FRIDAY THE 13^TH

By: Kelli Parker

An interesting activity to work on with students is how many Friday the 13^ths can occur in one year. How do you discover the answer to the question? You can look at lots of old calendars and try to predict what will happen in the years to come, or you can look at it mathematically. Will there always be a Friday the 13^th each year? What is the maximum number of Friday the 13^ths you could have in one year? Some work with modular arithmetic will help us answer this question. (Note: we are working with a non-leap year.)

We really only need to look at twelve days: the 13^th day of each month. We can determine what day of the week each 13^th will fall on, without even specifying what actual day it is! We do this by working modulo 7. Since there are only 7 days in a week, each day would have a number 0 through 6. We can then go through and see what day of the year (out of 365) each 13^th of every month is, and then reduce it “mod 7”. That just means we divide the number by 7, and see what the remainder is. If the number is a multiple of 7, the remainder is 0 and so that number is equivalent to 0. If the remainder is 3, then the number is equivalent to 3 and we call it 3.

Each of the numbers in the second column was obtained by adding the total number of days in each month to the day the 13^th fell on. So January 13^th + 31 days in January gives us that the 13^th of February will be on the 44^th day of the year. Adding 28 days for February to 44 gives us 72 for the day of the year of March 13^th. Keep doing this and you get all the days of the year for the 13^th of each month! (This is great practice in remembering how many days are in each month!)

The numbers in the third column were obtained by dividing the corresponding number in the second column by 7 and listing the remainder. Excel will actually do the modular arithmetic for you!

To get Excel to do modular arithmetic, you click on the cell you want the mod reduction to be in, and then type “=”. Next you want to click the arrow next to the Summation sign on the menu bar () and choose the “more functions” option. Select “Math & Trig” in the menu on the left, and then scroll down until you see “MOD” in the box on the right.

So what do the modular arithmetic numbers in the third column tell us? They tell us how many possible Friday the 13ths there could be in each year. Each number 0-6 represents one of the 7 days of the week, and each number appears at least once on the 13^th. So the 13^th day of at least one month each year will be a Friday.

Depending on what the first day of the year is, as in what day of the week January 1^st falls on, the number of Friday the 13ths will vary. We can see this in the fact that some of the numbers from 0-6 appear more than once in the table. Let’s look at an example, and use the numbers that excel generated for us. We’ll assign each day of the week a number, with 0 being the first day of the week, which we’ll call Sunday. So if Sunday is 0, Monday is 1, Tuesday is 2, and so on. Let’s look at the table and see how many Fridays will be on the 13^th of a month.

So if Sunday is 0, then Friday is 5. So using the table, we only have two 5’s, one in the month of April and one in July. So in a non-leap year, if January 1^st is on Sunday, we will have 2 Friday the 13ths, in April and July. If this were a classroom activity with students, a teacher could ask all kinds of questions like this to prompt students to use the Excel-generated table to think about the problems.

Students are no doubt going to ask, out of curiosity, “But what would happen in a leap year?” We can easily answer that question, just making a few modifications to our Excel spreadsheet.

So we see here in the new table that when we add 29 days for February instead of 28, there are no longer 3 occurrences of any of the days of the week. So the most Friday the 13ths we could have in a leap year is 2.

An extension activity for students would be discussing if a year has 3 Friday the 13ths, how many years it would be until this happens again? You can do this concretely by saying, “In the year 2000, there were 3 Friday the 13ths. When will this happen again?” A further extension would be giving students a particular year and asking them to determine how many Friday the 13ths would occur in that year.

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