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Optimizing Potency (Posted on 2015-11-19) Difficulty: 4 of 5
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?

See The Solution Submitted by Steve Herman    
Rating: 5.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: I think I got it. | Comment 4 of 10 |
(In reply to I think I got it. by Jer)

Excellent work, Jer!!


In answer to your question in the queue, this is an original problem. When I solved it (after submitting) I was struck by the solution structure.  Since you have already worked it out starting from n = 1, I suggest that you try starting with a 4 week supply (28 pills), and see if you notice anything interesting.

Or, you could wait for me to post a solution with all of my insights.  You certainly deserve a rest after solving this!

  Posted by Steve Herman on 2015-11-20 16:51:16
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