Ilkka Törmä and Ville Salo, a pair of researchers at the University of Turku in Finland, have found a finite configuration in Conway’s Game of Life such that, if it occurs within a universe at time T, it must have existed in that same position at time T−1 (and therefore, by induction, at time 0). Here is the configuration of live and dead cells, surrounded by an infinite background of grey “don’t care” cells:
The configuration was discovered by experimenting with finite patches of repeating ‘agar’ and using a SAT solver to check whether any of them possess this property. Similarly, one can use a SAT solver to verify that Törmä and Salo’s result is correct.
Since this configuration can be stabilised (by the addition of further live cells, shown in yellow) into a finite still-life, this demonstrates that not every still-life can be constructed by colliding gliders.
The first finite stabilisation was 374 cells, but this was promptly reduced to 334 cells by Danielle Conway and then to the 306-cell configuration above by Oscar Cunningham. Oscar moreover proved, again using SAT solvers, that this is the minimum-population stabilisation of the Törmä-Salo configuration.
Consequently, we have the following pair of bounds:
- Every strict still-life with ≤ 20 cells can be synthesised by gliders.
- There exists a strict still-life with 306 cells that cannot be synthesised.
More importantly, the Törmä-Salo result positively answers a question first posed by John Conway hi
