Treebeard's Stumper Answer 13 November 98

Friday the 13th

Today is Friday the 13th! Why is this particular day considered unlucky, and just how uncommon is it? Are there years with no Friday on the 13th? Can there be two in a row in successive months? What's the most there can be in a single year? This must be a good omen for our Joshua Tree trip next week, since the Leonid Meteor Shower might give us a spectacular sky show very early in the morning on Tuesday, November 17. We will have no moon and (hopefully) clear desert skies for a perfect view.

It was overcast on the first night of the DMS Joshua Tree trip on November 17, but the clouds parted long enough to give us a great view of the promised Leonid Meteor Shower. We saw many shooting stars, including several fireballs that lit up the sky and left glowing trails! See the next stumper.

If Friday is bad and 13 is bad, then Friday the 13th must be the worst. There was Good Friday, and there were 13 at the Last Supper. But I think these fears probably go back even further. I figure bad luck comes when we look for it! Every year has at least one Friday the 13th. This year we had three, the most possible, in February, March, and November. That's also two in a row. It's odd, but because of a quirk in the leap year rule, the 13th is slightly more likely to occur on a Friday than any other day of the week! The details are below.

Note: The number 13 is unlucky, but 7 is lucky. Being prime isn't the issue. Could it be that 13 is unlucky because 13 full moons don't quite fit in a (solar) year? The conflict between solar/agricultural and lunar/nomadic societies is a deep one, and the victors write the history....

Every calendar year has at least one Friday the 13th, but Graybear found another way by spanning two years:

There are no years without a Friday the Thirteenth. However, you can go thirteen (how interesting) months without one (e.g. if you have a July F/13, and the next year is not a leap year, the next F/13 will be the following September; OR, if you have an August F/13, and the next year is a leap year, the next F/13 will be the following October.) The next occurrence of this will be no F/13s between the one in August 1999 and October 2000.

You can have at most three in one year. In a non-leap year they will occur in February, March, and November, as happened this year (1998). In a leap year, they will occur in January, April, and July. This leap year configuration results in the highest concentration of three F/13s in seven months, which can also happen between a non-leap year September and a leap year March (e.g. September, December, March). This last happened in 1991/1992, and will happen again in 2012.

Graybear also reminded me that the Last Supper was on Friday, not the day before:

The most plausible reason for the superstition is that Jesus Christ was the thirteenth figure at the Last Supper, which took place on Friday (we'd call it Thursday night, but the Jewish day starts at sundown).

Why the 13th is most likely to fall on a Friday

The 13th of the month is slightly more likely to occur on a Friday than any other day of the week! I found this claim on a Web page by Drew Larson at www.furrfu.com/magpies/friday_13.html. But I had to work through it myself to appreciate it. Thanks for getting me thinking about this Drew, this was fun!

Along the way, I wrote a BASIC program that counts Friday the 13ths and figures which day of the week the first day of the year falls on for any year or span of years. You can download my FRI13 program for your own explorations. It includes source code for QBasic/QB/PDS and an executable for DOS/Win.

The number of Friday the 13ths in a year depends only on which day of the week the first day of the year falls, and whether the year is a leap year with an extra day in February. The rest is fixed by the distribution of days and months in the calendar. For example, in 1998, we had three Fridays on the 13th in February, March, and November. This can only happen in a non-leap year that begins on a Thursday. Trace the days on a calendar to see why.

There are only 14 different yearly calendars, one for each day of the week and whether or not it's a leap year. This is how perpetual calendars are possible. These different calendars repeat in a sequence of 28 years called the Solar Cycle. Each year has 365 or 366 days . Since 52 x 7 = 364 days, this is 52 weeks and 1 or 2 extra days. So New Year's day falls 1 or 2 days later each year. Here's the entire sequence:

Year	First day of year	Leap Year?	# Fri 13ths	Months with Fri 13th
1972	Saturday	Yes	1	Oct
1973	Monday		2	Apr, Jul
1974	Tuesday		2	Sep, Dec
1975	Wednesday		1	Jun
1976	Thursday	Yes	2	Feb, Aug
1977	Saturday		1	May
1978	Sunday		2	Jan, Oct
1979	Monday		2	Apr, Jul
1980	Tuesday	Yes	1	Jun
1981	Thursday		3	Feb, Mar, Nov
1982	Friday		1	Aug
1983	Saturday		1	May
1984	Sunday	Yes	3	Jan, Apr, Jul
1985	Tuesday		2	Sep, Dec
1986	Wednesday		1	Jun
1987	Thursday		3	Feb, Mar, Nov
1988	Friday	Yes	1	May
1989	Sunday		2	Jan, Oct
1990	Monday		2	Apr, Jul
1991	Tuesday		2	Sep, Dec
1992	Wednesday	Yes	2	Mar, Nov
1993	Friday		1	Aug
1994	Saturday		1	May
1995	Sunday		2	Jan, Oct
1996	Monday	Yes	2	Sep, Dec
1997	Wednesday		1	Jun
1998	Thursday		3	Feb, Mar, Nov
1999	Friday		1	Aug
2000	Saturday	Yes	1	Oct
2001	Monday		2	Apr, Jul
(etc.)

