TOOHARD

Since interaction happens only between 2 neighbors, let’s focus on the interaction when the ball is between 2 neighbors

Claim: When a ball is between two neighbors, the left neighbor must be of form A
Proof: Try out both cases of A and B

Claim: When after the ball left the right neighbor, it will never come back to the this gap
Proof: Consider both cases AB and AA, it will be in the form of AA and BA, respectively, after it left, and we run into the same case again! By induction this means the ball will move from left to right gaps, one by one


Claim: After ball left the whole sequence, the state of each one is the negation of the next one before it starts
Proof: By the previous claim, we know once we leave the current gap, we are done. So we look at AA and AB gap, and we notice that the left item, A in this case, is just the negation of right item pre-action, i.e., we proof by complete search

Note that the last one will always be A, as shown in the AA and AB case