Non Leap YearLeap Year DateDay NumberMod 7Day NumberMod 7 13-Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Friday the 13th To see in which months Friday the 13 th can occur in the same calendar year, it’s helpful to number the dates 1 – 365. In the table below, the 13 th of every month is numbered as it falls in the year and each 13 th has its equivalent mod 7 day number: From the table, it’s evident that in a non-leap year, February and March 13 th are both equivalent to 2 mod 7, so a Friday the 13 th in February will be followed by a Friday the 13 th in March. These are the only months that have consecutive equivalent values for the 13 th in mod 7. In order for the 13 th 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 1 st would need to fall on a Thursday. From this table, we can see that there must be a Friday the 13 th every year, because the 13 th of the months fall on some value 0 – 6 at some point in the calendar year – this means that the 13 th 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.

To forecast the years where January 1 st will fall on a Thursday, we can look at recent years and find a pattern: Number of Days in the Following Year # Days in Followi ng Year mod 7 Sum of Days Mod 7 Since The Last Year that Started on Thursday 2004Thursday Saturday Sunday Monday Tuesday Thursday Friday Saturday Sunday Tuesday Wednesday Thursday Friday Sunday Monday Tuesday Wednesday Friday Saturday Sunday Monday Wednesday Thursday Friday Saturday Monday Tuesday Wednesday Thursday Saturday Sunday Monday Thursday3651

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