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**FRIDAY THE 13 ^{TH}**

**By: Kelli Parker**

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** ^{ }** An
interesting activity to work on with students is how many Friday the 13

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 Òmod 7Ó. 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 Ò=Ó. Next you want to click
the arrow next to the Summation sign on the menu bar () and choose the Òmore functionsÓ option. Select ÒMath & TrigÓ in the menu on
the left, and then scroll down until you see ÒMODÓ 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. LetÕs look at an example, and use the
numbers that excel generated for us.
WeÕll assign each day of the week a number, with 0 being the first day
of the week, which weÕll call Sunday.
So if Sunday is 0, Monday is 1, Tuesday is 2, and so on. LetÕs 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
5Õs, 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, ÒBut what would
happen in a leap year?Ó We can
easily answer that question, just making a few modifications to our Excel
spreadsheet.

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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,
ÒIn the year 2000, there were 3 Friday the 13ths. When will this happen again?Ó 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.