Easter's early this year. Deal with it.

Family holidays ruined by earliest Easter in 90 years (from the Daily Mail).

About halfway down, a formula is given:

It may look daunting to non-mathematicians but the fiendishly complex formula used to work out when Easter actually falls is:

((19*t+u-w-(u-(u+8)\25)+1)\3)+15)mod30)+(32+2*x+2*y-(19*t+u-w- (u-(u+8)\25)+1)\3)+15)mod30)-z)mod7)-7*(t+11*(19*t+u-w(u- (u+8)\25)+1)\3)+15)mod30)+22*(32+2*x+2*y-(19*t+u-w-(u- (u+8)\25)+1)\3)+15)mod30)-g)mod7)+114)\31

Um, do you understand that formula? I think I know why some of the numbers are there -- the 31 at the end probably has something to do with the length of months, the 7 with the length of weeks, and the 19 with the Metonic cycle. Also, any sane mathematician wouldn't write the formula like that. First, there are repeated subexpressions like that ((u + 8) \ 25 + 1); I'd just call that by some other name and be done with it. Second, the formula just sits there in the middle of the article; this gives people the idea that mathematicians are freaks of nature who think in formula. What do the variables mean?

If you're curious, there is an algorithm at the Calendar FAQ. Easter is the first Sunday after the first (computed) full moon on or after the vernal equinox (calculated, and assumed to be March 21). The algorithm reflects this. First, assume that the Metonic cycle, which says that lunar phases repeat every 19 solar years, is exactly correct in the Julian calendar. (The algorithm was invented back when the Julian calendar was used.) Then make two corrections, one for the fact that the Julian calendar includes leap years that the Gregorian doesn't (years divisible by 100 but not 400) and one for the fact that the Metonic cycle's a bit off. (The expression "(u+8)\25" in the formula above comes from the second correction.) This gives the date of the full moon. Presumably if you've gotten this far you already know what the days of the week are.

Anyway, the cycle of Easter dates repeat themselves every 5,700,000 years. The cycle of epacts (which encode the date of the full moon) in the Julian calendar repeat every nineteen years. There are two corrections made to the epact, each of which depend only on the century; one repeats (modulo 30, which is what matters) every 120 centuries, the other every 375 centuries, so the air of them repeat every 300,000 years. The days of the week are on a 400-year cycle, which doesn't matter because that's a factor of 300,000. So the Easter cycle has length the least common multiple of 19 and 300,000, which is 5,700,000.

This whole computation is known as the computus (Latin for "computation"; I guess it was just that important at the time). Not surprisingly, Gauss had an algorithm which is much easier. Let Y be the current year. Then take:

a = Y mod 19

b = Y mod 4

c = Y mod 7

d = (19a + M) mod 30

e = (2b + 4c + 6d + N) mod 7

where M and N are constants depending on the century that don't look that hard to calculate, and which I assume are the corrections I alluded to above; the Wikipedia article gives them in a table. Then Easter falls on the d+e+22 of March or the d+e-9 of April, with certain exceptions which move it up a week when this algorithm gives a very late date for Easter. Basically, d finds the date of the full moon (so M is something like the epact) and e find the day of the week. In the case of this year you get a = 13, b = 0, c = 6; a table gives M = 24, N = 5 for this century, so d = 1, e = 0, and Easter is on the 23rd of March.

As for when Easter usually falls, well, go back to the original description: Easter is the date of the first Sunday after the first full moon on or after March 21. To me this seems like adding two random variables -- the number of days between March 21 and the first full moon, which is roughly uniformly distributed over [0, 29], and the number of days between that moon and the next Sunday, which is uniformly distributed over [1, 7]. There are 210 ordered pairs in ([0, 29] × [1, 7]). One of them sums to 1, giving an Easter date of March 22 in about one year out of 210. Two sum to 2, giving an Easter date of March 23 in two years out of 210. Three sum to 3 (March 24), ..., six sum to 6 (March 27). Seven sum to each of 7 through 30, giving Easter dates of each of March 28 through April 20 in seven years out of 210. Six sum to 31, giving April 21 in six years out of 210, ..., one sums to 36, giving April 26 in one year out of 210.

Indeed, this is basically what computations show, except that for some reason, when the methods given above call for Easter to be on April 26 it gets moved up to April 19. But basically the distribution of Easter dates is just a convolution of two uniform distributions! The Wikipedia article on the computus has a nice graph.

And I have no sympathy for the people quoted in that article. They've known this was coming since 1752, when the UK changed over to the Gregorian calendar. (It perhaps says something about me that I have more sympathy for the bakeries with lots of Irish patrons that are unhappy because Easter was only six days after St. Patrick's day this year.)