Problem:
1. Show that for any year there must be atleast 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.
Observe that in 1998 both February and March have a Friday the 13th.
2. Prove that Friday the 13th can occur in two consecutive months only in February and March in a year that is not a leap. On what day of the week must January 1 occur for February and March to have Friday the 13ths?
3. What is the next year that this will occur again?
4. Is there a pattern or cycle by which you can determine which years between 2000 and 2100 that this will occur?
The first thing that came
to my mind when initially reading the problem is to use a spreadsheet.
This would enable me to organize the input so that I would clearly
be able to see what is being asked. In the problem statement,
it is suggested to number the days of the year using mod 7.
Using modular
arithmetic is a useful way of organizing the days of the
year, especially when we will be looking at all 365 or 366.
I have started the problem in two ways. If I were non-math
oriented, I may not have seen to use modular arithmetic, and need
another method of finding how many times in a year that Friday
the 13th falls on. I created a spreadsheet using excel that
lists each day of the year, and the corresponding number of that
day of the year. Two things are important, the day of the
week (Friday), and the day of the months (13th). On
January 1st, there are seven possibilities for what day of the
week it is, so all seven must be examined.
Click here to view Spreadsheet
I
organized the spreasheet starting with Jan.1 on a Monday, then
finding the Friday for each day thereafter. This process
was then repeated for Jan.1 starting on Tuesday, Wednesday, and
so on. Once all the Fridays were found, I could then look
for which Fridays fell on the 13th of a given month. By
looking at the spreadsheet, it is obvious that for Jan.1 falling
on any seven possible days of the week, Friday the 13th occured
at least once, and at most three times.
Note:
I have only illustrated this method usng a non-leap year.
The same method can be used for a leap year.
Now I want to illustrate
a solution to the first problem using modular arithmetic.
Click above to review modular arithmetic. To use modular
arithmetic, we know there are seven possible days in a week, so
if we count by seven and use the remainders, we could certainly
be able to tell which day of the week each Friday 13th fell on
once we have examined the possibilities for Jan.1 just as was
done above. I'll use excel for this graph as well.
Click here to view
Spreadsheet 2
From the spreadsheet, we can
see exactly what day of the year the 13 falls on as well as illustrating
that there must be atleast one month and at most three months
for which the 13th of the month falls on Friday (this can be seen
by looking at which days have the same remainders). By using
modular arithmetic, it is apparent at the amount of time saved
versus the spreadsheet above, and modular arithmetic will help
me to solve the next set of problems.
To finish this first problem,
all that remains is to examine the seven different cases for which
day of the week Jan.1. Below is a table for both non-leap
year and leap-year.
Non-Leap Year Leap Year
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For a brief explanation
of the table, If Jan.1 falls on Sunday, this gives us a remainder
of 6, and so on for the other six possible days of the week.
Once we have completed this for both non-leap and leap years,
we can then match this to our earlier table to see what month
these remainders occured.
Now we want to see when
this occurs again. It may be easier to ask when is the next
year that that January 1st will occur on Thursday in a non-leap
year. We are using 1998 as the last year that Friday the
13th occured in two consecutive months in a year that is not a
leap year. We said earlier that we are looking for the next
time that January 1st will occur on Thursday, and since there
are seven days in a week, we would think that this may occur again
seven years from now, but we must consider when a leap year occurs
to offset the days of the week beginning on January 1st.
A leap year occurs every 4 years and when a lepa year does occur,
the day of the week for the upcoming year is pushed ahead two
years instead of only one. To solve this question, in 1998,
Jan 1st occurs on Thursday, and in 1999, Jan 1st occurs on Friday,
etc. 2004 would be the next year Jan 1st occurs, but this
is a leap year with an extra day in February, thus giving us 31
days between Friday the 13ths in the two months. The following
year, 2005, Jan 1st occurs on Saturday, Jan 1st occurs on Sunday
in 2006, and so on until 2008. This leap year moves the
next year, 2009, to January 1st falling on Thursday, which will
give us Friday the 13ths falling in February and March.