Problem

An open problem is a problem for which no solution has been found. A former 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, until DOGun SaGaQR, no quadratic growth replicator had been constructed, but a workable blueprint was available, and no really large new technical problems would have had 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.
• Minimization problems: Find an object that satisfies some criterion that fits within a certain bounding box/minimum population. 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:

Category	Problem	Status	Year posed	Posed by	Year solved	Solved by
Description
Infinite growth	Replicators	solved	?	?	1982	Elwyn R. Berlekamp, John Conway, Richard K. Guy
	The question “do replicators exist in Conway's Game of Life?” was answered in the affirmative.
Infinite growth	Minimal-population infinite growth	solved	?	?	1997	Paul Callahan
	The question "What is the minimum population necessary for a pattern to exhibit infinite growth?" was found to be 10 cells by an exhaustive search. There are 24 such patterns, with 17 unique evolution sequences. (The equivalent question for quadratic growth is thus between 11 and 20.)
Infinite growth	One cell thick infinite growth	solved	?	Nick Gotts	1998	Stephen Silver
	The question “is there a one-cell-thick pattern exhibiting infinite growth?” was answered in the affirmative.
Still lifes	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.
Still lifes	Still life finitization problem	open	?	Dean Hickerson	—	—
	The question "given a still life without finite boundaries, can any MxN finite window of it be preserved within a finite still life by adding appropriate unchanging cells around it?" remains open. It has been shown that the margin around a window needed to stabilise it might be arbitrarily large.
Density	Still life conjecture	solved	?	?	1999	Noam D. Elkies
	The question "What is the maximum density of a still life agar?" has been answered to be 12. The proof requires only that no living cell has more than three living neighbours (including itself), so provides that the maximal density is ≤ 12 in all rules with survival conditions constrained to a subset of S0123.
Density	Oscillator conjecture	open	?	?	—	—
	What is the maximum average density of an oscillating agar? The conjectured value is 12, the maximum that has been achieved. It was proven to be ≤ 813 ≈ 0.6153... on September 18, 1992, which was improved to ≤ 1728 ≈ 0.6071... on December 16, 2023 by 400spartans, then to ≤ 11762087 ≈ 0.5634... on January 9, 2024 by Keith Amling.
Density	Still life density problem	solved	?	?	2012	Geoffrey Chu, Peter J. Stuckey
	The question "What is the maximum number of cells in a still life fitting in an n × n bounding box?" has been solved. As a sequence with respect to n, after the first 60 terms, the sequence's second differences oscillate with period 54.
Oscillators	Omniperiodicity	solved	?	?	2023	various people
	The question “do oscillators of all periods exist?” has been answered in the affirmative.
Oscillators	Non-p2 phoenix	solved	?	?	2024	Adam P. Goucher, Keith Amling, Alex Greason, Stephen Silver
	The question “Does there exist a finite oscillator of period > 2 such that no cell remains on contiguously for more than one generation?” was answered in the negative. (The question for infinite agars and wicks was previously answered in the affirmative.)
Predecessors	Garden of Eden	solved	?	?	<1970	Edward Moore
	The existence of a Garden of Eden in Conway's Game of Life was known from the start because of a 1962 theorem by Edward Moore that guarantees their existence in a wide class of cellular automata.
Predecessors	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.
Predecessors	Unique father problem	solved	1972	John Conway	2022	Ilkka Törmä, Ville Salo
	The question “is there a stable configuration whose only father is itself?” was answered in the affirmative.
Computation	Universal computer / Universal constructor	solved	?	John Conway	1982	Elwyn R. Berlekamp, John Conway, Richard K. Guy
	The question “does Conway's Game of Life support universal computation and universal construction?” was answered in the affirmative.
Computation	Universality of predecessor-finding	solved	?	?	2023	Ville Salo and Ilkka Törmä
	The question "can problems of finding predecessors of specific patterns be equivalent to arbitrary Boolean satisfiability problems?" was answered in the affirmative, with the corollary that the process of reversing a single iteration (or otherwise proving that a pattern is a Garden of Eden) is computationally universal.
Circuitry	Small glider reflector / duplicator	open	2013, 2020	Mike Playle	—	—
	After the discovery of the color-preserving Snark in 2013, Mike Playle offered 100 USD for a stable 90-degree colour-changing glider reflector that fits in a 25x25 bounding box and has a repeat time of 50 generations or less. When the 0-degree Bandersnatch was found in 2020, he released another prize of 200 USD for a stable pattern which fits in a 25x25 bounding box, has a repeat time of 50 generations or less, and produces two output gliders that emerge in different directions and do not affect input gliders from infinity.
Agar crawlers	Minimum speed	open	?	?	—	—
	Does there exist a negative spaceship in an agar of density 12 that travels at a speed ≤ c/2? In zebra stripes, a with-the-grain agar crawler must travel at c (making it a lightspeed bubble), but Hartmut Holzwart proved in 2006 that an against-the-grain agar crawler's speed is ≤ 2c/3.
Agar crawlers	Bound-height lightspeed bubbles	solved	?	?	2017	Arie Paap
	The question "Does there exist an n such that a lightspeed bubble containing an empty region of any width may be constructed with height ≤ n?" was answered in the positive, with n ≤ 37.
Indestructibility	The Immovable Object Problem	open	1972	John Conway	—	—
	Does there exist a bounded pattern such that the central cell will never change state no matter what is placed outside the pattern's boundary?
Indestructibility	The Single-Glider-Proof Object Problem	open	?	?	—	—
	Does there exist a bounded pattern such that a collision with a nonempty glider approaching from infinity (at any direction and offset) results in the glider's destruction and the pattern's eventual evolution being unaffected?

External links

• Unproven conjectures (discussion thread) at the ConwayLife.com forums
• Life has been proven omniperiodic. What next? (discussion thread) at the ConwayLife.com forums