Update: See detailed info on Fri 13th at bottom. Also, several revisions have been made.
Update on Contest:
All teams' Answer Forms have been submitted and scored. The highest score was a perfect 11 out of 11. There were several other outstanding scores as well. More details to follow...
If you suffer from triskaidekaphobia, uh, well, maybe you can be Rip Van Winkle starting tonight, Thu Mar 12th, and wake up on August 14th next year. Next year will be much less scary with only one Friday the 13th!
Note: Triskaidekaphobia only refers to fear of the number 13. See the Wikipedia article for the very long word denoting fear of Friday the 13th!
Can this be a learning experience for our students?
Here are some questions to ask:
Feb 13th 2009 happens to fall on a Friday.
Q: What day of the week will it be 28 days later? Explain.
Explanation: The days of the week repeat every 7 days and 28 is a multiple of 7.
Q: What day of the month will it be 28 days after Feb 13th? Explain.
Ans: It will be the 13th of the next month, in this case, March.
If today is the nth day of a given month and there are k days in this month, then k days from today will always be the nth day of the following month!
Why? Well, we can write k = (k-n) + n, which can be interpreted to mean that k days from today will be the nth day of the following month.
Ok, an example would help here:
Let's say today is the 5th day of some month and there are 31 days in that month. We can write 31 = (31-5) + 5 = 26 + 5. the "26" brings us to the end of the month and then we add 5 more days which brings us to the 5th of the next month. Make sense?
Putting this all together, 28 days from Friday Feb 13th will be Friday March 13th. The key was that 28 is BOTH the number of days in the month AND a multiple of 7!
That wasn't hard, but now for the tougher question:
Q: Why is there going to be a third Friday the 13th in 2009, namely Nov 13th?
Mar-31 Cum. Total = 31 (not div by 7)
Apr-30 Cum. Total = 61 (not div by 7)
May-31 Cum. Total = 92 (not div by 7)
Jun-30 Cum. Total = 122 (not div by 7)
Jul-31 Cum. Total = 153 (not div by 7)
Aug-31 Cum. Total = 184 (not div by 7)
Sep-30 Cum. Total = 214 (not div by 7)
Oct-31 Cum. Total = 245 (Div by 7!!)
Thus, 245 days after Fri Mar 13th will not only be the 13th of Nov but it will also land on a Friday!
Why? Because, 245 is the cumulative total of the number of days in the 8 months starting in Mar but it is also a multiple of 7. No simple formula here, just grinding it out.
Suggested Additional Questions:
(1) Is 3 the maximum number of Friday the 13ths in any calendar year?
(2) When is the next calendar year in which this 3-peat (three Friday 13ths in one calendar year) will occur?
Ans: See below!
(3) Is 1 the minimum number, i.e, could there be a calendar year in which there are no Fri the 13ths?
Ans: See below!
(4) In which months could a 3-peat occur in a non-leap year? A leap year?
Ans: See below!
(5) Related to (4): 31+29+31 = 91: What does this tell you about Friday 13ths in a leap year?
(6) Other questions: From the students!
The following is from the excellent Wikipedia article on Friday the 13th...
The following months have a Friday the 13th:
The following years have Fridays the 13th in these months:
This sequence, here given for 2001–2028, repeats every 28 years from 1901 to 2099. The months with a Friday the 13th are determined by the Dominical letter (G, F, GF, etc.) of the year. Any month that begins on a Sunday will contain a Friday the 13th, and there is at least one Friday the 13th in every calendar year.
The longest period that can occur without a Friday the 13th is fourteen months, either from July to September the following year (e.g. in 2001/2002 and 2012/13), or from August to October in a leap year (e.g. in 2027/28).
Patterns for non leap-years:
Patterns for leap years: