My pills, which I buy 31 at a time, come in an airtight canister. Last month (December), I opened the canister 31 times, removing (and swallowing) one pill every day. The problem is that the last pill I took on December 31 was exposed to the air 31 times, which diminishes its potency. The December 1 pill was only exposed to air once. On average, the pills that I took were exposed to the air 16 times, calculated as (1 + 31)/2.
But I can do better in January, because now I have an empty canister!
On January 1, I could swallow one pill and transfer 15 to the empty canister. If I make no more transfers, the remaining 30 pills will be exposed an additional 8 times on average, i.e (15+1)/2, in addition to the once that they have already been exposed. Average time exposed for all 31 pills is (1 + 30*(8+1))/31 = 271/31 = 8.742. A significant improvement from 16! I feel healthier already!
I think that I can do better if I transfer pills between my two canisters more frequently. What is the best I can do? How?
(In reply to Well, I was wrong!
Thanks, broll. I thought you made a great start. Counting total exposures instead of average exposures was a very useful insight, as was the Current vs. Reserve nomenclature. I plan to use both of those when I write up an official solution. There is also one other key concept that I am hoping shows up in somebody's post, which you already sort of touched on. Please forgive the mystery.