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/
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