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An open problem is a problem for which no solution has been found. An example is "Do oscillators of all periods exist in Conway's Game of Life?".

Unsolved problems can be subdivided into several basic categories:

  • Periods: Do oscillators, spaceships, guns or puffers exist of a particular period?
  • Unusual-growth patterns: What is the long-term effect of a predefined pattern? For example, it is unknown whether the Fermat prime calculator grows indefinitely.
  • Solvable problems: Some problems are known to have a solution, but as yet no pattern has been built. For instance, no Conway's Life quadratic-growth replicator has been built to date, but a workable blueprint is available, and no really large new technical problems would have to be overcome to complete the construction.
  • Construction and destruction problems: These include finding the smallest Garden of Eden, building a stable eater that can absorb any single glider aimed at it, determining whether a particular object has a glider synthesis, or discovering an unstoppable-growth pattern.
  • Spatial minimization problems: Find an object that satisfies some criterion that fits within a certain bounding box. Examples include Mike Playle's prize for a small stable reflector.
  • Temporal minimization problems: As above, but concentrating on speed rather than size.

List of problems

Open and previously open problems include:

Problem Status Year posed Posed by Year solved Solved by
1 cell thick infinite growth solved  ? Nick Gotts 1998 Stephen Silver
The question “is there a 1 cell thick pattern exhibiting infinite growth?” was answered in the affirmative.
Coolout Conjecture solved <1992 Richard Schroeppel 2001 Richard Schroeppel
The question “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?” was answered in the negative.
Garden of Eden solved  ?  ? <1970 Edward Moore
The existence of a Garden of Eden in Life was known from the start because of a theorem by Edward Moore that guarantees their existence in a wide class of cellular automata.
Grandfather problem solved 1972 John Conway 2016 mtve
The question “is there a configuration which has a father but no grandfather?” was answered in the affirmative.
Omniperiodicity open  ?  ?
The question “do oscillators of all periods exist?” remains open for Life; no oscillators are known are periods 19, 23, 38 and 41, and no non-trivial oscillators are known for period 34.
Replicators solved  ?  ? 1982 Elwyn R. Berlekamp, John Conway, Richard K. Guy
The question “do replicators exist in Life?” was answered in the affirmative.[1]
Unique father problem open 1972 John Conway
The question “is there a stable configuration whose only father is itself?” remains open.
Universal computer / Universal constructor solved  ? John Conway 1982 Elwyn R. Berlekamp, John Conway, Richard K. Guy
The question “does Life support universal computation and universal construction?” was answered in the affirmative.[1]


  1. 1.0 1.1 Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (2004), Winning Ways for Your Mathematical Plays, 4 (2nd ed.), pp. 927-961

External links