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 HavilŐs 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 
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.

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.