The year 2000 is a leap year (but see below) that begins on a Saturday. With this year, the 28 year cycle repeats again and calendars can be reused in sequence. The number of Friday the 13ths within each unique year is:

	Sun	Mon	Tue	Wed	Thu	Fri	Sat	Total
normal year:	2	2	2	1	3	1	1	12
leap year:	3	2	1	2	2	1	1	12

Within these 28 years, three normal years and one leap year begin on each day of the week. There are 48 Fridays on the 13th, four in each month. There are also 48 Mondays on the 13th, and Tuesdays, etc. In 28 years there are 12*28=336 months, and 48*7=336 13ths. The 13ths are evenly distributed among the days and months.

At least that's how it used to be.

The old Julian Calendar added a leap year every fourth year according to the rule that a year is a leap year if it is evenly divisible by four. But over the centuries, this proved to be a bit too much and the calendar slowly grew out of sync with the traditional date for the Vernal Equinox. Finally in 1582, the Gregorian Calendar was proposed. The new calendar eliminated the ten extra days that had accumulated, and changed the leap year rule to eliminate three leap years every 400 years.

According to the new rule, century years (divisible by 100) are not leap years unless they are evenly divisible by 400. The century years 1800 and 1900 were not leap years, but the year 2000 will be, as will the year 2400. This reform was adopted in Catholic countries in the year 1582, but it was not adopted in Britain (or it's then colony America) until 1752. One odd result is that the year 1700 was a leap year in America, but not in Spain. See my When is Washington's Birthday? stumper for another. Check section 2 of the Calendar FAQ for the whole story.

The new Gregorian Calendar reform also messes up the 28 year Solar Cycle whenever a cycle crosses a century that is not a leap year. Some years will have more or less Friday the 13ths than they should by Julian Calendar rules depending on where the 28 year cycle is broken.

The new leap year rule creates an odd pattern that I first found as problem 516 in Henry Ernest Dudeney's 536 Puzzles & Curious Problems (edited by Martin Gardner, 1967): The first day of a century year can never occur on a Sunday, Tuesday, or a Thursday! And the first day of a leap year century will always be a Saturday. (I'm taking the year 2000 as the century only for convenience. Dudeney correctly figures from 2001.) My FRI13 computer program verifies this:

Date	New Year's Day	Leap Year?
1 Jan 2000	Saturday	Yes
1 Jan 2100	Friday	no
1 Jan 2200	Wednesday	no
1 Jan 2300	Monday	no
1 Jan 2400	Saturday	Yes
1 Jan 2500	Friday	no
1 Jan 2600	Wednesday	no
1 Jan 2700	Monday	no
1 Jan 2800	Saturday	Yes
1 Jan 2900	Friday	no
1 Jan 3000	Wednesday	no
1 Jan 3100	Monday	no

This shows that a full Gregorian calendar cycle is 400 years. In fact, any date 400 years later will fall on the same day of the week. Dudeney's puzzle about century years is just a special case of this. Within this period, there are:

Kind of Years:	Count	Extra	Days Total
normal non-leap years	300	x 1 =	300
century non-leap years	3	x 1 =	3
normal leap years	96	x 2 =	192
century leap years	1	x 2 =	2
total	400 x 52 weeks		497 days

The extra 497 days / 7 days per week = 71 extra weeks with no remainder. So in 400 years there will be exactly 400 x 52 + 71 = 20,871 weeks with nothing left over. This comes to 146,097 days. I expected this number to also be evenly divisible by 28, but it's 7 days short. But at least the same days are missing from the Solar Cycle every 400 years.

Here's the summary of a 400 year run from my FRI13 program. The results are the same for any 400 year period (after 1752). This shows that the 13th is slightly more likely to fall on a Friday, and least likely to fall on a Thursday or Saturday.

     ==========================================================================
     There are 688 Friday the 13ths in the years 2000 to 2399.

     Day distributions of the 13th of the month in these 400 years:
     --------------------------------------------------------------------------
     Sun       Mon       Tue       Wed       Thu       Fri       Sat      Total
     687       685       685       687       684       688       684       4800                   
                                             min       max       min
     
     First day of year distributions:
     --------------------------------------------------------------------------
                  Years:      Centuries:     All Fri 13:    Fri 13 per:              
               Normal Leap    Normal Leap    Normal Leap    Normal Leap              
     Sunday      43    15        0     0       86    45        2     3                
     Monday      42    13        1     0       86    26        2     2                
     Tuesday     44    14        0     0       88    14        2     1                
     Wednesday   42    14        1     0       43    28        1     2                
     Thursday    44    13        0     0      132    26        3     2                
     Friday      42    15        1     0       43    15        1     1                
     Saturday    43    12        0     1       43    13        1     1                
     ==========================================================================
     Totals:    300    96        3     1      521   167       
     ==========================================================================

I don't know if it's good luck or bad that the 13th is more likely to fall on a Friday than any other day of the week. (I'm a Libra and we Librans aren't superstitious!)   I'm sure this unexpected result is absolutely unimportant. But I love these trivial questions that expose deep patterns in things we take for granted. That's what stumpers are for!

Back to Stumper

Last modified .