Everyone knows that Christmas Day is on 25th December, but Easter is more complicated. For a start, it always falls on a Sunday, so it can't have the same date every year. The date of Easter can vary by even more than a week, though, because its date is fixed by a combination of the sun's and the moon's behaviour.

Working out the positions of the sun and the moon on different dates is fairly complicated and gives what looks like quite an unpredictable result. In fact, in 1928 a law was passed in Britain to fix Easter as the first Sunday after the second Saturday in April, to make it closer to being the same date every year. But it was never enforced and we've carried on using the sun and the moon as our indicators.

To avoid actually having to work out the behaviour of the sun and the moon, over the years various people (including the famous mathematician Carl Gauss) have worked out "recipes" for Easter day which tell you what calculations to do on the year number to find out the date of Easter that year. Some of these recipes involve looking up numbers and letters in tables (a method like this is given in the Book of Common Prayer ), but I'm going to describe one which doesn't need any tables - just a pencil and paper and a calculator. Oh, and your 19-times table...

First, Choose Your Calendar

As you know, it takes the earth a bit over 365 days to travel round the sun. So if our years were always 365 days long, after a few years our calendar would start to get out-of-synch with the earth's orbit and we would notice the seasons (which depend on where the earth is in its orbit) starting on different dates. To stop this happening, in about 45 BC Julius Caesar introduced the rule that every 4 years we add an extra day to the end of February so that the seasons can "catch up" with us. This way of deciding the date is still called the Julian calendar.

In the Julian calendar, years are 365.25 days long on average. However, the time it takes for the Earth to orbit the sun is actually slightly less than this - about 365.2422 days. So after several hundred years the seasons had "caught up" with the calendar too much, and were starting earlier (according to the calendar) than they should have been. To solve the problem, in 1582 Pope Gregory introduced the Gregorian calendar, which is the calendar we still use today.

The Gregorian calendar has a leap year:
So the leap years we've skipped since 1582 are 1700, 1800 and 1900.

The Gregorian calendar gives you 97 leap years every 400 years, so the average calendar yearis 365.2425 days long, which is very close to the actual time it takes us to go round the sun. To make up for all the leap years we'd had since the Julian calendar was introduced, but which by the Gregorian calendar we should have skipped, when the new calendar was introduced some days had to be missed out. In France, for example, people went to bed on Sunday, December 9th, 1582 and got up again on Monday, December 20th 1582.

Not everyone adopted the Gregorian calendar straight away. Britain did not start using it until 1752 and Russia not until 1918. In some counties, although the Gregorian calendar is used for most things, to this day it is not used to calculate the dates of festivals including Christmas and Easter. That's why there's often more than one date for Christmas and Easter, used in different countries.

The method I'm using to calculate Easter here works for the Gregorian calendar, so you can use it to work out when Easter will be celebrated in Britain and other Western European countries. I've tested it using a spreadsheet I made*, and it works almost all the time, but got it wrong once or twice in 100 years or so. If anyone knows what's wrong, I'd be very interested to hear from you!

* If you like writing spreadsheets or computer programs, a program to calculate the date of Easter in any year is quite an interesting thing to make. Then you can do things like work out how long it takes for the sequence of dates to start all over again, and what the frequency of each date is over the whole cyle.

What Day of the Week is It?

We're going to need to know how to work out what day of the week different dates are, so that we can make sure our date for Easter falls on a Sunday. Here's a way to do it:

Every year has a special day of the week called a domesday. The domesday is the day of the week on which the last day of February falls that year. For example, 2007 wasn't a leap year so the last day of February was 28th, which was a Wednesday. Wednesday is the domesday for 2007. Because 2004 was a leap year, there were 29 days in February that year. February 29th 2004 was a Sunday, so the domesday for 2004 is Sunday.

If the last day of February is a domesday, then it will be the domesday again exactly one week later, on 7th March, and exactly one week after that, on 14th March, and so on. We can write down one date each month which will be a domesday:

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Domesday 31 or 32 28 or 29 7 4 9 6 11 8 5 10 7 12

In most months (March to December), the domesday is always on the same date. In February it is on 28th normally, or 29th in a leap year. In January it is on 31st normally. In a leap year, it falls on "32nd January", which is another way of saying 1st February.

You can calculate the domesday for any year you like. First, we give each day of the week a number:

Sunday MondayTuesdayWednesdayThursdayFridaySaturday
0 1 234 5 6

(I know I said there would be no tables, but you could have worked these ones out for yourself...)

Next, you need to know the domesday in some other, earlier year - say the century year, the year at the start of a century. The century year for this century (21st) was 2000, and the domesday was Tuesday. The domesday in 1900 was Wednesday and in 1000 was Thursday.

To find the domesday for a given year, you add to the domesday for a century year:
the number of dozens after that year
the remainder after this and
the number of fours in the remainder.

So what was the domesday in 1066?
The domesday in 1000 was Thursday (4 ).
There were 66 years after that, which is 5 whole dozens.
The remainder after the 5 dozens is 66 - 60 = 6.
The number of whole fours in 6 is 1.

The domesday in 1066 was 4+5+6+1=16.
But because the days of the week come round again every 7 days, 16 is the same as 16-7=9, which is the same as 9-7=2.
The domesday in 1066 was Tuesday.

Now you can use the domesday to calculate the day of the week for any date. The battle of Hastings was fought on October 14th 1066. We know that October 10th was a Tuesday, because the table above says October 10th was a domesday. So October 14th, four days later, must have been a Saturday.

If you'd like to try some more, you could calculate the day the First World War ended (November 11th 1918), 1st January 2000, or the day you were born.

Be careful with years around the time we switched to the Gregorian calendar, though. When we changed the date to catch up with the seasons, we carried on with the days of the week as normal. So this way of finding the day of the week won't work if the calendar changed between your century year and the date you're interested in.

The Sun, the Moon and the Calendar

The day chosen for Easter day is the first Sunday after the first full moon after March 21st. (March 21st is more or less the spring equinox.) In other words, to get to the date of Easter you start on the spring equinox - the day in (the Northern hemisphere's) spring when the day and the night are the same length - then you wait for the next full moon, and then you wait for the next Sunday.

The problem is that the sun and the moon don't run to the same timetable. As we said, it takes the earth about 365.242 days to travel once round the sun (a tropical year ), so that's how often the spring equinox happens. There are about 29.53 days from one full moon to the next (a synodic month ). Because a tropical year doesn't divide into a whole number of synodic months but has about 11 days left over, the number of days between the equinox and the next full moon changes by 11 days or so every year.

So if we want to know when Easter is, we have totake into account how far out-of-synch the sun and the moon are this year. Here's a procedure which does that, and gives you the date of the first full moon after the equinox - the Paschal full moon.
Let's take 1985 as an example:
    • (1985÷19)=104  r9, so the Golden number is 10.
    • 11×10=110
    • The year begins "19", so we add -6: 110-6=104
    • 104÷30=3  r14, so write down 14.
The number you've got written down tells you how much earlier the Paschal full moon was than 19th April (which you could also write as 50th March).

So let's find the date of the Paschal full moon in 1985:
The Paschal full moon in 1985 was on April 5th.

So When's Easter?

Got that? If so, now (at last) you can find out the date of Easter day. I'll demonstrate with 1985 again:

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