Some consider Friday the 13th (any Friday that falls on the 13th day of a month) unlucky.

In a calendar year, how many Friday the 13th can occur? Find both the minimum and maximum.

#### Solution

A year has at least one and at most three Friday the 13th.

- Start by figuring out how many days are between the beginning of the year and the 13th of every month.
- January 13 is the 13th day of the year, February 13 is the 44th day, etc.
- We get 13, 44, 72, 103, 133, 164, 194, 225, 256, 286, 317, and 347 for normal years
- And 13, 44, 73, 104, 134, 165, 195, 226, 257, 287, 318, and 348 for leap years

- Divide by 7 and look at the remainders. Each remainder represents a day of the week (but each number may represent a different day of the week depending on the year).
- We get remainders of 6, 2, 2, 5, 0, 3, 5, 1, 4, 6, 2, and 4 for January-December, respectively, for normal years
- And 6, 2, 3, 6, 1, 4, 6, 2, 5, 0, 3, and 5 for leap years

- All possible days of the week (0-6) show up at least once among the remainders, even in leap years, so there must be at least one Friday the 13th every year.
- The most common remainder appears at most three times (February, March, and December all have remainder of 2), even in leap years (January, April, and July all have remainder of 6), so there can be at most three Friday the 13th each year.