Good Friday on the 13th

Date: 04/10/2001 at 08:12:26
From: Susan Melanson
Subject: Good Friday on the Thirteenth

How many times has Good Friday fallen on the 13th? I have looked at 
your formulas and don't seem to find one that fits this question, 
since Good Friday doesn't fall on the same date each year. I've looked 
at some other sources as well, to no avail.

Thanks for your help on this,
Susan Melanson

Date: 04/10/2001 at 13:39:36
From: Doctor Rick
Subject: Re: Good Friday on the Thirteenth

Hi, Susan.

Easter can fall on one of 35 dates, March 22 to April 25. Thus Good 
Friday can fall on 35 dates also, March 20 to April 23. It will fall 
on a Friday the 13th if, and only if, Easter falls on April 15 (tax 
day in the US - which makes for good luck, because people get an 
extra day to file their tax returns).

If we suppose that the dates of Easter are evenly distributed among 
these 35 dates, then we expect 1/35 of all Good Fridays to fall on 
Friday the 13th. It may be that the probability of Easter falling on 
the very early or very late dates is reduced somewhat. Without doing 
fancy calculations, I would expect that there is about a 1/29 
probability for the middle dates, because the date of Easter depends 
on when in the lunar cycle of about 29 days the March equinox falls, 
and I do expect the equinox to be evenly distributed throughout the 
lunar cycle. Thus I predict that Good Friday will fall on the 13th in 
1 of every 29 years, on average.

I checked by looking at this Web page:

   Easter Dating Method, by Ronald W. Mallen 
   http://www.assa.org.au/edm.html   

It lists Easter dates for years from 1700 to 2299. In these 600 years, 
Easter fell or will fall on April 15 in these years:

     1770, 1759, 1781
     1827, 1838
     1900, 1906, 1979, 1990
     2001, 2063, 2074, 2085, 2096
     2131, 2142, 2153
     2210, 2221, 2283, 2294

That's a total of 20 times in 600 years, for a probability of 21/600, 
or 1 in 28.6 days. That's pretty close to my estimate!

You'll notice that there are patterns in the numbers: lots of dates 
separated by 11 years (including 1979, 1990, and 2001), and one pair 
separated by only 6 years: 1900 and 1906. I'll bet this anomaly is due 
to the fact that 1900, which would normally be a leap year, was not 
because of the Gregorian calendar's rule that years divisible by 100 
(but not by 400) are not leap years. (Note that according to this 
rule, 2000 was a leap year.)

To make up for these times of increased concentration of Good Friday 
the 13ths, there are long periods with none: the next time it happens 
won't be for 73 more years. The "1 in 29 years" rule only holds in the 
long run: it may be that Good Friday the 13ths are never 29 years 
apart.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 04/10/2001 at 14:34:21
From: Doctor Rob
Subject: Re: Good Friday on the Thirteenth

Thanks for writing to Ask Dr. Math, Susan.

Your question is equivalent to asking when Easter falls on April 15th.
 
Easter is the first Sunday after the first full moon on or after 
March 21st, according to a decision by the Council of Nicea in 325 
A.D. This means that both the tropical solar year of 365 days, 
5 hours, 48 minutes, and 46 seconds, and the length of a lunation, 
29 days, 12 hours, 44 minutes, and 2.8 seconds, must be taken into 
account.

It was noticed by the ancient Babylonians and other cultures that 19 
tropical solar years is very nearly equal to 235 lunations, the 
difference being about 2 hours, 4 minutes. That and other very 
complicated considerations led Carl Friederich Gauss to give the 
following rule for the computation of the date of Easter:

   Let N be the number of the year. Let C = [[N/100]], where [[x]] 
   means the greatest integer less than or equal to x (that is, x 
   rounded down to the nearest integer). Set:

      L = 4 + C - [[C/4]],
      M = 11 + L - [[(8*C+13)/25]]

   Divide N by 4, 7, and 19, and call the resulting remainders a, b,
   and c. Divide 19*c + M by 30, calling the remainder d. Divide 
   L + 2*a + 4*b + 6*d by 7, calling the remainder e. Then the date of 
   Easter is either March 22+d+e, or April d+e-9, with the following 
   two exceptions:

      (1) If e = 6 and d = 29, Easter is on April 19; and
      (2) If e = 6, d = 28, and M = 2, 5, 10, 13, 16, 21, 24, or 29, 
          then Easter is on April 18.

The explanation and proof of this rule is given by J. V. Uspensky and 
M. A. Heaslet, _Elementary Number Theory_ (New York: McGraw-Hill, 
1939), pages 212-221.

To apply this to the problem of in what years did Easter fall on 
April 15th, we have to do this century by century. Let's start with 
the 1900s, for which C = 19, L = 19, M = 24. We can ignore the two 
exceptions. We need d + e - 9 = 15, so d = 24 - e, and

     19*c + 24 = d = 24 - e (mod 30)

     2*a + 4*b + 6*d + L = 2*a + 4*b + 6*(24-e) + 19 = e (mod 7)

     c = 11*e (mod 30)

     b = 3 + 3*a (mod 7)

Now considering e = 0, 1, 2, 3, 4, 5, and 6, we find that we can 
eliminate e = 2 or e = 5 because they lead to c >= 19. Thus:

     e = 0, c = 0
     e = 1, c = 11
     e = 3, c = 3
     e = 4, c = 14
     e = 6, c = 6

To find N, we must solve the system of congruencies:

     N = a (mod 4)
     N = 3 + 3*a (mod 7)
     N = c (mod 19)

Using the Chinese Remainder Theorem, one gets:

     N = 304 + 437*a + 476*c (mod 4*7*19)

There are five values of c above, and four values of a (0, 1, 2 and 
3), so there are 20 possible values of N (mod 532):

      19,  30,  41,  52, 114, 125, 136, 209, 215, 220, 299, 304, 310, 
     383, 394, 405, 467, 478, 489, 500

Of these only four lead to years with C = 19, that is, with 
1900 <= N <= 1999:

     1900, 1906, 1979, 1990

Repeating this with C = 20, one finds that the years:

     2001, 2063, 2074, 2085, 2096

are the ones that have the same property. Repeating this with C = 18, 
one finds that the years

     1837, 1838, 1849

are the ones that have the same property.

I leave further calculations to you.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 04/13/2001 at 13:27:27
From: Doctor TWE
Subject: Re: Good Friday on the Thirteenth

I used Gauss' Easter formula documented in:

   Formula for Easter
   http://mathforum.org/dr.math/problems/john.6.12.99.html   

(and explained in Dr. Rob's answer above) in a spreadsheet and 
calculated all Good Friday the 13th's for the years 1 AD to 5000 AD. 
There were 28 such dates in the first Millennium (1 AD - 1000 AD; of 
course, the Gregorian calendar wasn't in use back then) and 29 such 
dates in the second Millennium (1001 AD - 2000 AD). There will be 40 
such dates in the third Millennium, 42 such dates in the fourth 
Millennium, and 35 such dates in the fifth Millennium. That's 174 
occurrences in 5000 years, or an average of once every 28.74 years.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/