### Introduction

Regardless of your particular religious faith, working out the date of Easter in your head for a given year can be an impressive feat.

It's well known that the date for Easter moves around quite a bit, and many people aren't even sure exactly how the Easter date is determined. Despite the rules, it is still possible to mentally calculate the date for a given year.

The Rules

Easter is defined as the first Sunday after the first full moon after the first day of spring in a given year.

It's important to note that, while the official rules for calculating the date for Easter in a given year involve the full moon date and the first day of spring, these are not necessarily tied to the astronomical definitions.

For example, depending on the year, the first day of spring can happen on March 20th or March 21st. For the purposes of Easter calculation, however, the first day of spring is *always* considered to be March 21st. The first full moon after the first day of spring, then, must always happen *after* March 21st.

This may seem like a small detail, but it is important to understand the difference between this and the astronomical definition. For example, in 2038, the first day of spring will happen on March 20th, and the next full moon will happen the next day, on March 21st. This full moon doesn't count for Easter calculation, though, because March 21st is always considered to be the first day of spring, so it's the first full moon *after* that date that is considered.

Finally, if the full moon occurs on a Sunday, then that Sunday is not considered to be Easter, but rather the following Sunday.

So, to sum up the rules for Easter, it falls on the first Sunday *after* the first full moon *after* March 21st, the first day of spring. Yes, this is why the Easter date varies so widely from year to year.

### Getting Started

Despite the complexity of these rules, the calculation is still feasible to perform mentally, thank largely to the work of John Conway, whose contributions to mathematics I've noted many times on this blog. This tutorial is adapted from his instructions in “Winning Ways for Your Mathematical Plays, Volume 4”.

First, you do need to be able to perform some version of a calendar calculation. Here on Grey Matters, you can find my *Quick Calendar Month Creation* and the *Day of the Week For Any Date (Revised)* tutorials. Elsewhere, this Doomsday algorithm tutorial, this first Sunday Doomsday variation, and my commercial *Day One* notes will also work.

You'll need to be able to calculate given dates to determine the day of the week on which a given date falls, so that you'll know when the next Sunday occurs.

You should also be familiar with the basics of moon phases and the importance of the various numbers in moon calculations. You don't need to be able to calculate the moon phase for a given date; it's more important that you reasons for the numbers.

If you're up to speed on that, then you're ready to begin learning how to calculate the Easter date for a given year!
### Easter Calculations

Since the first day of spring is always considered to be March 21st, the first full moon must happen by April 19th. The first step in Easter calculations, then, is working out how many days before April 19th the full moon occurs.

The only information you need to start is the year in question. For this part of the tutorial, we'll assume that you're working with a year from 1900 to 2099. Since

the moon has a 19-year cycle, you begin by finding out where it is in that 19-year cycle. First, you simply subtract 1900 from the year. Next,

assuming you known your multiples of 19, you will subtract the largest multiple of 19 equal to or less than this number.

For example, let's say you're given the year 1988. The first step is easy: 1988 - 1900 = 88. Next, you subtract the largest multiple of 19 which is equal to or less than 88. In this case, that happens to be 76, so 88 - 76 = 12. In other words, 1988 is 12 years into the

moon's 19-year Metonic cycle.

The next step is to take this result, and multiply it by 11. If you're worried about multiplying by 11 in your head, you can pick this skill up quickly:

The best part is that the only 2-digit numbers you'll have to deal with in this feat are 10 through 18, so you'll never have to worry about carrying that 1!

After multiplying the number by 11, there are just 2 more steps: Add 5, and then subtract the largest multiple of 30 (due to the

moon's roughly 30-day cycle) that is equal to or less than this current total.

Returning to our 1988 example, you'll remember we reduced that to 12. 12 × 11 = 132, and 132 + 5 = 137. The largest multiple of 30 that is equal to or less than 137 is 120, so we subtract 137 - 120 = 17.

The result you get at the end of this process is the number of days before April 19th. In our 1988 example, our result of 17 means that the first full moon after the first day of spring falls on April 2nd (because April 19th - 17 days = April 2nd).

What happens if you get a number of 19 or more? In that case, you think of April 19th as being the same as the imaginary date of “March 50th” So, if you came up with a total of 24, for example, you'd think of March 50th - 24 days = March 26th.

As you'll only need to work with the numbers 0 through 18,

here's all the possible answers you can get by working through the above calculations.

**EXCEPTIONS**
There are 2 small exceptions. If you begin with a total of 5, you'll get a result of 0, which suggests that the first full moon after the first day of spring is April 19th (0 days before April 19th). You need to make a correcting by moving 1 day back to April 18th, in this case.

The other special case is when you have a total of 16, which gives a result of 1, which suggests that the date is April 18th (1 day before April 19th). This one is also incorrect, and has the same solution; you move 1 more day back to April 17th.

To become comfortable with the full moon calculations before proceeding,

go over to the *Easter Date For A Given Year Quiz*, and click on the

*Full Moon Date* button in the

*Roman Catholic Easter, 1900-2099* section.

