FRIDAY 13

By

Rhonda Iler

Prove that there is at least one and no more than three Friday the 13ths during each year.

We know that if there is a Friday 13th in any month, the first day of the month is on a Sunday.

January, March, May, July, August, October, and December all have 31 days.  We can write the number of days as

3(mod7) since the remainder is 3 when 31 is divided by 7.

April, June, September, and November have 30 days.  The number of days in each of these months is written 2(mod 7).

We know February has 28 days in a normal year and can be written as 0(mod 7) and during leap year as 1(mod 7).

Let January 1 be x.

In a normal year we have the following:

Jan. 1 = x (mod 7)

Feb. 1 = Jan. 1 + 31 days of January

= x (mod 7) +3(mod 7)

= (x+3) (mod 7)

Mar. 1 = Feb. 1 + 28 days

= (x+3) mod 7 + 0(mod 7)

=(x+3) mod 7

Continuing the pattern of adding the number of days we get:

Apr. 1 = (x+6) mod 7

May 1 = (x+1) mod 7

June 1 = (x+4) mod 7

July 1 = (x+6) mod 7

Aug. 1 = (x+2) mod 7

Sept.1 = (x+5) mod 7

Oct. 1 = x (mod 7)

Nov. 1 = (x+3) mod 7

Dec. 1 = (x+5) mod 7

Similarly for leap year:

Jan. 1 = x (mod 7)

Feb. 1 =(x+3) mod 7

Mar. 1 = (x+4) mod 7

Apr. 1 = x (mod 7)

May 1 = (x+2) mod 7

June 1 = (x+5) mod 7

July 1 = x (mod 7)

Aug. 1 = (x+3) mod 7

Sept.1 = (x+6) mod 7

Oct. 1 = (x+1) mod 7

Nov. 1 = (x+4) mod 7

Dec. 1 = (x+6) mod 7

So If January 1st is a Sunday then January is x mod 7 and we have a Friday the 13th in January.

If January 1st is a Saturday, then the months with (x+1) mod 7 will have a Friday the 13th.

If January 1st is a Friday, then the months with (x+2) mod 7 will have a Friday the 13th.  The pattern continues as seen in the table below:

 Day of the week for January 1st Months that will have a Friday the 13th Sunday X mod 7 Saturday (x+1)mod 7 Friday (x+2)mod 7 Thursday (x+3)mod 7 Wednesday (x+4)mod 7 Tuesday (x+5)mod 7 Monday (x+6)mod 7

Thus we have the following calendar:

 Sunday Monday Tues. Wed. Thurs. Friday Saturday. Jan. n,l Feb. n,l Mar. l n Apr. n May l n June n l n July l Aug. l n Sept. l n Oct. n l Nov. l n Dec. l n

(n=normal year, l=leap year)

To clarify the table: In a normal year, if the year begins on a Sunday, there are Friday the 13ths in January and October.  If the year begins on a Monday, then there are Friday the 13ths in April and July.  For leap year, if the year begins on a Wednesday, there are Friday the 13ths in March and November.

We can see that every year has at least one Friday the 13th because each column has all n and l.

During a normal year, there is never a Friday the 13th in July.

If the year begins on Thursday, there will be 3 Friday the 13ths in a normal year.  For every other starting day, there will be 2 or 1 Friday the 13th in normal years and in leap years.

Other information I found while working on this problem: This information came from Julian HavilŐs book Nonplussed.  I am sharing it because I thought it was interesting, but because I developed it.

We can further show that the 13th day of the month falls more frequently on a Friday that any other day of the week.

Gauss formulated a calendar formula for the day of the week.  W = 1 corresponds to Monday, w = 2 to Tuesday and so on.  D corresponds to the day of the month.  And the variable y corresponds to the year. C = y/100 and is the 2 digit century.  The month is represented by the letter e given in the following table where m is the number of the month:

 M 1 2 3 4 5 6 7 8 9 10 11 12 E 0 3 2 5 0 3 5 1 4 6 2 4

G = y – 100c if the 2 digit year of the century.

If m = 1 or 2, then y is replaced by y – 1 in the calculations of c and g.

The century c is associated with the variable f and given the following values:

 C mod 4 C f 0 16, 20, etc 0 1 17, 21, etc 5 2 18, 22, etc 3 3 19, 23, etc 1

GaussŐs formula for the day of the week of any date in the Gregorian calendar was

W = (d + e + f + g + [.25g]) mod 7 where [] represents the greatest integer function.

The formula was used to generate the following table using a computer.

## The following table illustrates the frequency that the 13th of the month falls on each day of the week over a course of 400 years:

 Mon Tues Wed Thur. Fri Sat Sun Total Jan 57 57 58 56 58 56 58 400 Feb 58 56 58 57 57 58 56 400 Mar 56 58 57 57 58 56 58 400 Apr 58 56 58 56 58 57 57 400 May 57 57 58 56 58 56 58 400 June 58 56 58 57 57 58 56 400 July 58 56 58 56 58 57 57 400 Aug 58 57 57 58 56 58 56 400 Sept 56 58 56 58 57 57 58 400 Oct 57 58 56 58 56 58 57 400 Nov 56 58 57 57 58 56 58 400 Dec 56 58 56 58 57 57 58 400 Total 685 685 687 684 688 684 687 4800

Clearly, there appears to be more 13ths on Fridays than any other day of the week.