Coolout Conjecture

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Coolout Conjecture
Coolout Conjecture image
Pattern type Problem
Number of cells 8
Bounding box 6×2
Discovered by Richard Schroeppel
Year of discovery 2001

The Coolout Conjecture is a conjecture proposed by Richard Schroeppel before 1992, and disproven by counterexample in 2001. The conjecture has been stated as:

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?

or alternatively,

If a configuration C is locally stable over a rectangle R, does there exist a configuration C* such that:
  1. C* is locally equal to C over R; and
  2. C* is globally stable?

In August 2001, Schroeppel published the following 6x2 pattern as a counterexample to the conjecture:

x=6, y = 2, rule = B3/S23:P6,2 2o2b2o$ob2obo! [[ THEME 6 GRID GRIDMAJOR 0 VIEWONLY THUMBNAIL ]]

The row above the top edge must have six consecutive OFF cells; if it does not, the ON cells in the second and/or fifth columns will turn OFF. However, six consecutive OFF cells prevent the OFF cells in the third and fourth columns from being stabilized: without an ON neighbor above the top row, they will turn ON.

By similar logic, this pattern can be shown to be a counterexample to the conjecture:[1]

x=5, y = 2, rule = B3/S23:P5,2 2ob2o$obobo! [[ THEME 6 GRID GRIDMAJOR 0 VIEWONLY THUMBNAIL ]]

Similarly to the original counterexample, the cells in the top row allow the center OFF cell to turn on if they are OFF and turn OFF the flanking ON cells if they are ON. Some other counterexamples have also been found[2], including this one:

x=6, y = 2, rule = B3/S23:P6,2 2ob3o$2o2bo! [[ THEME 6 GRID GRIDMAJOR 0 VIEWONLY THUMBNAIL ]]

It has been shown via a small brute-force search that a 2-by-5 rectangle is the smallest bounding box that allows a counterexample to exist.

References

  1. BlinkerSpawn (June 16, 2017). "Re: Random posts". ConwayLife.com forums. Retrieved on July 2, 2017.
  2. wwei23 (June 24, 2017). "Coolout Conjecture Counterexamples". ConwayLife.com forums. Retrieved on July 2, 2017.