**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.