Article 36326 of rec.puzzles:
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Newsgroups: comp.theory.cell-automata,rec.puzzles
Subject: Re: Smallest dying "life" group?
Followup-To: comp.theory.cell-automata
Date: 01 Dec 1995 22:43:47 GMT
Organization: Navy Center for Artificial Intelligence
Distribution: inet
In-reply-to: wft@math.canterbury.ac.nz's message of 28 Nov 1995 23:17:45 GMT
To: wft@math.canterbury.ac.nz (Bill Taylor)

wft@math.canterbury.ac.nz (Bill Taylor) asked for the smallest dying
group in Conway's Life, specifying

> ... a group of stones that completely dies with no progeny, in one
> generation.  To exclude trivia it must include at least one stone that
> dies by overcrowding.

I also exclude non-minimal groups--groups that contain a nontrivial
dying group as a proper subset.

Bill gives the population-9 and -10 examples

  . . x . .         . . . x .
  . . x . .         x . x . .         . . x x . .
  x x x x x         . x x x .         x x x x x x
  . . x . .         . . x . x         . . x x . .
  . . x . .         . x . . .

and asks for anything else of population 10 or less.  A
disappointingly large number of duplicate and incorrect examples were
posted....

I wrote a program to search for minimal examples.  There are no other
groups of population 9, and only the three population-10 groups:

  . . . x . x       . . . x . x       . . . x . x
  . . x . . .       x . x . . .       x . x . . .
  x x x x x .       . x x x x .       . x x x x .
  . . x . . .       . . x . . .       . . x . . .
  . . x . . .       . . x . . .       . x . . . .

You may consider these to be inessential modifications of the first
two above, but that's all there are.

Similar inessential modifications yield the three minimal
population-11 groups:

  . . . . x . x       . . . x . x      . . x x . x
  . . . x . . .       . . x . . .      x x x x x .
  . x x x x x .       x x x x x .      . . x x . .
  . . . x . . .       . . x . . .      . . . . . x
  x . x . . . .       . . . x . x

It is not until population 12 that we see the significantly new form
(in three variants)

  . . . x . .       . . . x . .       . . . x . .
  . . . x . .       . . . x . .       . . . x . .
  . . x x x x       x . x x x x       x . x x x .
  x x x x . .       . x x x . .       . x x x . x
  . . x . . .       . . x . . .       . . x . . .
  . . x . . .       . . x . . .       . . x . . .

in addition to four more modifications of the population-9 and -10
examples.

Dan Hoey                     posted and e-mailed
Hoey@AIC.NRL.Navy.Mil