Source: Adapted from *American Mathematical Monthly*, Problem E 1541, September 1963.

**Hint:** Pull out some past calendars....start gathering some data. Do you see some patterns developing?

**Solution Commentary:** First, look at the day of a week that each month starts. For a nonleap year, months can be separated into seven "equivalence" classes: {January, October}, {February, March, November}, {April, July}, {May}, {June}, {August}, and {September, December}.

Using the same criteria, for a leap year, the seven "equivalence" classes are: {January, April, July), {February, August}, {March, November}, {September, December}, {May}, {June}, and {October}.

Thus, based on these classes and on whatever day a Friday-the-13th happens to fall, the minimum number and maximum number of times it can occur are one and three respectively.

**Follow-up Question:** So, is there a pattern of years for the cases of one time, two times, or three times?