Thursday, February 12, 2009

Friday the 13th Parts I, II and III in 2009: Feb, Mar and Nov. Is this unusual?

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.
Ans: Friday
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?
Ans: Yes!
(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:

Month Years Dominical
January 2006, 2012, 2017, 2023 A, AG
February 2004, 2009, 2015, 2026 D, DC
March 2009, 2015, 2020, 2026 D, ED
April 2001, 2007, 2012, 2018 G, AG
May 2005, 2011, 2016, 2022 B, CB
June 2003, 2008, 2014, 2025 E, FE
July 2001, 2007, 2012, 2018 G, AG
August 2004, 2010, 2021, 2027 C, DC
September 2002, 2013, 2019, 2024 F, GF
October 2006, 2017, 2023, 2028 A, BA
November 2009, 2015, 2020, 2026 D, ED
December 2002, 2013, 2019, 2024 F, GF

The following years have Fridays the 13th in these months:

Year Months Dominical
2001 April, July G
2002 September, December F
2003 June E
2004 February, August DC
2005 May B
2006 January, October A
2007 April, July G
2008 June FE
2009 February, March, November D
2010 August C
2011 May B
2012 January, April, July AG
2013 September, December F
2014 June E
2015 February, March, November D
2016 May CB
2017 January, October A
2018 April, July G
2019 September, December F
2020 March, November ED
2021 August C
2022 May B
2023 January, October A
2024 September, December GF
2025 June E
2026 February, March, November D
2027 August C
2028 October BA

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:

First month occurring Second month Third month
January October
February March November
April July



September December

Patterns for leap years:

First month occurring Second month Third month
January April July
February August
March November


September December


Eric Jablow said...

I recommend Klaus Tonderling's Calendar FAQ.

Dave Marain said...

Thanks, Eric.
Klaus' website is a wealth of information about the historical and scientific origins of various calendars, explained clearly and thoroughly. I read it for just a few moments and then got hooked! He answers pretty much any question anyone might have regarding calendars, holidays, origins for various terms, etc.

~Steph~ said...

Thank you so much for posting this. I thought I was the only one noticing that there is a pattern for Friday the 13th! :)

Dave Marain said...

You're welcome, Steph...
It's nice to know someone enjoyed this phenomenon as much as I did!