3 ms·

After years of watching numberphile on Youtube, my guess would be that there's a cutoff number n0 for which can be inductively shown that if n0 is a period then

After years of watching numberphile on Youtube, my guess would be that there's a cutoff number n0 for which can be inductively shown that if n0 is a period then n0+1 is also a period; that would then eliminate almost all natural numbers from the search. Such a proof might perhaps use combining periodic patterns to obtain longer periods. It seems to be difficult to construct a periodic GoL pattern with a given period, so that then would explain why of all numbers two very small ones held out over the decades; this is truly astounding, especially given how incredibly inventive and industrious people have gotten with the Game of Life, simulating entire computers on the canvas.

AeiumNE 3y ago
As a fellow numberphile viewer, I accept your authority on the topic, and will probably repeat this guess as fact later.
mhink 3y ago
Actually, progress looks like it's been very steady. You're right regarding inductive-style proofs, but the ones for _all_ natural numbers greater than a certain value came surprisingly late. There appear to have been many more oscillators with periods of x + yn (for constant x and y, any natural number n). In 1996 there was a paper showing that it was possible to use a particular family of patterns called "Herschel loops" to create oscillators of any period >= 61. From there, the only missing oscillators were 17, 19, 22, 23, 27, 31, 33, 34, 37, 38, 39, and 41, 49, 51, 53, 57, and 59. There were gradual discoveries of new oscillators over the next several years, then in 2013 there was another pattern discovered which lowered that upper bound to >= 43. At this point, there were only five oscillators missing: 23, 34, 38, 19, and 41. There appear to have been a few years where progress stalled; 23 was found in 2019, and then the last four were found over the course of these past couple years.

AeiumNE 3y ago

As a fellow numberphile viewer, I accept your authority on the topic, and will probably repeat this guess as fact later.

mhink 3y ago

Actually, progress looks like it's been very steady. You're right regarding inductive-style proofs, but the ones for _all_ natural numbers greater than a certain value came surprisingly late. There appear to have been many more oscillators with periods of x + yn (for constant x and y, any natural number n). In 1996 there was a paper showing that it was possible to use a particular family of patterns called "Herschel loops" to create oscillators of any period >= 61. From there, the only missing oscillators were 17, 19, 22, 23, 27, 31, 33, 34, 37, 38, 39, and 41, 49, 51, 53, 57, and 59. There were gradual discoveries of new oscillators over the next several years, then in 2013 there was another pattern discovered which lowered that upper bound to >= 43. At this point, there were only five oscillators missing: 23, 34, 38, 19, and 41. There appear to have been a few years where progress stalled; 23 was found in 2019, and then the last four were found over the course of these past couple years.