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:
1st part
-------
(20-7)=13
2nd part
------------
(13%3) =1, the second pile
floor(13/3)=4, the position in the second pile
4+1*7=11
(11-0)=11
3rd part
----------------------
(11%3) =2, the third pile
floor(11/3)=3, the position in the pile
3+2*7=17
(17-7)=10
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...)