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The Pills problem revisited
Abstract
We revisit the pills problem proposed by Knuth and McCarthy. In a bottle there are m large pills and n small pills. The large pill is equivalent to two small pills. Every day a person chooses a pill
at random. If a small pill is chosen, it is eaten up, if a large pill is chosen it is broken into two halves, one half is eaten and the other half which is now
considered to be a small pill is returned to the bottle. How many pills are left, on average, when the last large pill has disappeared? We show how to
compute the moments, in particular the variance, and then generalize the problem in various ways.
Mathematics Subject Classification (2000): Primary: 05A15; Secondary: 60C05.
Key words: Pills problem, recursions, harmonic numbers, generating functions.
Quaestiones Mathematicae 26(2003), 427–439
at random. If a small pill is chosen, it is eaten up, if a large pill is chosen it is broken into two halves, one half is eaten and the other half which is now
considered to be a small pill is returned to the bottle. How many pills are left, on average, when the last large pill has disappeared? We show how to
compute the moments, in particular the variance, and then generalize the problem in various ways.
Mathematics Subject Classification (2000): Primary: 05A15; Secondary: 60C05.
Key words: Pills problem, recursions, harmonic numbers, generating functions.
Quaestiones Mathematicae 26(2003), 427–439
