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 13^{th} 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 13^{th} 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 13^{th} 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 13^{th} 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 13^{th} by three days, so
February 13^{th} 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 13^{th}. Also, we can see that we
encounter several classes repeatedly (and at most 3 times). Depending upon what
day January 1^{st} falls on, this means that we could possibly have up
to three Fridays that fall on the 13^{th} of the month.

LetŐs
take a look at when we might encounter multiple Fridays on the 13^{th}
of the month. We will look at years that are not Leap Years. LetŐs say that
January 1^{st} falls on a Sunday. Then, January 13^{th} 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 1^{st} falls on a
Monday. We can put January 13^{th} into our 0 mod 7 class, but then we
would be looking for those months that have the 13^{th} in the 6 mod 7
class. We see that we have Friday the 13^{th} 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 1^{st} to fall on different
days of the week.

If
you notice, we only have two consecutive months that can have the 13^{th}
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 13^{th} 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 1^{st} must fall on a
Thursday.

If
we were to look at past calendars, we would find that in 1998, we had Friday
the 13^{th} 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 21^{st} 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 13^{th}.
We can also take this information and determine a very easy way to check
whether a given year has three Fridays on the 13^{th}. 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 13^{th}. 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 13^{th}s.