Fun With Math & Numbers  The Calendar

You will
often encounter problems that sound a little bit like "Given two
consecutive integers, the square of which..., etc." While these are fun to solve, they sometimes
strike a few students as having no relationship to "real world
problems." I just recently was
given a real problem that reads similar to this, as it involved having to
deduce and work with integer relationships. A group I
belong to meets once a week for most of the year. It always meets on the same day of each week in each month and
typing up the schedule for the year is so simple  you merely copy the
schedule from last year, modify the dates to correspond to the correct day of
the week and, perhaps, change the name of the individual in charge of that
meeting. It is so simple that we
assigned the task to one of our youngest members who was in his early
twenties. He did
an excellent job of typing up the schedule for the year until we all noticed
that his only "goof" was in the month of February 2004. He totally ignored the fact that February
2004 had five Sundays in the month
and, once flagged, he kept shaking his head in disbelief that such a short
month could actually have five Sundays.
In fact, he was heard to utter "I have never experienced a
FiveSundayFebruary in my life."
We will shortly show that his intuition was correct in that regard. The rest
of the group also became fascinated with the discovery of a FiveSundayFebruary
and even the eldest were claiming that "there never had been such an
event before." Almost in unison,
they turned, pointed to me and assigned me, as the "resident
mathematician," with the challenge of explaining this
"mystery". Their charge, if
I understood it correctly, was to predict whether and when this could ever
happen again. The
topic for the meeting wasn't that interesting, so I accepted the charge and
sat in the back of the room and began scribbling trial equations on a piece of
paper. I surprised myself by
"cracking the code" of the calendar in less than 15 minutes. A. Recasting The
Problem Mathematically. For any problem that your
boss, or a math instructor, or others, present to you, that sounds
mathematical, you should initially spend some time trying to recast the
problem into a precise mathematical statement. The way they presented it to me it sounded like a research
problem you MIGHT locate on the Internet.
But they wanted the solution today and, of course, I had a reputation
to maintain. At the
rear of the room, I opened my brief case and at least located my check book
which has three years of calendars on the back of the record pages. The first thing I noticed was what I
dubbed "1stdaycreep." I
noticed that if the 1st of February is on a Sunday during one year, it
becomes a Monday on the following year, a Tuesday on the second year, etc.
but only for a normal year.
Immediately following a leap year, and 2004 was a leap year, the 1st
day jumps by two days. That is
because 365 divided by 7 (days in a week) has a remainder of 1 and 366
divided by 7 has a remainder of 2. Next I
noticed that "squeezing" five Sundays into such a short month is
only possible when two conditions are met  (a) the 1st of February is itself
a Sunday and (b) we are in a leap year so that the 29th
of February forms the fifth, and final, Sunday of the month. So I was no able to recast their request
into more concise, mathematical expression. Find the next occurrence of a leap year when February 1 is
also a Sunday. B. Next, work
towards a formula that will force a solution.
My Astronomy background told me that there are really 365 ¼ days in a
year, we only observe 365 of them for three years, and then observe a 366
dayyear to make up for the error. I
was NOT about to work with any fractions in my equations, so I came up with
my own "time interval" which I dubbed a "leap interval,"
or four years, and noticed that the number of days in my "leap
interval" was 365 + 365 + 365 + 366 = 1461 days. Now I already know the number of days that
must pass, in order to have a FiveSundayFebruary, is an integral multiple of 1461, or I can
express that as: n x 1461 = # of days
required. (where n = # of "leap
intervals") I already
know that 2008 will be a leap year and the 1st of February will then be on a
Friday (moved by 5 days) so n = 1
will not work. In order
for the above formula to work, this same number must also be an integral
multiple of 7, since there are 7 days in a week and the1st of February must
have moved a number of days equal to m x 7 where m is some integer (in order
to be back on Sunday again). Hence, I
can now write: n x 1461 = m x 7 Normally
you would expect not to obtain a solution from one equation with two
unknowns, m and n.
However, the difference here is we are dealing with integers that
force additional conditions on the equation.
Since "n" represents a solution to our problem, we proceed
undaunted and write: n = m x 7 1461 We
picture "m" as ever increasing, marching off into the future, until
it reaches a condition that the entire equation involves integers. The earliest that can occur is when
"m" itself is 1461. Then n = 7 leap intervals or 7 x 4
= 28 years. That means we are predicting that there will be five
Sundays in February 2032 and there were five Sundays in February 1976 and
2004 was the only such occurrence in between those years. C. Check the solution by recasting the problem into
intuitive terms you best understand. That helps you to know
"deep in your heart" that the mathematical solution is correct,
even if your intuitive explanation is not one that would explain the problem
to some other person. I began making
"tick marks" to track the "1stdaycreep" for
February. There are five tick marks
per row corresponding to the 5day advance February 1 makes in one 4year
"leap interval" with 7 rows corresponding to the 7 "leap
intervals" required. 4
years
      
    
    
    
    
    
   7 leap intervals of 4 years
each. Start
with the upper most "green" tick and count the ticks as 1,2,3...7,
1,2,3,...7, etc. In other words,
count the multiples of 7. You should
say "seven" when you hit the red tick. That means we are on February 1 of 2004 + 28 = 2032 and the
"7" said it was a Sunday and since 2032 is a leap year there will
be five Sundays that month. D. Even More
Intuitive. Sometimes, when you have spent this much
time on a problem, you can express it in even more intuitive and simple
terms. By now we recognize we are
dealing with "two rhythms"  one a rhythm of 7 and another rhythm
of 4. Imagine you are a symphony
conductor. With one hand you are
conducting the orchestra with a 77 beat and with the other hand you are
directing the choir with a 44 rhythm.
Your hands come down in synchronization at the start, but from there
on your hands will be out of synchronization, called a "syncopated
rhythm." As every musician
knows, your hands will eventually be synchronized. When? When the number
of beats is evenly divisible by both 7 and 4 and that first occurs at 28
beats, the same number we calculated above.
(Much like finding a common denominator for two fractions.) E. Other Checks.
Only when I got home was I able to use the Internet to reassure me my
calculations were correct. I located
an electronic calendar on the Internet and used "fast forward" and
"fast reverse" to make the calendar give me the months of February
in those two years. 
February 2032 
February 1976 
Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 
Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29






February 1900 
February 2300 

Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
26 27 28 
Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
25 26
27 28 


Any guesses
as to why February 22, 1900 is highlighted but February 22, 2300 is not? But
then, every 2500 years this correction overcorrects the other way and....oh,
never mind. 

H. Problem for you to solve.
The same group mentioned above, noticed in the year 2006 that January
1st was on a Sunday which totally "screwed up" the schedule for
football bowl games (the one thing that a group of men might notice). They realized it occurred more frequently
than the FiveSunday February but could not remember when that last
happened. Remembering that January 1
will "creep" forward in the week by one day, unless this year is a
leap year when it will occur 2 days later in the week next year, locate the
years for the next two occurrences of this event, beyond 2006, and then
"induce" a general mathematical pattern. Hint: 6 + 5 + 6 + ..... 
Then, If "n" represents the integers
1, 2, 3, 4, etc. write a general formula containing "n" so that the
exact year beyond 2006 when this will occur again can be obtained by merely
insert an integer value for "n", that is: Year = 2006 + f(n) where f(n) is your function of
"n". Click solution key
when stumped 
Try an automatic Calendar Calculator to see other calendar patterns.