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Many people shudder at the thought of Friday the 13th. Myths, legends and horror films have turned it into an omen of bad luck.

History has also had plenty of bad Friday the 13ths. On Friday, September 13, 1940, Nazi forces bombed Buckingham Palace in London. On Friday, January 13, 2012, the Italian cruise ship Costa Concordia struck a rocky outcropping and capsized, resulting in a total evacuation and 32 deaths. And on Friday, September 13, 1996, Tupac Shakur succumbed to his gunshot wounds from six days prior.

But the 13th day of a month that happens to fall on a Friday is just a day. Superstitions about it can be dispelled with mathematics. Using number theory, you can easily demonstrate that there isn’t a single year without this ominous date. In fact, the 13th day of a month falls on a Friday more often than on any other day of the week.

Adventures in Number Theory and Calendars

To keep things simple, let’s first focus on years with 365 days. We can begin by calculating which sequentially counted day of the year the 13th of a month will fall on, using the number of days in each month as a guide. So January 13 is the 13th

day of the year, February 13 is the 44th day, March 13 is the 72nd day, and so on. Here’s a table summarizing what this approach reveals.

During a year with 365 days, January 13 falls on the 13th day of the year. February 13 falls on the 44th day of the year; March 13 falls on the 72nd day; April 13 falls on the 103rd day; May 13 falls on the 133rd day; June 13 falls on the 164th day; July 13 falls on the 194th day; August 13 falls on the 225thday; September 13 falls on the 256thday; October 13 falls on the 286th day; November 13 falls on the 317th day; and December 13 falls on the 347th day

A week is a repeating pattern of seven days. That means, for example, that the first, eighth, 15th, 22nd, and so on always fall on the same day of the week. Therefore, it’s possible to determine which 13th days of a month fall on a given day of the week. To do this, simply divide the number of days in the year that have already passed on the date you’re investigating by 7. The remainder will tell you which day of the first week of the year that date matches.

This might sound complicated, but it’s actually quite simple: when you divide the 13th day of the year by 7, for example, you get 1 with a remainder of 6. This means that the 13th day of the year falls on the same day of the week as the sixth day of the year. You can repeat this process for the 13th day of each month, which result in the following.

During a year with 365 days, January 13 and October 13 will fall on the same day of the week as the sixth day of the year. February 13, March 13, and November 13 will fall on the same day of the week as the second day of the year. April 13 and July 13 will fall on the same day of the week as the fifth day of the year. May 13 will fall on the same day of the week as the seventh day of the year, recorded in the chart as the zeroth day of the year. June 13 will fall on the same day as the third day of the year. August 13 will fall on the same day of the week as the first day of the year. September 13 and December 13 will fall on the same day of the week as the fourth day of the year.

Each day of the week, labeled 0 to 6, appears at least once in the list. (And for those skipping the math and skimming the tables, the 0thday of the year is actually the 7th day.) This means that in a year with 365 days, each day of the week will be the 13th of a month at least once (days 0, 1, and 3 each appear only once). Days of the week can also be the 13th of a month twice (days 4, 5, and 6). And there is one day of the week that will be the 13th of a month three times (day 2).

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