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First of all, see Langton's ant Wikipedia page.

If we place a pair of ants looking north (using Golly or any another prog) on the coordinates $(x_1,y_1)$ and $(x_2,y_2)$ under the conditions:

  • $p=|x_1-x_2|$, $q=|y_1-y_2|$,
  • $p$ odd, $q$ even,

then I conjecture that, if the pair of ants constitutes an oscillator, its periodicity has the form $4(2n+1)$.

Is there some intuition behind this? Is there a way to prove it?

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    $\begingroup$ The question seems interesting. Since you seem to lack both an intuition and a proof of this, I assume the conjecture comes from some numerical simulation of the automaton, right? If so, could you please tell something more on these results? $\endgroup$
    – Alessandro Della Corte
    Commented Sep 25, 2021 at 11:28

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