March 13, 2026 5 min read Add Us On Google Add SciAm Why Friday the 13th is a mathematical inevitability No bad luck here—just lots of fascinating math that explains why the 13th of a month so often falls on a Friday By Manon Bischoff edited by Daisy Yuhas Liudmila Chernetska/Getty Images 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. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing . By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. 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. View Detailed Image Description 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. View Detailed Image Description 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). So if the second day of a regular year is a Friday, then there will be three Friday the 13ths. This is the case in 2026. Let’s Talk about Leap Years But what if the year is a leap year, with 366 days? The previous calculations can be performed analogously, except in this case, February has 29 days. View Detailed Image Description Here, too, we must now examine which of these dates corresponds to which days of the week. Again, divide the number of days by 7 and calculate the remainder. View Detailed Image Description During leap years, the days of the week shift, but the distribution of days of the week remains the same. Each day of the week appears at least once (days 0, 1 and 4). And there are still days of the week that are the 13th of the month twice (days 2, 3 and 5), as well as one day of the week that occurs three times (day 6). If Friday is the sixth day of the first week during a leap year, then there will be three Fridays on the13th of a month. With this analysis, we can show that there will be at least one and at most three Fridays each year that are the 13th of a month. Superstitious people must face their fears every year; there’s no way around it. The 13th of a Month Is Most Often a Friday Statistically speaking, the 13th of a month falls on a Friday more often than on any other day of the week. This may seem surprising, but it, too, can be demonstrated with mathematics. To do this, we need to consider the peculiarities of the Gregorian calendar. If our years were always exactly 365 days, then the distribution of weekdays would repeat every seven years. To put this in a more formal way, every seven years would begin a new periodic cycle. After seven years, every date would align with the same d