Friday the 13th

1. Friday the 13th To see in which months Friday the 13th can occur in the same calendar year, it’s helpful to number the dates 1 – 365. In the table below, the 13th of every month is numbered as it falls in the year and each 13th has its equivalent mod 7 day number: From this table, we can see that there must be a Friday the 13th every year, because the 13th of the months fall on some value 0 – 6 at some point in the calendar year – this means that the 13th will fall on every day of the week at some point in the calendar, and this must include Friday. The most Friday the 13ths that can occur in one year would occur in February, March and November in a non-leap year. In a leap year, three Friday the 13ths could occur in the months January, April and July. From the table, it’s evident that in a non-leap year, February and March 13th are both equivalent to 2 mod 7, so a Friday the 13th in February will be followed by a Friday the 13th in March. These are the only months that have consecutive equivalent values for the 13th in mod 7. In order for the 13th of February and March to occur on a Friday, January 13th must be on a Tuesday (6 mod 7 is on a Tuesday, 0 mod 7 is on Wednesday, 1 mod 7 is on a Thursday and 2 mod 7 is on a Friday). This would indicate that January 1st would need to fall on a Thursday.

2. To forecast the years where January 1st will fall on a Thursday, we can look at recent years and find a pattern:

3. By using the sum of days in each year mod 7, we can predict the years that will have consecutive months with Friday the 13th (in February and March, i.e. when January 1st is on Thursday): 2009, 2015, 2026, 2032, 2037, 2043, 2054, 2060, 2065, 2071, 2082, 2088, 2093, 2099.