### Full Moon to Easter

This is the point at which you'll need to be comfortable with some version of the Day of the Week For Any Date feat,

as I mentioned in the previous tab.

Obviously, you'll need to work out the day of the week of date you determined in the previous step. In our 1988 example, we worked out that the first full moon after the first day of spring occurred on April 2nd, 1988. If you know your calendar calculations, you'll be able to work out that this was a Saturday. Obviously, the first Sunday after this date, and therefore Easter, is the very next day, April 3rd, 1988.

One important note at this point: If the full moon date occurs on a Sunday itself, then Easter is the

*following* Sunday.

As another example that will help highlight a few key points, let's work out the date of Easter in 1994. 1994 - 1900 = 94. Subtracting the largest multiple of 19 equal to or less than than gives us 94 - 76 = 18.

Continuing the calculation, we have 18 × 11 = 198, and 198 + 5 = 203. Casting out the largest multiple of 30 from 203 gives us 203 - 180 = 23. 23 days before April 19th, or rather March 50th in this case, is March 27th (March 50th - 23 days = March 27th).

You then work out that March 27th, 1994 is a Sunday. That means that the following Sunday is Easter. March 27th plus 7 days is April 3rd (March 27th + 7 days = March 34th, minus 31 days in March, gives you April 3rd).

We can check the answer with Wolfram|Alpha, which

confirms that Easter in 1994 did occur on April 3rd!

At this point, you should practice what you've learned over

in the quiz section. Click on the

*Full Moon and Easter Dates* button in the

*Roman Catholic Easter, 1900-2099* section to become comfortable calculating the full moon and Easter dates together.

Once you get comfortable with the calculations, it's time to

learn a few tips on presenting this feat before others.

### Presentation

The simplest way to start is by asking for the year. Since you have to know some version of the day of the week for any date feat, you can likely use this information to say something about the year. For example, in my

*Day One* routine, you can instantly give the days of the week for Christmas and Halloween that year, as well as which months in that year will have a Friday the 13th.

Such lead-in information given instantly makes a good introduction to calendar feats, but isn't absolutely necessary. In versions of the date feat such as the Doomsday algorithm, which give you a key day for the year, this information can be helpful to work out in advance of the Easter calculations.

You can go directly into the Easter feat, as well. The trick to presenting the Easter calculation before an audience, however, is talking through the major steps while you do the mental calculations.

For example, let's say someone gives you the year 1982. Doing a quick mental subtraction of 82 - 76 (do you remember why?), you get an answer of 6, and can immediately say, “In 1982, the moon was 6 years into its 19-year

Metonic cycle...” As you say that sentence, you can then quickly multiply 6 by 11, getting 66, add 5 to get 71, and subtract 71 - 60 to get 11. Since you know you need to subtract this number from April 19th/March 50th, you can quickly finish by saying, “...which means that April 8th was the date of the first full moon after the first day of spring.”

Already, this feat should seem impressive to the audience. At this point, as you already know, you'll have to work out the day of the week for April 8th, 1982 in your head. In this particular case, you should be able to say, “Now, April 8th, 1982, was a Thursday, so the following Sunday was only 3 days later, on April 11th, which makes that the date of Easter in 1982!”

Have your audience

verify the date of Easter in 1982 with Wolfram|Alpha, or a similar source, and they should be suitably impressed!. At that point, you've done the whole Easter feat in front of the audience.

### Tips

• Working through the equations while talking is a bit of a challenge, but with practice, it's not that difficult.

• If you can master a day of the week for any date feat, this Easter feat, and

giving the phase of the moon for any date, you'll have enough material for a human calendar routine!

• The method you've learned is actually the way to calculate the date for the Roman Catholic Easter. There's an alternative calculation for what's known as Orthodox Easter, celebrated primarily by Greek Orthodox and Russian Orthodox churches, and can sometimes fall on a different date. Once you've mastered the basic calculation, you can

learn how to calculate the date for Orthodox Easter here.

### Orthodox Easter

Easter has been around longer than the

Gregorian calendar, of course, so the original calculations were done using the

Julian calendar. Some churches, such as the Russian and Greek Orthodox churches, still use the Julian calendar in their Easter date calculations. This means that the date for Orthodox Easter can sometimes be a different date entirely than the Roman Catholic Easter, sometimes as much as a month apart!

Fortunately, if you can work through the original Easter calculations, calculating the Orthodox Easter date isn't that much different.

To calculate Orthodox Easter, you don't have to start over again. For example, returning to

our 1982 example from the Presentation tab, you start by recalling that you calculated the full moon date as 11 days before April 19th/March 50th.

To this number, you need to add 9. Sometimes, this will give you a number larger than 30, in which case you need to subtract 30 from that number. You can then give this new date as the date of the first full moon after the first day of spring in the Julian calendar.

Continuing with our 1982 example, then, you'd add 11 + 9 to get 20, and work out that April 19th/March 50th minus 20 days is March 30th, which is the date of the first full moon after the first day of spring in 1982 is March 30th

*in the Julian calendar*.

