half pills [ Atwood's machine ] [ bag of marbles ] [ balance moon stone ] [ ball up/down ] [ bead parabola accelerometer ] [ boat anchor lake ] [ boat time ] [ bobbin on incline ] [ bosun's chair ] [ bouncing ball ] [ bowling ball rolling ] [ bug on band ] [ bursting shell ] [ falling chain ] [ Feynman's restaurant problem ] [ five pills ] [ flying cable ] [ forced pendulum ] [ gold mountain ] [ half pills ] [ hallway pole ] [ impelled rod ] [ inelastic relativistic collision ] [ infinite pulleys ] [ mass on an incline ] [ maximum angle of deflection ] [ packs of shirts ] [ particle in bowl ] [ particle in cone ] [ particle on sphere ] [ particle points parabola ] [ pile of bricks ] [ pion muon neutrino ] [ piston ramp spring ] [ plank weight trough ] [ rocket vs. jet ] [ roll without slipping ] [ rough inclined plane ] [ shooting marbles ] [ speedometer test ] [ three balls ] [ three logs ] [ turntable cart ] [ two rolling balls ] [ wheel and block ] [ whirling pendulum ] [ worlds fair ornament ]

This interesting problem was told to me by Jim Farned.

You have a prescription to take one half of a pill  per day for 20 days, but the pharmacist (who is too busy to divide pills for you) gives you 10 whole pills in a bottle. On day 1, you remove a pill from the bottle, break it into two half-pills, take one, and return the other half-pill to the bottle. On all subsequent days you shake the bottle thoroughly and pour something out - whatever comes out first  - either a half pill or a whole pill; if it's a half pill you take it and you're done for that day; if it's a whole pill, you split it into two half-pills, take one, and put the other back in the bottle, exactly like you did on day 1.

On day 20 there can be only one half pill left in the bottle, but on day 19 there are two possibilities: either there is one whole pill or there are two half-pills left in the bottle. What is the probability that there are two half-pills in the bottle on day 19?

### Solutions (listed by author)

(in the order devised -- MAG)

Michael A. Gottlieb (approximation, with difference equations) (pdf, 74K)  (nb, 12K)

Michael A. Gottlieb (exact, with recursive functions) (pdf, 108K)  (nb, 36K)

Michael A. Gottlieb (exact, with Markov chains) (pdf, 95K)  (nb, 24K)

Michael A. Gottlieb (fast & accurate approximation, with differential equations) (pdf,104K)