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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: 4.6667 (3 votes)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Small typo in the official solutionSteve Herman2015-11-28 08:57:31
re(2): Triangular numbers and Pascal's Triangle (spoiler)broll2015-11-27 21:42:19
re: Triangular numbers and Pascal's Triangle (spoiler)Jer2015-11-26 22:26:07
Triangular numbers and Pascal's Triangle (spoiler)Steve Herman2015-11-26 09:14:27
re: Well, I was wrong!Steve Herman2015-11-21 09:26:18
Well, I was wrong!broll2015-11-20 20:16:56
re: I think I got it.Steve Herman2015-11-20 16:51:16
SolutionI think I got it.Jer2015-11-20 15:32:57
Possible improvementJer2015-11-20 14:12:58
Possible approach.broll2015-11-20 03:06:03
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