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I meant one of these: http://en.wikipedia.org/wiki/Oscillator_(CA)

So cells keep dying and being born forever, yet the game is stable.

Ah, good point! You'd have to keep quite a few digests of the 'state' to figure out that you're caught in a loop, the 'halting problem' in disguise.

You could work out the size of the cycle by checking the lower and upper limit of the populations though, if the lower and upper limits have not changed for a number of cycles it could also be end-of-game.

Store evey state, and if you ever get a state you've already stored you know you're in a loop. Should be possible to store it in very few bytes in "raw" form, and then compress it for good measure.

hash the state..

even then, if the period is very long it might be problematic

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