George Rebane
RR readers were introduced to the Split Pill Problem (SPP) in a passing Scattershots entry last summer. The problem involved giving our puppy Puna her pain meds after her second knee repair surgery, and that evoked an interesting problem the solution to which turned out to have several features which generalize to and can illuminate other realworld problems. Puna’s daily dose from the pill bottle was a half pill. If I shook out a whole pill, I would split it and give her a half, returning the other half to the bottle. On a day when a half pill emerged, I would just give the half.
The vet said to do that until the pill bottle was empty. Given the random way in which whole and half pills emerged, I wondered how soon could I expect to get to a day when only half pills remained in the bottle and I would no longer have to split pills. The solution turned out to be very interesting, which I have documented in a somewhat whimsical technical note that requires no math beyond grade school arithmetic to understand. You can access the pdf here – Download TN1910-1_Split Pill Problem Solved
I’ll leave you with a piece of eye candy from the report that illustrates the graphical beauty of the SPP solution where it will be fully explained. Enjoy.
George, your split pill analysis was truly quite fun. We had a similar experience - splitting pain meds for our Lab with a bad knee. We resolved the issue on day one by splitting all the pills and putting them back into the bottle. Thus on each succeeding day, our odds of shaking out a whole pill were zero, and of shaking out a half pill were 100%. On the 20th day I got the same results as you - a half pill. My attempts at charting did not turn out well.
Posted by: Bob Hobert | 13 October 2019 at 06:29 PM
One night I put the sheep in the pen and spend the night doing a mental exercise on how to solve this problem. Sleep came "real soon" and the problem went unsolved. I tried again several nights later and discovered it beats counting sheep in getting to sleep.
Posted by: Russ | 13 October 2019 at 07:15 PM
Bob & Russ - Well gentlemen, I figgered there was probably a better way to solve the problem by doing all the splitting at the gitgo; even my daughter suggested that I wouldn't have to do all this tomfoolery if I had done this. But then Russ does identify another use for such problems for people with insomnia.
BTW, since you both are technically oriented, the numerical solution to that and other such stochastic problems is in the form of a probabilistic structure known as a Markov DAG (directed acyclic graph).
Posted by: George Rebane | 13 October 2019 at 08:57 PM
Problem? What problem?
The lifetime task of splitting meds is minimized by doing it only as required.
Posted by: Gregory | 13 October 2019 at 10:37 PM
This kind of thing is one of the several reasons I still read your blog, George. It's sad that so many refuse to see the beauty and joy of math exercises.
Posted by: Michael R. Kesti | 14 October 2019 at 08:26 AM
George...Ya lost me in the weeds, but an impressive proof never the less. I will stick with whole pills to keep the math simple!
Posted by: Peter | 15 October 2019 at 05:23 PM
Nice graph. Which software package did you use to produce it?
Posted by: Wayne Hullett | 18 October 2019 at 07:30 AM
WayneH 730am - Matlab. In my dotage I have retreated to doing everything in Matlab and Simulink. Powerful stuff.
Posted by: George Rebane | 18 October 2019 at 08:48 AM