## When is Easter likely to happen?

It’s a natural question to wonder this time of year. The date when Easter falls is calculated by some tricky numerical rules. These come from the desire to make Easter an early-spring (in the Northern hemisphere) holiday, while tying it to the date of Passover, as worked out by people who did not know the exact rules by which the Jewish calendar worked. The result is that some dates are more likely than others to be Easter.

A few years ago I wrote a piece finding how often Easter would be on different dates, in the possible range from the 22nd of March through the 25th of April. And discussed some challenges in the problem. Calendars are full of surprising subtle problems. Easter creates a host of new challenges.

## What’s the most probable date for Easter? What’s the least?

This is a nice chance to highlight one of my older pieces. I had been wondering about the most and the least likely dates for Easter. And I calculated several hundred years’ worth of Easters, to find when they’re most and least likely to happen.

The 22nd of March is the least probable date for Easter. That date was last Easter in 1818, and will next be Easter in 2285. The 12th of April, though? That’s one of the most likely dates for Easter. To say what is “the” most probable date for Easter requires some thought. First, what it means to talk about the chance of an algorithmically defined quantity. Second, what it means to look at Easter. The holiday is intended to happen early in the European spring. But the start of European spring is moving through the calendar. Someday we will abandon the Gregorian calendar, or radically change the calculation of Easter. This makes it harder to say how often each possible date turns up. But we can make some rough answers.

The 15th of April is the most probable date for Easter, if we look at a 532-year span. (There are astronomical reasons to look at 532 years.) If we look at a more limited stretch, 1925 to 2100, on the assumption that that’s the maximum spread of dates that anyone alive today can be expected to see, then we have ten dates equally common, the 12th of April among them.

## What Dates Are Most Likely For Easter?

I had a slight nagging feeling about this. A couple years back I calculated the most and least probable dates for Easter, on the Gregorian calendar, using the current computus. That essay’s here, with results about how often we can expect Easter and when. It also holds some thoughts about whether the probable dates of Easter are even a thing that can be meaningfully calculated. And it turns out, uncharacteristically, that I forgot to do a follow-up calculating the dates of Easter on the Julian calendar. Maybe I’ll get to it yet.

## What Is The Most Probable Date For Easter? What Is The Least?

If I’d started pondering the question a week earlier I’d have a nice timely post. Too bad. Shouldn’t wait nearly a year to use this one, though.

My love and I got talking about early and late Easters. We know that we’re all but certainly not going to be alive to see the earliest possible Easter, at least not unless the rule for setting the date of Easter changes. Easter can be as early as the 22nd of March or as late as the 25th of April. Nobody presently alive has seen a 22nd of March Easter; the last one was in 1818. Nobody presently alive will; the next will be 2285. The last time Easter was its latest date was 1943; the next time will be 2038. I know people who’ve seen the one in 1943 and hope to make it at least through 2038.

But that invites the question: what dates are most likely to be Easter? What ones are least? In a sense the question is nonsense. The rules establishing Easter and the Gregorian calendar are known. To speak of the “chance” of a particular day being Easter is like asking the probability that Grover Cleveland was president of the United States in 1894. Technically there’s a probability distribution there. But it’s different in some way from asking the chance of rolling at least a nine on a pair of dice.

But as with the question about what day is most likely to be Thanksgiving we can make the question sensible. We have to take the question to mean “given a month and day, and no information about what year it is, what is the chance that this as Easter?” (I’m still not quite happy with that formulation. I’d be open to a more careful phrasing, if someone’s got one.)

When we’ve got that, though, we can tackle the problem. We could do as I did for working out what days are most likely to be Thanksgiving. Run through all the possible configurations of the calendar, tally how often each of the days in the range is Easter, and see what comes up most often. There’s a hassle here. Working out the date of Easter follows a rule, yes. The rule is that it’s the first Sunday after the first full moon after the spring equinox. There are wrinkles, mostly because the Moon is complicated. A notional Moon that’s a little more predictable gets used instead. There are algorithms you can use to work out when Easter is. They all look like some kind of trick being used to put something over on you. No matter. They seem to work, as far as we know. I found some Matlab code that uses the Easter-computing routine that Karl Friedrich Gauss developed and that’ll do.

Problem. The Moon and the Earth follow cycles around the sun, yes. Wait long enough and the positions of the Earth and Moon and Sun. This takes 532 years and is known as the Paschal Cycle. In the Julian calendar Easter this year is the same date it was in the year 1485, and the same it will be in 2549. It’s no particular problem to set a computer program to run a calculation, even a tedious one, 532 times. But it’s not meaningful like that either.

