## Tonya C. Brooks

Almost everyone has certain numbers they are attracted to, and other numbers that they try to avoid. Some people even go so far as to let certain numbers rule their lives. If you have seen the movie The Number 23 with Jim Carrey, then you know what I am talking about. Even if you do not have a favorite number, I am sure that you have heard about Friday the 13th being an unlucky day.

You also might have noticed that sometimes we encounter two, and sometimes three, unlucky Fridays within a given year. Ever wonder why that is? LetŐs take a look and see if we can figure out exactly what is up with that.

It might help us to know how many days in the month we have for each of our months. Below you will see a table giving the number of days in each of the months.

 January 31 February 28 (or 29 on leap years) March 31 April 30 May 31 June 30 July 31 August 31 September 30 October 31 November 30 December 31

Along with this, we really are only interested in which day of the week our Friday the 13th falls on. Obviously, it has to be on Friday and we want to see how many times this happens within a given year. Therefore, if we name the days of the week by numbers 0 through 7 (in essence, we would like to work within modular arithmetic and use mod 7 to simplify things for us), we can look at the FridayŐs fairly easily.

LetŐs say that January 13th falls on whatever day we want to classify as 0 mod 7. It doesnŐt really matter what day this is at this point. Just using this and the number of days in our months, we can determine how many days after our 0 class we have the 13th of every month following. For example, since four weeks is only 28 days and January has 31 days, we know that the 3 extra days in January ŇpushÓ February 13th by three days, so February 13th would fall within the 3 mod 7 class. We do this for the rest of the months to get the following chart:

 Month Friday the 13th class Leap Year January 0 (mod 7) 0 (mod 7) February 3 (mod 7) 3 (mod 7) March 3 (mod 7) 4 (mod 7) April 6 (mod 7) 0 (mod 7) May 1 (mod 7) 2 (mod 7) June 4 (mod 7) 5 (mod 7) July 6 (mod 7) 0 (mod 7) August 2 (mod 7) 3 (mod 7) September 5 (mod 7) 6 (mod 7) October 0 (mod 7) 1 (mod 7) November 3 (mod 7) 4 (mod 7) December 5 (mod 7) 6 (mod 7)

If you notice, in each case (Leap Years and those that are not), each class is encountered at least once. This tells us that no matter what year it is, we can expect to get at least one Friday the 13th. Also, we can see that we encounter several classes repeatedly (and at most 3 times). Depending upon what day January 1st falls on, this means that we could possibly have up to three Fridays that fall on the 13th of the month.

LetŐs take a look at when we might encounter multiple Fridays on the 13th of the month. We will look at years that are not Leap Years. LetŐs say that January 1st falls on a Sunday. Then, January 13th is actually on a Friday. If we let this be our 0 class, then we look for others within the 0 mod 7 class. We see that we have one in October (as well as the obvious one in January). LetŐs look at when January 1st falls on a Monday. We can put January 13th into our 0 mod 7 class, but then we would be looking for those months that have the 13th in the 6 mod 7 class. We see that we have Friday the 13th in April and July. We can continue this to get:

 January 1st Months with 13th on Friday (non-leap years) Months with 13th on Friday (Leap Years) Sunday January, October January, April, July Monday April, July September, December Tuesday September, December June Wednesday June March, November Thursday February, March, November February, August Friday August May Saturday May October

As you might have noticed, your birthday doesnŐt always fall on the same day every year. This is due to the fact that we generally have 365 days in our year, which turns out to be 52 full weeks and one day, which is what pushes your birthday a day later (or two days if the year is following a February with a leap day). This is what causes our January 1st to fall on different days of the week.

If you notice, we only have two consecutive months that can have the 13th on a Friday, February and March, and this only happens on years that are not Leap Years. This is due to the fact that the 13th for the two months must be exactly 4 weeks apart (28 days) and this only happens when the first month has 28 days total, which February does. In order for this to happen, we can see from the data above that January 1st must fall on a Thursday.

If we were to look at past calendars, we would find that in 1998, we had Friday the 13th in both February and March, which doesnŐt happen again until 2009 (11 years later). Doing some research using calendars for future years, we can see that this phenomenon happens in 2015, 2026, 2037, 2043, 2054, 2065, 2071, 2082, 2093, and 2099 as well. We can see a pattern in the difference between our years of 6 years, 11 years, and 11 years which repeats throughout the 21st century. Taking a closer look at why this is, we find that when we have a span of 11 years, we see that we have a leap year within one or two years, which causes us to have three leap years before we end with February and March having Friday the 13ths, whereas when we have six years spanning between our phenomenon, Leap Year will not be encountered for three more years, which gives only one Leap Year before the next time our phenomenon occurs.

We cycle through the same pattern every 28 years, which we can use to determine when the next year is that we will have three Fridays fall on the 13th. We can also take this information and determine a very easy way to check whether a given year has three Fridays on the 13th. We can once again set our years up in modular, only we will be looking at mod 28. Within mod 28, we have four classes that generate three Friday the 13ths Đ 2 mod 28, 13 mod 28, 16 mod 28 (which is the case of a Leap Year), and 19 mod 28- if we allow the year 2000 to be in our 0 mod 28 class. Using this, we can determine whether this is the case or not. For example, if we want to look at the year 2038, we can look at what class 38 falls into, which we see is 10 mod 28. This equivalence class is not one of the four listed as giving our desired result, therefore 2038 does not have three Fridays on the 13th. It is not for another three years (since 10 mod 28 is 3 away from the closest case of 13 mod 28) or until 2041 that we have three Friday the 13ths.