How many Friday the 13^{th}’s are there
in a year?
By
Carly Coffman
Prerequisite Skills
Ø Mod Mathematics
Ø Micrsoft Excel
Problem 1: Show that for any year there must be at least one
month and at most three
months for which the 13th of the month falls on Friday.
Suggestion: Consider using a spreadsheet and numbering the days of the year by mod 7.
1. Prove that Friday the 13th can occur in
two consecutive months only in February and March in a year that is not a leap
year. On what day of the week must January 1 occur for February and March to
have Friday the 13ths?
2. What is the next year in which this will
occur again?
3. Is there a pattern or cycle by which you
can determine which years between 2000 and 2100 that this will occur?
*Problem stated by Joe Hooten, Jr.
Solutions:
In order to prove
that there must be at least one month that contains a Friday the 13^{th}
and at most three months where the 13^{th} falls on a Friday, we will
look at a spreadsheet. The following
link will provide a spreadsheet using Microsoft Excel.
Friday
the 13^{th} Spreadsheet
From the spreadsheet,
I condensed the information into the chart below stating the number of days
between the 13^{th} of each month.
I then converted that number to mod 7.
NonLeap Year
Month 
Day of 13^{th} (Mod 7) 
January 
6 
February 
2 
March 
2 
April 
5 
May 
0 
June 
3 
July 
5 
August 
1 
September 
4 
October 
6 
November 
2 
December 
4 

Days
until 


Next
13th 
Mod 7 
Jan  Feb 
31 
3 
Feb  Mar 
28 
0 
Mar  Apr 
31 
3 
Apr  May 
30 
2 
May 
June 
31 
3 
June 
July 
30 
2 
July 
Aug 
31 
3 
Aug  Sep 
31 
3 
Sep  Oct 
30 
2 
Oct  Nov 
31 
3 
Nov  Dec 
30 
2 
From the charts
above, I drew the following conclusions:
·
Since February has 28 days,
which is 0 using mod 7, the 13^{th} will always be on the same day of
the week in February and March.
·
Since April through July
total 7 additional days using mod 7, April and July will have the same day of
the week for the 13^{th}.
·
Since September through
December total 7 additional days using mod 7, September and December have the
same day of the week for the 13^{th}.
·
March and November have the
13^{th} on the same day of the week because the additional days equal
21, or mod 7. So, February, March, and
November have the 13^{th} fall on the same day of the week. (Here is our maximum of three Friday the 13^{th}s
in a year).
·
January and October have the
13^{th} fall on the same day of the week since the additional days add
up to 21, or 0 using mod 7.
·
May, June, and August have no
combinations of mod 7 with any other months.
Therefore, the day of the week that the 13^{th} falls on is
unique for the year. (Here is our
minimum of one Friday the 13^{th} in a year).
Above, there are
7 combinations of months that either have a unique day of the week for the 13^{th}
or share a day of the week for the 13^{th}. Thus, each of the seven days of the week has
a month that the 13^{th} will fall on.
If you look back at the spreadsheet, I labeled 1 for Thursday 2 for
Friday, and so forth. You could change
the labels around as long as the days of the week are in the correct
order. Let’s look at the chart of the days
the 13^{th} falls on each month and assign different days of the week
to different mod numbers.
Ø
In order to get one Friday
the 13^{th }in a nonleap year, January 1^{st} must be on a
Wednesday, Friday, or Saturday.
Ø
In order to get two Friday
the 13^{th}s in a nonleap year, January 1^{st} must be on a
Sunday, Monday, or Tuesday.
Ø
In order to get three Friday
the 13^{th}s in a nonleap year, January 1^{st} must be on a
Thursday.
Now, let’s look
at the case for a leap year. Since
February has an extra day each leap year, the 13^{th} for each month
after February would move forward one day. Thus, we get the following chart.
Leap Year
Month 
Day of 13^{th} (mod 7) 
January 
6 
February 
2 
March 
3 
April 
6 
May 
1 
June 
4 
July 
6 
August 
2 
September 
5 
October 
0 
November 
3 
December 
5 

Days
until 


Next
13th 
Mod 7 
Jan  Feb 
31 
3 
Feb  Mar 
29 
1 
Mar  Apr 
31 
3 
Apr  May 
30 
2 
May 
June 
31 
3 
June 
July 
30 
2 
July 
Aug 
31 
3 
Aug  Sep 
31 
3 
Sep  Oct 
30 
2 
Oct  Nov 
31 
3 
Nov  Dec 
30 
2 
Now, we get the
following conclusions:
·
May, June, and October have
unique days of the week for the 13^{th}.
·
February and August have the
same day of the week for the 13^{th}, but it is unique from the other
months.
·
March and November have the
same day of the week for the 13^{th}, but it is unique from the other
months.
·
September and December have
the same day of the week for the 13^{th}, but it is unique from the
other months.
·
January and July have the
same day of the week for the 13^{th}, but it is unique from the other
months.
So, during a leap
year, there must be at least one Friday the 13^{th}, but there will be
no more than two.
Solutions, Continued:
In order to find the next
year that a Friday the 13^{th} will occur in February and March, we
need to figure out the numbering scheme for the first day of the year. Does the first day of the year always start
on the same day or is there a pattern in figuring out the first day of the
year?
If we look at how many days
are in a year, 365, and divide that by 7, we get 52 with a remainder of 1. So, using mod mathematics every new year will
start one day later than the previous year with the exception of years
following leap years. We will use
another spreadsheet to chart out the day that January first falls on each year. Since the pattern for years that are not leap
years are the same, we will be looking for the next year that is not a leap
year and that has the same first day as January did in 1998. Leap years also have 366 days, so when we
divide 366 by 7, we get 52 with a remainder of 2. Thus, the year after a leap year starts two
days ahead of the day of the week that the leap year started. For example, if the leap year 2000 started on
Tuesday, 2001 would start on Thursday.
*From our investigations above, a year that contains a
Friday the 13^{th} for February and March must start on a Thursday, so
we will assign 6 to Thursday.
Mod 
Day of Week 
1 
Sat 
2 
Sun 
3 
Mon 
4 
Tues 
5 
Wed 
6 
Thurs 
0 
Fri 

First Day of 

January (Mod 7) 


1998 
6 
1999 
0 
2000 (leap year) 
1 
2001 
3 
2002 
4 
2003 
5 
2004 (leap year) 
6 
2005 
1 
2006 
2 
2007 
3 
2008 
4 
2009 
6 
2010 
0 
Since February and March had
a Friday the 13^{th} in 1998, the next year this will occur will be
2009. Since 2004 is a leap year,
February and March will have different days for the 13^{th}.
So, will this occur every 11
years?
Let’s look at an extended
spreadsheet: Yearly Spreadsheet
If you find a pattern for the
years in which February and March will both have Friday the 13^{th}’s,
please email me at carly_coffman@gwinnett.k12.ga.us
(I will only receive them during the
school year.)