Friday the 13th

Essay 1

by: Maggie Hendricks

The occurrence of a Friday that falls on the 13th day of the month is an event of quite some interest to many people. This brings up the question of how often we can expect such a day to happen in any given year. To begin, we know that there are always 12 months in the year; we also know that each of these months occurs in the same order and with the same number of days every year (with the exception of February during Leap Years, but we’ll get to that later). In addition to our knowledge of the 12 months of the year, we also know that there are seven days in any given week, and these also occur in the same order every week. The issue now is relating how the number of days in a month (anywhere from 28 to 31) can be connected to the number of days in a week (always seven). Since the number of week days is always the same, it would be helpful if we could find some way of naming the days of the month in terms of their day of the week. We must find a way to do this in a general way, though, since the days of the month do not always fall on the same day of the week every year. Those familiar with modular arithmetic might agree that it could be useful to “rename” the days of the month using a counting system in mod7. 

For instance, the days of the week can be thought of as day 0, day 1, day 2, day 3, day 4, day 5, and day 6. If January the 1st fall on a Thursday we would call it a 4, and this would make January 2nd be 5, January 3rd be 6, and January 4th be 0 (starting over since 7 = 0mod7). Since January 1st does not always fall on a Thursday, let us instead say that January 1st fall on day x (where x is any integer such that 0 ≤ x ≤ 6). Then January 2nd falls on day x+1, January 3rd falls on x+2, January 3rd on x+3, January 7th on x+6, and January 8th on day x again. Note that for a known value of x, each of these days would be simplified according to their value in mod7. We could use this labeling mechanism to go on and name all the days in the month of January in mod7, and we would find that there are three “extra” days at the end of the month that keep up from being able to say February starts on day x (since January has 31 days instead of 28, and 31 = 3mod7). To add a little clarity, let’s see if we can “name” what day each of the months begins on if we decide to say that January 1st is on day x.

January 1st: x
February 1st: x+3
March 1st: x+3 (since February has exactly 28 days in regular years, and 28 = 0mod7)
April 1st: x+6 (since March has 31 days and 31 = 3mod7, we get x+3+3)
May 1st: x+1 (April has 30 days, 30 = 2mod7, and x+6+2 = x+8 = x+1mod7)
June 1st: x+4 (May has 31 days, 31 = 3mod7, and x+1+3 = x+4)
July 1st: x+6 (June has 30 days, 30 = 2mod7, and x+4+2 = x+6)
August 1st: x+2 (July has 31 days, 31 = 3mod7, and x+6+3 = x+9 = x+2mod7)
September 1st: x+5 (August has 31 days, 31 = 3mod7, and x+2+3 = x+5)

Note that without going any further, we have now seen all of the possible days of the week show up as start days for the month (x, x+1, x+2, x+3, x+4, x+5, x+6). This is important because it tells us that from January to September, at least ONE month must begin on a Sunday (which would result in the 13th of that month falling on a Friday). So we know that there must be AT LEAST ONE Friday the 13th in any regular year (regular here indicating a non-leap year). Let’s continue with our naming of the first of each month just so we can see a full year.

October 1st: x (September has 30 days, 30 = 2mod7, and x+5+2 = x+7 = x+0mod7)
November 1st: x+3 (October has 31 days, 31 = 3mod7, and this gives us x+3)
December 1st: x+5 (November has 30 days, 30 = 2mod7 ,and x+3+2 = x+5)

Now let’s see that in a little more organized fashion:
January 1st: x
February 1st: x+3
March 1st: x+3
April 1st: x+6
May 1st: x+1
June 1st: x+4
July 1st: x+6
August 1st: x+2
September 1st: x+5
October 1st: x
November 1st: x+3
December 1st: x+5

From this, we can see that in a regular year, January and October will start on the same day of the week, February, March, and November will all begin on the same day of the week, April and July will start on the same day of the week, September and December will begin on the same day of the week, and May, June, and August will all begin on a day of the week that is not the same as any other month that year. This is important because when two or more months begin on the same day of the week, the 13th of those months will also fall on the same day of the week. Looking specifically at the months that begin on the same day when that day happens to be Sunday, we can say four things:
1. There can be only ONE Friday the 13th in any regular year in which May, June, or August begin on a Sunday.
2. There will be exactly TWO Friday the 13th occurrences in any regular year in which January, April, July, September, October, or December begin on a Sunday.
3. There will be exactly THREE Friday the 13th occurrences in any regular year in which February, March, and November begin on a Sunday.
4. There will never be more than three Friday the 13th occurrences in any regular year.

Now let’s go back to the idea of a leap year. Every fourth year there is an extra day added to the end of February, and this means that we need to relook at how we named the start day for the months of a leap year. January 1st is still on day x, and February 1st is still on day x+3, but each month after February in a leap year starts one day later than how we named them above. So let’s see a full leap year’s worth of start days just for the sake of visualization.

