypothesis to try is "1-cell boundary consistency": that the pattern have at least one possible 1-cell thick extension that's consistent with stability.

There probably also need to be some topological restrictions on the pattern: connected, or perhaps some kind of convexity.