Friday the 13^th

By

Jeffrey R. Frye

Problem 1: Show that for any year there must be at least one month and at most three months for which the 13th of the month falls on Friday.

By using modular arithmetic, we can derive a method to find when the 13^th of the month will occur. Since each week has 7 days, using Mod 7 will give the answer we are searching for. Since 13 is not evenly divisible by 7, a remainder of 6 results. 6 mod 7 will represent the date of 13th during the month. With this approach, we find that there is at least one and at most 3 months for which the 13^th of the month falls on Friday. Click here to see the table and results for a non-leap year. Click here to see the results for a leap year.

Observe that in 1998 both February and March have a Friday the 13th.

Problem 2a: Prove that Friday the 13th can occur in two consecutive months only in February and March in a year that is not a leap year. On what day of the week must January 1 occur for February and March to have Friday the 13ths?

The only month that is evenly divisible by 7 is February during a non leap year. In non leap years, all March dates will fall on the same day of the week as they did during February. So, if Friday the 13^th occurred in February, then it will also occur during March. It will not occur in any other consecutive months, since all other months are either 30 or 31 days which are not evenly divisible by 7. During a leap year, since February has 29 days, no consecutive months are possible. As we can see from the table above for a non leap year, when January 1 is on Thursday, February and March will both have a Friday the 13^th.

Problem 2b: What is the next year in which this will occur again? Is there a pattern or cycle by which you can determine which years between 2000 and 2100 that this will occur?

We know that the next year that back to back months occurred was 2009, since it has just occurred. The next year that this will occur is in 2015, followed by 2026, 2037, 2043, 2054, 2065, 2071, 2082, 2093, and 2099. Click here to see the 1998 calendar and an explanation of the pattern that is used to determine when the back to back years will occur.