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

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

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

Day of the week for January 1^st	Months that will have a Friday the 13^th
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 13^th in normal years and in leap years.

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

We can further show that the 13^th 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

Gauss’s 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 13^th 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 13^ths on Fridays than any other day of the week.