January 1st: x
February 1st: x+3
March 1st: x+4
April 1st: x
May 1st: x+2
June 1st: x+5
July 1st: x
August 1st: x+3
September 1st: x+6
October 1st: x+1
November 1st: x+4
December 1st: x+6

This changes things slightly, but there are still some commonalities. Now January, April, and July begin on the same day of the week; February and August being on the same day; March and November begin on the same day; September and December begin on the same day; and May, June, and October all begin on days of the week that are not the same to the start date for any other month that year. This means that there will still be three possibilities that would result in only one Friday the 13th in a leap year (when the 1st fall on a Sunday in May, June, or October), there will still be three possibilities that result in two Friday the 13th occurrences in one year (when the 1st falls on a Sunday in February/August, March/November, or September/December), and there will still be one possibility that would result in three Friday the 13th occurrences in one year (when the 1st falls on a Sunday in January, April, and July).

In addition, we could now also say that the only time there will be back-to-back Friday the 13th occurrences two months in a row is if February the 1st falls on a Sunday in a non-leap year (since this is the only time that two consecutive months begin on the same day). We could even go a step further and say that we know January 1st must be on a Thursday in a non-leap year in order for there to be two consecutive months with a Friday the 13th (see a quick proof of this below).

For there to be two consecutive months with a Friday the 13th, they must take place in February and March during a non-leap year where February 1st falls on a Sunday. We have shown above that during non-leap years, February the 1st can be represented with the expression x+3, and this day will be a Sunday when x+3 = 0. Some algebraic manipulation leads us to the fact that x = -3 = 4 mod7, so January 1st (which begins on day x) must fall on day 4 (which is a Thursday).

Another (somewhat simpler) view of this can be seen by taking a look at what day the 13th of each month occurs instead of considering the first of each month. Regardless of what day of the week January 1st is on, the 13th of that month will be on the 13th day of the year. Accordingly, the 13th of February will be on the 44th day of the year. If we continue numbering the 13th day of each month in this way, we get the list shown below.

January 13: 13th day
February 13:  44th day
March 13:  72nd day
April 13:  103rd day
May 13:  133rd day
June 13:  164th day
July 13:  194th day
August 13:  225th day
September 13:  256th day
October 13:  286th day
November 13:  317th day
December 13:  347th day

If we renumber this list now in mod 7 (where 0-6 represent the seven days of the week), we get a new list:

January 13: 6
February 13: 2
March 13: 2
April 13: 5
May 13: 0
June 13: 3
July 13: 5
August 13: 1
September 13: 4
October 13: 6
November 13: 2
December 13: 4

From this list we can see that each day (0-6) shows up at least once in any given year. We also see that no number shows up more than three times in any given year (nor would it in a leap year either). These two facts tell us (as we've already seen) that the 13th MUST fall on a Friday at least once in a year and it will never fall on a Friday more than three times in one year.

From observation, we can notice that January 1st fell on a Thursday during a non-leap year in both 1998 and 2009. We could also look ahead a little and notice that next year (2015) will also have January 1st being on a Thursday in a non-leap year. But is there some kind of pattern to when this happens that could help us predict when else it might happen? Let’s start by saying that when January 1st is on day x in a non-leap year, December 31st will be on day x as well (since December 1st starts on day x+5, and x+5+30 = x+35 = x+0mod7). This would mean we should add 1 day to the weekday for January 1st for the year after a non-leap year. For the year after a leap year, we would need to add 2 days to the weekday for January 1st since there is one extra day in the calendar that year. We also know that leap years only occur in years that are divisible by four (2000, 2004, 2008, and so on). Using these things, let’s start a list to look for a pattern.

We can observe in this table that it was 11 years after 1998 that January 1st again fell on a Thursday in a non-leap year, but then it was only six years after that when we will see it happen again (2015). So what’s the pattern? Let’s extend our table.

Now we see a difference of 11 years after 1998, 6 years after that year, then another 11 years after that. From here, we might be tempted to say that it must be an alternating pattern of 11 years, then 6 years, then 11, then 6, and so on. But look what happens if we assume this:

1998+11=2009
2009+6=2015
2015+11=2026
2026+6=2032

But wait! 2032 is a leap year! So our idea fails. Back to the list?

Now we see a pattern of 11 years, 6 years, 11 years, 11 years, 6 years. We could see this pattern of 11-11-6-11-11-6 repeated if we continued on through year 2100. See this list below for all of the years between 2000 and 2100 with consecutive Friday the 13th occurrences in February and March.

2009
2015
2026
2037
2043
2054
2065
2071
2082
2093

Is this enough to say our pattern works all the time? That’s for you to decide; but for now at least, we can say that we know for certain all of the years between 2000 and 2100 when there was/will be a January 1st that falls on a Thursday in a non-leap year (which means there will be consecutive months with a Friday the 13th).

For an Excel spreadsheet with the data from this essay shown, click here.

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