# Langton's ant questions

I'm a mathematician currently working on the Langton's ant conjecture, just for fun. I have some result but I don't know if they are meaningless. So that is why I'm asking.

1) Is there a mathematical proof that the Langton's ant will stay forever on the highway once it's enter in it ?

2) Is there a proof that for only one initial condition the Langton's ant will always build the highway ?

• 1) a quite informal proof is by "induction on the shape of the highway": after some steps (104 according to Mathworld site) the last portion of the highway is cloned (+2 pixels horizontally and +2 pixels vertically) and the ant is in the same relative position; so if there are no "obstacles" it will grow forever. 2) It is false: many initial conditions we can test generate a highway ...and -indeeed- the conjecture is "Does every finite initial configuration generates a highway?" (we only know, by the Cohen-Kung theorem, that for every finite initial configuration its trajectory is unbounded) Apr 2, 2016 at 15:26
• hi, thanks a lot for replying, in the 2nd question I'm referring to the basis of induction for proving the conjecture. If someone already proof if you put one square black anywhere the ant will make the highway.thanks again ;) Apr 2, 2016 at 18:26
• ok! (you can specify that you're asking if it is known whether the ant always builds a highway if it is started on a board with another active "cell" at an arbitrary position) I'm not aware of such result ... and IMO it could be a nice "intermediate" result by itself; probably there are cases in which when the highway "hits" a block the ant goes back to the "root" of the highway (and the highway itself is messed up) ... I just made a few simulations with a short highway and it happens :-) Apr 2, 2016 at 19:46
• It's a few years old now, but Ian Stewart discusses a possible approach in 'Figments of Reality'. Apr 17, 2016 at 18:35