# Wild About Math!Making Math fun and accessible

12Apr/084

## We have a winner for Monday Math Madness #3

We have a winner for the third Monday Math Madness contest. It's Johan Potums. Congratulations, Johan! I'll be contacting you about your prize.

Blinkdagger has a very interesting new contest problem that they'll be posting Monday.

Seventeen people submitted entries. Everyone got the right answer and explained their answer well. Everyone realized (or figured out) that every year does indeed have a Friday the 16th. Proofs varied but everyone used modular arithmetic to demonstrate that regardless of what day of the week the year begins on, some month will have a Friday the 16th.

Johan's answer was concise. Here it is:

1. List the number of days between each 16th of the month.
2. Take the remainder of each of those numbers divided by 7
3. Check whether each of the possible remainders is available (the day of the week is represented by 7n+k, where k is the number of days away from the first day on which it was the 16th of the month).
4. If so, there is always a fryday the 16th

Johann included a spreadsheet as well, that shows the modular arithmetic he did.

Here is the list of everyone who submitted the answer. If you have a blog or web-site leave me your URL as a comment and I'll update this post to give you some link love:

Maurizio Codogno
John Potums
Henno Brandsma (editor of "Ask a Topologist")
Roger Armstrong
Richard Berlin
Danny Lawson
Brandon Smith
Chin Hui Han
Pat Ballew
Anneleen Van Geenhoven
Sameer Shah
Alane Tentoni
Trey Goesh
Marijn Jongerden
Jonathan
Joe Fruchey
Doan Minh Dang

Some people noted that there are only 14 possible calendars for a year since each year starts on one of 7 days and is a leap year or not. But, there's a shortcut. Some people noticed that leap years don't matter, and that if you look at the months between April and November that with just those 8 months you cover all 7 starting weekdays. (April and July always start on the same day of the week.) This means that one of these 8 months starts on a Thursday and whichever that month is that starts on Thursday, 15 days later is the 16th and 15 days after Thursday is Friday so you always get a Friday the 16th.

Pat Ballew had (again) a very interesting approach to the problem:

You can ignore leap years because there has to be a Friday the sixteenth AFTER Feb 29th each year.

This can be shown using the pigeon hole principle.

For any year, the dates of June 6, Aug 8, Oct 10, and Dec 12 all show up on the same day of the week. John H Conway called this the Doomsday for that year. The dates of May 9, Sept 5, Nov 7 and July 11 also always fall on that same Doomsday as June 6 etc.

That accounts for days number 6, 7,8,9,10,11, and 12 of some month, and if we add 7 to each months dooms day, we see that June 13, Nov 14, Aug 15, and May 16 all fall on that same day of the week...

So if DOOMSDAY is a Friday, then May 16th will be Friday the 16th, AS IT IS THIS YEAR.

If Doomsday is a Thursday, then Aug 16th will be Friday the 16th (again in 2013)
If Doomsday is a Wednesday, Nov 16th will be Friday the 16th (as in 2007, and will be again in 2012)
If Doomsday is a Tuesday, June 16th will be Friday the 16th (2006 and again in 2017)
If Doomsday is a Monday, Dec 16th will be Friday the 16th
If Doomsday is a Sunday, July 16th will be Friday the 16th
If Doomsday is a Saturday, Oct 16th will be Friday the 16th.

Joe Fruchey also made an interesting observation:

Furthermore, it can also be seen that there is a N-day the Mth in
every calendar year as well, for all M<29. ... in every case, February begins on the same day as
at least one other month. So February can be discarded, and all days
are still represented. Now we can extend M to include 29 and 30. The
same does not hold true when omitting the 30-day months no longer
leaves us with a full set of days. (There would be no Saturday the
31st for the regular year example, and no Sunday the 31st for the leap
year example.)

So, there you have it -- the cousin to the Friday the 13th problem solved and extended to the general case.

I hope you enjoyed this problem. Check Monday for Monday Math Madness #4 at Blinkdagger.

1. hi, i actually have relatively recently started a math teaching blog, so i’d love some link love!

samjshah.wordpress.com

congrats johan!

2. Sam – you are now loved 🙂

3. I bought myself:

The Music of the Primes by Marcus du Sautoy

with the price money 🙂

4. Johan,

Enjoy the book!