The problem is the Julian calendar repeats itself every 28 years, which fits nicely with the Paschal Cycle. The Gregorian calendar, with different rules about how to handle century years like 1900 and 2100, repeats itself only every 400 years. So it takes much longer to complete the cycle and get Earth, Moon, and calendar date back to the same position. To fully account for all the related cycles would take 5,700,000 years, estimates Duncan Steel in Marking Time: The Epic Quest To Invent The Perfect Calendar.

Write code to calculate Easter on a range of years and you can do that, of course. It’s no harder to calculate the dates of Easter for six million years than it is for six hundred years. It just takes longer to finish. The problem is that it is meaningless to do so. Over the course of a mere(!) 26,000 years the precession of the Earth’s axes will change the times of the seasons completely. If we still use the Gregorian calendar there will be a time that late September is the start of the Northern Hemisphere’s spring, and another time that early February is the heart of the Canadian summer. Within five thousand years we will have to change the calendar, change the rule for computing Easter, or change the idea of it as happening in Europe’s early spring. To calculate a date for Easter of the year 5,002,017 is to waste energy.

We probably don’t need it anyway, though. The differences between any blocks of 532 years are, I’m going to guess, minor things. I would be surprised if the frequency of any date’s appearance changed more than a quarter of a percent. That might scramble the rankings of dates if we have several nearly-as-common dates, but it won’t be much.

So let me do that. Here’s a table of how often each particular calendar date appears as Easter from the years 2000 to 5000, inclusive. And I don’t believe that by the year we would call 5000 we’ll still have the same calendar and Easter and expectations of Easter all together, so I’m comfortable overlooking that. Indeed, I expect we’ll have some different calendar or Easter or expectation of Easter by the year 4985 at the latest.

For this enormous date range, though, here’s the frequency of Easters on each possible date:

Date Number Of Occurrences, 2000 – 5000 Probability Of Occurence
22 March 12 0.400%
23 March 17 0.566%
24 March 41 1.366%
25 March 74 2.466%
26 March 75 2.499%
27 March 68 2.266%
28 March 90 2.999%
29 March 110 3.665%
30 March 114 3.799%
31 March 99 3.299%
1 April 87 2.899%
2 April 83 2.766%
3 April 106 3.532%
4 April 112 3.732%
5 April 110 3.665%
6 April 92 3.066%
7 April 86 2.866%
8 April 98 3.266%
9 April 112 3.732%
10 April 114 3.799%
11 April 96 3.199%
12 April 88 2.932%
13 April 90 2.999%
14 April 108 3.599%
15 April 117 3.899%
16 April 104 3.466%
17 April 90 2.999%
18 April 93 3.099%
19 April 114 3.799%
20 April 116 3.865%
21 April 93 3.099%
22 April 60 1.999%
23 April 46 1.533%
24 April 57 1.899%
25 April 29 0.966%

If I haven’t missed anything, this indicates that the 15th of April is the most likely date for Easter, with the 20th close behind and the 10th and 14th hardly rare. The least probable date is the 22nd of March, with the 23rd of March and the 25th of April almost as unlikely.

And since the date range does affect the results, here’s a smaller sampling, one closer fit to the dates of anyone alive to read this as I publish. For the years 1925 through 2100 the appearance of each Easter date are:

Date Number Of Occurrences, 1925 – 2100 Probability Of Occurence
22 March 0 0.000%
23 March 1 0.568%
24 March 1 0.568%
25 March 3 1.705%
26 March 6 3.409%
27 March 3 1.705%
28 March 5 2.841%
29 March 6 3.409%
30 March 7 3.977%
31 March 7 3.977%
1 April 6 3.409%
2 April 4 2.273%
3 April 6 3.409%
4 April 6 3.409%
5 April 7 3.977%
6 April 7 3.977%
7 April 4 2.273%
8 April 4 2.273%
9 April 6 3.409%
10 April 7 3.977%
11 April 7 3.977%
12 April 7 3.977%
13 April 4 2.273%
14 April 6 3.409%
15 April 7 3.977%
16 April 6 3.409%
17 April 7 3.977%
18 April 6 3.409%
19 April 6 3.409%
20 April 6 3.409%
21 April 7 3.977%
22 April 5 2.841%
23 April 2 1.136%
24 April 2 1.136%
25 April 2 1.136%

If we take this as the “working lifespan” of our common experience then the 22nd of March is the least likely Easter we’ll see, as we never do. The 23rd and 24th are the next least likely Easter. There’s a ten-way tie for the most common date of Easter, if I haven’t missed one or more. But the 30th and 31st of March, and the 5th, 6th, 10th, 11th, 12th, 15th, 17th, and 21st of April each turn up seven times in this range.

The Julian calendar Easter dates are different and perhaps I’ll look at that sometime.