And that's exactly how it was solved. The puffers are now spaced prime distances apart and that took care of that, it would be harder for a longer belt because then you'd start wasting an awful lot of space next to the belt.
Since the pieces fall onto the faster conveyor belt with random spacing, isn't it possible that two consecutive pieces will have the spacing of the puffers they are destined for (within margins), regardless of the puffer placements?
In the end the error rate went down a lot because of a less predictable spacing. But you are right that if the pieces would fall with random spacings that it would not matter what the distance between the puffer stations would be.
The funny thing is that I spent a lot of time measuring out the vertical spacing on the conveyor in the hopper. If I had done that more sloppily it would have worked better :)
At any snapshot, the pieces are lying with a Gaussian distribution around x multiples along the belt i.e having sigma at x, 2x, 3x,...nx....
So for the bins to not overlap:
1) their width/span-along-the-belt should be lesser than x
2) And they can be placed at x, 3x/2, 5x/2, 7x/2 (i.e. prime multiples of x/2)
Wow! Learnt something useful today. Thank you. :)
Edit: I realize after posting that, my solution won't work! If somebody can explain how the prime thing works will be great. I can imagine, though, that the bin placements should be such that, at any given time the piece is only in front of a single bin. Meaning, no pair of bins should have a distance of x-multiple. I can guess, perhaps heuristics which work well, can be devised. But it will be great to know the mathematical solution for this.
"more than one puffer would be active at once" — sounds like a job for prime numbers!