I had a proof but here, but it was wildly wrong.
So here's the right proof! I'm so proud of myself (core pats self on back)
It utilizes the power of modulus.. ooooohh :)
Take, say, position 20:
(13%3) =1, the second pile
floor(13/3)=4, the position in the second pile
(11%3) =2, the third pile
floor(11/3)=3, the position in the pile
Tada! Rinse and repeat, for every other starting position. You'll see it works out nicely.
Interestingly enough, the fact that this works out so nicely mathematically is completely irrelevant - you could pick the most erratic behaviour you like, and no matter how painful the math was, the proof will be trivial - just prove it for each possibility. When there are a finite, very small number of possibilities, things become really, really easy.
*note: there is still an error hidden in this proof. Find it and win a bon bon. (even with the error, the proof is still valid. Whenever modulus is involved, you have a lot of leeway...)