PROBLEM: Friday 13
Prove that there is at least one Friday the 13th in some month during each year.
HINT: Number the days of the year from 1 through 365 (or 366). Determine which days would be the 13th of the successive months. Try a spreadsheet :
Prove that there are at most three Friday the 13ths any given year.
Related Problem:
Joe Hooten, Professor Emeritus
at the University of Georgia, observed
that in
1998 both February and March have a Friday the 13th. That is there
is a Friday the 13th in two consecutive months.
First of all, this problem looks as if it will take some time and lots of effort to prepare for...What the teacher will need to have present are:
Secondly, the first four above necessities are obvious. The last one, patience, is not so obvious to the teacher. She/he needs to have this in place even before this activity takes place because the students really need to understand what this problem is asking for in the first place. The first part of the lesson should be focused on looking through calendars that the teacher provides or allow some groups of students (or the entire class) to go to the library to work on finding the information on their own in old calendars or whatever other resources may be available to them at that time.
Third, after the kids rummage through old calendars, newspapers, etc., take the kids back to the room and ask them where they are, and what kind of information have they come up with or even have found. Also, ask if they have some sort of ideas they are messing around with as to how they will get started with this search...what kind of plan do they have in place? They need one! Make sure this information is stressed to them. If not they will not know where to go with any type of information they happen to find. Make sure the above questions are discussed within small groups before the whole group discussions take place...this will give others the opportunity (especially those who are shy) to share their thoughts and feelings about discussing the whole idea of Friday the Thirteenth as well as their ideas about how to go about solving the problem. This small group discussion will also lend itself to (hopefully) an enormous whole class discussion of the problem.
Now give them the hint that Dr. Wilson provided! Click here to see the hint again!
What can we say about the information in the spreadsheet? Does every month seem to have a Friday the thirteenth ? Are there any cases where there is more than one month of the year with more than one Friday the Thirteenth ? If you did notice any of these cases being the same or possibly different or possibly....something else, what was it? What about the number of the month? Does that have some sort of bearing on this problem at hand? Should we just look at this problem from the point of view of just a spreadsheet? Or should we look more deeply...like just writing out all the days of the year that Friday the Thirteenth fell on? Should we also take a look at Dr. Hooten's problem to see if we can find some relationship among all this information?
Go back up to read Joe Hooten's problem. (Make sure to click on Back to Where I was when you finish rereading about what he found)!
Okay, I must interject now because there is an insatiable need to let you know something. (We all need to know a little something about everyone)! I looked at this problem from the list and I just knew I HAD to work this problem! It seemed like so much fun! As I began working on this problem, I realized that there was a difficult task ahead of me. I had to look for resources...my friend Shae (who has been helping me the entire way), books, colleagues and other students. One classmate, Kelli Nipper, has helped me with understanding this new concept I have never even heard of. The concept is called Modular Algebra or Mathematics...I cannot remember, but I will certainly find out. Why don't you click here to see what Kelli has done and her definition of this concept. It has helped me with this Friday the Thirteenth problem.
Finally, the answer to the Friday the Thirteenth problem:



January  13  13  
February  44  2 mod 7  44  2 mod 7 
March  72  2 mod 7  73  3 mod 7 
April  103  5 mod 7  104  
May  133  0 mod 7  134  1 mod 7 
June  164  3 mod 7  165  4 mod 7 
July  194  5 mod 7  195  
August  225  1 mod 7  226  2 mod 7 
September  256  4 mod 7  257  5 mod 7 
October  286  257  0 mod 7  
November  317  2 mod 7  318  3 mod 7 
December  347  4 mod 7  348  5 mod 7 
What can you conclude about the information in this use of modular arithmetics?