The Department of Mathematics Education

FRIDAY THE 13 TH.
By Godfried Lawson

First, I will 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.


Second:

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.

From my observation I realize that Friday the 13 appears when the first of the month is Sunday.
This is true because, the previous Friday before friday the 13th is Friday the 6th, therefore Sunday must be the first.
I will focus my research on the first of the month.
Is the first of the month falls at least once each day of the week?
Let conduct some investigation.
If each month has 28 days, the first of the month will be the same day for the rest of the year.
Since we have some months with 31 days and some with 30 or 29, then the following condition appears:
The first day in January will be advanced by 3. That means if January the 1st is Monday, then February the 1st will be  Thursday.
This led me to construct the following table:

+3 ___ +0___+3___+2 ____+3___+2___+3___+3___+2___+3___+2___+3___

or ( for leap years.)

+3___+1____+3___+2____+3___+2___+3___+3___+2___+3___+2___+3____
 
 

Now, using the chart above I will construct a table that shows that there exists at least one day of the week which is the first of the month.
 
 
1st 1st 1st 1st 1st 1st 1st 1st 1st 1st 1st
  Jan   Feb   Mar   Apr   May   June  July Aug   Sep Oct Nov Dec
    +3     +0     +3     +2     +3     +2     +3     +3     +2     +3     +2     +3
1st Mon Thur Thur Sun Tue Fri Sun Wed Sat Mon Thur Sat
1st Tue Fri Fri Mon Wed Sat Mon Thur Sun Tue Fri Sun
1st Wed Sat Sat Tue Thur Sun Tue Fri Mon Wed Sat Mon
1str Thur Sun Sun Wed Fri Mon Wed Sat Tue Thur Sun Tue
1st Fri Mon Mon Thur Sat Tue Thur Sun Wed Fri Mon Wed
1st Sat Tue Tue Fri Sun Wed Fri Mon Thur Sat Tue Thur
1st Sun Wed Wed Sat Mon Thur Sat Tue Fri Sun Wed Fri

The importance of this table is to show the sequence of the first the of the month if the firs of January falls on Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, or sunday.
From the above table i can say that:

For a non leap year:

When the first of January falls on Wednesday, Friday, Saturday, or Sunday we expect to have one Friday the 13th.
When the first of January falls on Monday or Tuesday, we expect to have two Fridays the 13th.
When the first of January falls on Thursday, we expect to have three Fridays the 13th.

For a leap year:

When the first of January falls on Tuesday, Thursday, Friday, or Saturday we expect to have one Friday the 13th.
When the first of January falls on Monday, Wednesday, or Sunday, we expect to have two Fridays the 13th.
 

From the table, we can conclude that there exists at least one of the seven days of the week that represent the first of the month.
Also the table shows that each year follows the same pattern as the day of the week.
During a leap year the table will follow the same sequence; expect that all Saturdays become Sundays.

Friday the 13th occurs in two consecutive months only in February and March in a
year that is not a leap year. This happen when January the first falls on Thursday

 Observe that in 1998 both February and March have a Friday the 13th.
What is the next year in which this will occur again?

Let's construct a table that shows the day of first of January from 1997 to 2027. I will observe the resulting pattern, then draw a conclusion.The leap years are in red.
 
Year 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
MON 1 1 ## 1 1
TUE 1 1 1 1 ##
WED 1 1 ## 1 1 1
THU 1 1 1 1 ## 1
FRI 1 ## 1 1 1 1
SAT 1 1 1 ## 1 1
SUN ## 1 1 1 1 ##

In 1998, the first of January fell on Thursday and in 2026 the first on January will fall on Thusday. Notice that those two dates have the same pattern. The first of January after the leap year is Wednesday and if we had two consecutive fridays the 13th in 1998, we must have also two consecutive fridays the 13th in 2026.
This even happens every 28 years.
The equation of the frequency of this event is:

  y =  28x +1998,          where y, is the year the event happens and x, the  number of events. x is a positive integer.
  y =  - 28x + 1998
 

Source: IBM, July 1996, The Year 2000 and 2-Digit Dates: A Guide for Planning and Implementation, Fourth Edition, GC28-1251-03, Chapter 5. Testing Techniques for Year 2000 Changes.
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You can affect the system date change (on Intel-based PCs) as follows: use the configuration utility that sets the time and date or execute the DATE command in DOS Version 3 Release 3 or later.

Test the setting and display of special dates, including:
1900/2/29 should fail - the year 1900 is not a leap year
1996/2/29 should succeed - the year 1996 is a leap year
2000/2/29 should succeed - the year 2000 is a leap year
00/01/01 should display an unambiguous 4-digit-year date, the value of which depends on the application. For example, 1900/01/01, 2000/01/01, and so on.
This information must be used to test your computer.


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