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 Five-Sunday-February 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 Five-Sunday-February 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 "1st-day-creep." 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 days in a year, we only observe 365 of them for three years, and then observe a 366 day-year 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 Five-Sunday-February, 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 "1st-day-creep" for February. There are five tick marks per row corresponding to the 5-day advance February 1 makes in one 4-year "leap interval" with 7 rows corresponding to the 7 "leap intervals" required.

4 years

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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 7-7 beat and with the other hand you are directing the choir with a 4-4 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 

 

 

F. Other Discoveries. Always take the time to "savor" the steps in your solution. Sometimes you will discover you have produced "other discoveries" in the process. Having spent so much time working with the month of February, it was obvious that "as February goes, so goes the rest of the year." Since all the remaining months always have the same number of days, once February is "positioned" within the calendar year, all the remaining months simply fill up the remaining spaces. Hence, not only does a Five-Sunday-February reproduce itself every 28 years, the entire calendar is the same every 28 years. Hence, if you did not get a 2004 calendar for Christmas, and you had an old 1976 calendar still lying around, you could use that one just as well.

 

Although a couple of "old timers" in our group reminded us that if we used a calendar that was quite old we might not recognize a few holidays that would be listed there. I don't remember all the "old holidays" they mentioned, but I am wondering if any students have "been around long enough" to know the modern name of the old holidays which used to be called Decoration Day and Armistice Day? They are actually still with us.

 

By the way, the young man who made the original error was correct. February 2004 was the first Five-Sunday-February in his lifetime since he is only 24 years old and therefore born after 1976.

 

G. Trivia. The following may strike you as too trivial to worry about. The formula obtained above will be valid over any interval of time, either moving forward into the future or backward to past calendars - until we hit the years 1900 and 2300. Both numbers are divisible by four and should be leap years. After all, the years 1896 and 2304 were or will be leap years. How come not these years?

 

According to Astronomy, our adding one day every four years "overcorrects" the calendar by one day in 400 years. Hence, by international agreement, every 400 years, a "wannabe" leap year is cancelled and, by agreement, 1900 was the first cancellation and 1900 + 400 = 2300 will be the next.

New

 

 

 

 

 

 

 

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 Five-Sunday 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.