Many shudder at the thought of Friday the 13th. Myths, legends and horror films have made it a sign of bad luck.
History has also had a lot of bad Friday the 13th Friday the 13th. September 1940, Nazi forces bombed Buckingham Palace in London. Friday the 13th January 2012, the Italian cruise ship Costa Concordia hit a rocky outcrop and overturned, resulting in a total evacuation and 32 deaths. And Friday the 13th. On September 10, 1996, Tupac Shakur succumbed to his gunshot wounds from six days earlier.
But the 13th day of a month that happens to fall on a Friday is just a day. Superstitions can be dispelled with mathematics. Using number theory, you can easily demonstrate that there is not 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.
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Adventures in number theory and calendars
To keep things simple, let’s first focus on years with 365 days. We can start by calculating which sequentially counted day of the year the 13th of the month will fall on using the number of days in each month as a guide. So January 13th is the 13th day of the year, February 13th is the 44th day, March 13th is the 72nd day, and so on. Here is a table summarizing what this approach reveals.
A week is a repeating pattern of seven days. This means, for example, that the first, eighth, 15th, 22nd and so on always fall on the same day of the week. Therefore, it is possible to determine which 13th day of the month falls 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 examining by 7. The remainder will tell you which day of the first week of the year that date matches.
This may sound complicated, but it is actually quite simple: when you divide the 13th day of the year by, say, 7, you get 1 with the remainder of 6. This means that the 13th day of the year falls on the same day of the week as sixth day of the year. You can repeat this process on the 13th day of every month, resulting in the following.
Each day of the week, marked 0 to 6, appears at least once in the list. (And for those who skip the math and skim the tables, the 0th day of the year is actually the 7th day.) This means that in a 365-day year, every weekday will be the 13th of a month at least once (days 0, 1, and 3 only appear once each). Weekdays can also be the 13th of the month twice (day 4, 5 and 6). And there is one day of the week that will be the 13th of the month three times (day 2).
So if the second day of a normal year is a Friday, there will be three Fridays the 13th. 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.
Here, too, we must now investigate which of these dates correspond to which days of the week. Again, divide the number of days by 7 and calculate the remainder.
In leap years, the days of the week change, but the distribution of weekdays remains the same. Each day of the week is shown at least once (day 0, 1 and 4). And there are still weekdays that are the 13th of the month twice (day 2, 3 and 5), as well as one weekday that occurs three times (day 6). If Friday is the sixth day of the first week of a leap year, there will be three Fridays on the 13th of the 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 the month. Superstitious people must face their fears every year; there is no way around it.
The 13th of the month is usually a Friday
Statistically, the 13th of the month falls on a Friday more often than on any other day of the week. This may seem surprising, but it can also be demonstrated with mathematics.
To do this, we must consider the peculiarities of the Gregorian calendar. If our years were always exactly 365 days, the distribution of weekdays would repeat every seven years. To put this in a more formal way, every seven years would start a new periodic cycle. After seven years, each date would correspond to the same day of the week as it was during the first year. In this seven-year cycle scenario, the 13th day of a month will be the same number of times each weekday: each weekday falls on the 13th of the month exactly 12 times during a seven-year period.
The Gregorian calendar is more complicated. It is known that there is a leap day every four years – which shakes things up. Adding to that complication, the distribution of weekdays is such that the cycle only repeats itself every 28 years, a span in which there are seven leap days (in years 0, 4, 8, 12, 16, 20, and 24).
But wait – our calendar system has even more quirks. In principle, there is a leap year every four years, except every 100 years, when the leap day is omitted. And there is an exception that exception when the year is divisible by 400; in that case, the leap day is added as normal. That is why the year 2000 was still a leap year despite being divisible by 100.
With that information, you can calculate that a full periodic cycle, where the days of the week will fall on the same dates and the leap year patterns repeat, takes 400 years. This period comprises 365 ordinary days per year times 400 years, or 146,000 ordinary days plus 97 leap days (400 / 4 – 3, where 3 represents the leaped leap years of 100, 200 and 300). Unfortunately, these leap days are not evenly distributed, which makes it impossible to break this into a smaller cycle. As an example of this irregularity, between 2000 and 2099 there will be 25 leap days, including the start year, but in the next three centuries there will be only 24 leap days.
Using a computer program, you can perform a concrete calculation to work through the implications of this pattern. The year 2000 was a leap year, and 1 January fell on a Saturday. Based on this information, you can calculate how many 13ths of the month will fall on which days of the week for the next 400 years – that is, from January 1, 2000 to December 31, 2399.
The result is:
In other words, the 13th of the month will be a Friday more times than any other day of the week. It’s bad news for superstitious people. But it’s worth remembering how arbitrary all this is. If 1 January 2000 suddenly happened to be a Sunday, another weekday would line up with the 13th with greater frequency.
This article originally appeared in Spectrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the help of artificial intelligence and reviewed by our editors.
