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Paraskavedekatriaphobia, A Special Case of Triskaidekatriaphobia

This year (2006), two Friday-the-13ths will occur (i.e. in January and October).

What is the minimum and maximum number of least number of Friday-the-13ths that can occur in any year?

A Piece of Trivia: A long time ago, B.H. Brown established that the 13th of the month is more likely to be a Friday than any other day of the week.


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?