By Emily Gold Boutilier

Giving a cat a pill can lead to many things: frustration, anger, possibly stitches. For Professor Daniel Velleman, it inspired an award-winning math paper.

Here’s what happened: His cat Natasha was losing weight. The vet diagnosed a thyroid disorder and prescribed “one half a pill daily.” Every day, Velleman or his wife shook a pill from a bottle and broke it in half. One half went to the feline, the other back in the bottle. Eventually, the day came when they shook out a half pill, and they gave that to the cat.

“The pills solved Natasha’s medical problem,” Velleman—the Julian H. Gibbs 1946 Professor of Mathematics—writes. “But they created an interesting mathematical problem.”

He wondered: How does the mix of half and whole pills in the bottle change over the course of treatment? What is the expected number of whole pills removed before the first half pill is removed? What’s the expected number of half pills remaining after the last whole pill is taken?

In “A Drug-Induced Random Walk,” published in *The American Mathematical Monthly*, Velleman devotes 18 pages to proofs and theorems around these questions. The paper won a 2015 Mathematical Association of America Halmos-Ford Award for “articles of expository excellence.”

Velleman—a former editor of the journal—is drawn to questions that are “easy to understand but hard to answer,” he says.

To answer the cat-pill questions, he used principles of probability and differential equations. As he explains: If you flip a coin 20 times, it will land on heads roughly 10 times and on tails roughly 10 times. The actual results might be 11 and 9, but they probably won’t be 19 and 1. This is called the Law of Large Numbers.

Assuming all of the cat pills are equally likely to be chosen, the balance of pills in the bottle changes daily. That’s because if a whole pill is removed, a half pill is returned, while if a half pill is removed, nothing is returned.

So he turned to Euler’s Method, used to approximate solutions to differential equations. The result is a series of graphs that show the “path the distribution of pills in the bottle is going to follow.”

Velleman found that in a bottle of 100 pills, the expected number of whole pills removed before the first half pill is removed is around 12. The expected number of half pills after the last whole pill is roughly five.

Natasha died last year at the old age of 20. Happily, she lived to see the paper’s publication.

Illustration by Flavio Morias