Today is Friday the thirteenth. The only one of 2022. So your complete portion of misfortune for this year will be poured out on you today. Next year, the misfortune will spread over two days, because in 2023 there will be two Friday the 13ths: in January and in October. In 2026 there will even be three – just look it up.
Does Friday have a greater chance of being the 13th of the month than any of the other days of the week? That question sounds rather silly, because coincidence plays no role at all. After Monday comes Tuesday, after Tuesday comes Wednesday, and so on. The days of the week are neatly arranged. Perhaps I should phrase the question this way: if you pick the thirteenth of any month of any year, is the probability that this is a Friday exactly 1 in 7? Or bigger, or smaller?
The American mathematician Bancroft Huntington Brown (1894-1974) once came up with the idea of figuring this out. Brown was not a great mathematician. At Dartmouth College he brought his students’basic ideas‘ and ‘elementary skills‘ at, we read in a obituary in his university’s alumni magazine. You shouldn’t have gone to Brown for higher mathematics.
That doesn’t matter, because Brown knew exactly how to ask interesting questions at the ‘basic level’. Like the Friday the 13th question above. The answer surprised him so much that he thought it would be fun to put the question to a wider audience. He approached the editors of the magazine The American Mathematical Monthly and in the May issue of the year 1933 his question appeared in the section ‘Problems and Solutions’.
In December, the magazine published contributor Raphael Robinson’s solution. At the time, the 22-year-old Robinson was pursuing his master’s degree at the University of California at Berkeley. He would become a great mathematician, but a student of his whom he later married: Julia Bowman. A beautiful documentary was made about her in 2008: Julia Robinson and Hilbert’s Tenth Problem†
Back to Brown’s question. In his solution, Robinson begins by stating that leap years occur once every four years, omitting years that are divisible by 100 but not by 400. (So 2000 is a leap year, but 2100 is not.) A period of 400 years, which so contains 97 leap years, consists of 146,097 days. That’s exactly 20,871 weeks, and so, Robinson noted, the calendar repeats itself every 400 years.
There are 4,800 months in such a 400-year cycle and thus as many thirteenths of the month. Then the big tallying starts. 685 times the thirteenth falls on a Monday. Tuesday: also 685 times. Wednesday: 687 times. Thursday: 684. Friday: 688. Saturday: 684. Sunday: 687. Friday takes the cake! The chance that a randomly chosen thirteenth falls on a Friday is 688 in 4,800, or 14.33 percent. The chance is slightly smaller for the other days of the week.
Responsible for the fact that the thirteenth falls relatively more often on a Friday is the Pope who introduced the current calendar on October 15, 1582. What was the Pope’s name? Gregory. How many? The thirteenth!
A version of this article also appeared in the newspaper of May 13, 2022