How many Friday the 13th’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.

### Problem 2:Observe that in 1998 both February and March have a Friday the 13th.

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 13th and at most three months where the 13th falls on a Friday, we will look at a spreadsheet.  The following link will provide a spreadsheet using Microsoft Excel.

From the spreadsheet, I condensed the information into the chart below stating the number of days between the 13th of each month.  I then converted that number to mod 7.

Non-Leap Year

 Month Day of 13th (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 13th 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 13th.

·       Since September through December total 7 additional days using mod 7, September and December have the same day of the week for the 13th.

·       March and November have the 13th on the same day of the week because the additional days equal 21, or mod 7.  So, February, March, and November have the 13th fall on the same day of the week.  (Here is our maximum of three Friday the 13ths in a year).

·       January and October have the 13th 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 13th falls on is unique for the year.  (Here is our minimum of one Friday the 13th in a year).

Above, there are 7 combinations of months that either have a unique day of the week for the 13th or share a day of the week for the 13th.  Thus, each of the seven days of the week has a month that the 13th 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 13th falls on each month and assign different days of the week to different mod numbers.

Ø    In order to get one Friday the 13th in a non-leap year, January 1st must be on a Wednesday, Friday, or Saturday.

Ø    In order to get two Friday the 13ths in a non-leap year, January 1st must be on a Sunday, Monday, or Tuesday.

Ø    In order to get three Friday the 13ths in a non-leap year, January 1st 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 13th for each month after February would move forward one day. Thus, we get the following chart.

Leap Year

 Month Day of 13th (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 13th.

·        February and August have the same day of the week for the 13th, but it is unique from the other months.

·        March and November have the same day of the week for the 13th, but it is unique from the other months.

·        September and December have the same day of the week for the 13th, but it is unique from the other months.

·        January and July have the same day of the week for the 13th, but it is unique from the other months.

So, during a leap year, there must be at least one Friday the 13th, but there will be no more than two.

Solutions, Continued:

In order to find the next year that a Friday the 13th 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 13th 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 13th 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 13th.

So, will this occur every 11 years?

Let’s look at an extended spreadsheet:

If you find a pattern for the years in which February and March will both have Friday the 13th’s, please e-mail me at carly_coffman@gwinnett.k12.ga.us  (I will only receive them during the school year.)