We can solve this problem by an application of the pigeonhole principle. Get a calendar for the month of June and on each day record the running total number of pills I’ve taken so far in the month. So June 1 might read “1,” June 2 “5,” and so on. The entry for June 30 will read “48.” Write these numbers in red.
Now revisit each date, add 11 to the red number, and enter the result in blue. In the example above June 1 now reads “1 12,” June 2 “5 16,” and so on. June 30 reads “48 59.”
Now we’ve filled the calendar with 60 numbers, all of which fall in the range 1-59. This means that at least one number falls on two different days. In the interval between these two days I took 11 pills.
By similar arguments it can be shown that intervals must exist during which I took precisely 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 pills.
Adapted from Kenneth R. Rebman, “The Pigeonhole Principle (What It Is, How It Works, and How It Applies to Map Coloring),” The Two-Year College Mathematics Journal, January 1979.