To adjust this date to the currently-used Gregorian calendar, you need to add 13 days. The best way to think of this when starting with a date in March is to just add 13, giving the ridiculous date of March 43rd, and then subtract 31 (since March has 31 days), giving April 12th as the Gregorian date of the first full moon after the first day of spring by the standards of the Orthodox Church.

You can

quiz yourself on the Orthodox Julian and Gregorian full moon dates by clicking on the

*Full Moon Dates* button in the

*Orthodox Easter, 1900-2099* section. Because you need to work out the Roman Catholic full moon and Easter dates first, the quiz will ask you about those before asking about the Orthodox Julian and Gregorian full moon dates.

To find the next Sunday after that is also simple. In our 1982 example, you worked out that the Roman Catholic Easter fell on April 11th. Obviously, April 12th is after that, so the Orthodox Easter that year must fall on the next Sunday, which is April 18th. Clicking on the

*Orthodox Easter Sunday (Christianity)* button, your audience can verify that April 18th, 1992 is indeed the date of the Orthodox Easter.

Notice how the use of earlier calculations helps simplify the calculations for the Orthodox Easter, while calculating an entirely different Easter date with a different calendar seems much harder and more impressive to your audience!

You can

practice the full feat in the quiz section using the

*Full Moon and Easter Dates* button in the

*Orthodox Easter, 1900-2099* section. As with the above quiz, you'll be asked about the Roman Catholic full moon and Easter dates first, as you'll need those to make the Orthodox calculations.

### Presentation

After giving the Roman Catholic Easter date for a given year, you can point out that there's also an Orthodox Easter date, and that it always isn't the same. If they've

verified your given Easter date with Wolfram|Alpha, you can draw your audience's attention to the fact that Wolfram|Alpha has a little blurb that reads, “Assuming Easter Sunday (Christianity) | Use Orthodox Easter Sunday (Christianity) instead”. Explain to your audience that the Orthodox churches use an older calendar, as opposed to the currently-used one.

Recalling your original calculations for the number of days before April 19th/March 50th, you mentally add 9, subtract 30 if needed, and give this new date (March 30th, in our above 1982 example) as the old-style calendar date for the date of the first full moon after the first day of spring.

Next, mentally add 13 days, subtracting 31 if needed, and give this date (April 12th in our above 1982 example) as the new-style calendar date as the date of the first full moon after the first day of spring.

Finally, work out the date of the following Sunday (April 18th in our above 1982 example) and give that as the Orthodox Easter date. As always, have your audience verify this date for themselves.

### Tips

• Similar rules apply to both the Roman Catholic and Orthodox Easter dates. Especially important is the rule that, if a full moon date falls on a Sunday, then it's the

*following* Sunday on which Easter falls.

• Recall from the

*Easter* tab that some dates require adjustments in which you move 1 day back. When returning to the full moon date calculation, you want to add 9 to the

*unadjusted* number of days. For example, 1981 is 5 (81 - 76) years into the Metonic cycle, which results in a total of 0 days before April 19th/March 50th, but you have to adjust 1 day back. When you do the Orthodox calculation, you still want to start with 0 (the unadjusted amount), instead of 1 (the adjusted amount), and add 9 (0 + 9 = 9).

• Being able to calculate both the Roman Catholic and Orthodox Easter dates from 1900 to 2099 is quite an impressive feat. If you're curious about other years, you can

learn how to handle other centuries in the next tab!

### Other Centuries

Due to all the calculations involved, the easiest thing to do is to stay within the 1900 to 2099 range, as you've already learned. However, if you do wish to calculate Easter for other centuries, this section will give you the tools you'll need.

### The 2100s

The easiest extension of the Easter feat is to calculate Easter in the 2100s. As long as you can adjust your favorite day of the week for any date feat for dates in the 2100s, you can calculate Easter dates without any other changes!

### Roman Catholic Easter Dates: 1500s through 3000s

The

Gregorian calendar was first used in late 1582, so the Roman Catholic Easter date can only be calculated for years going back to 1583.

In the standard version you've learned, you start by subtracting 1900, and then subtracting multiples of 19 in order to learn how far the moon is into its

19-year Metonic cycle. For the following years, you can start by use the corresponding amounts below:

- 1583 to 1709: subtract 1520
- 1710 to 1899: subtract 1710
- 2100 to 2279: subtract 2090
- 2280 to 2469: subtract 2280
- 2470 to 2659: subtract 2470
- 2660 to 2849: subtract 2660
- 2850 to 3039: subtract 2850
- 3040 to 3099: subtract 3040

Subtracting these amounts can be made easier by subtracting the hundreds amount, followed by the tens. For example, subtracting 2280 can be done by subtracting 2200 and then subtracting 80. From this point, you reduce the year by multiples of 19 as you learned previously.

The next steps involve multiplying that amount by 11, and then adding 5. Adding 5, however, only works if you're dealing with the 1900s, 2000s, or 2100s. You'll need a different offset for other centuries. The following widget will calculate the required offset for centuries from the 1500s to the 3000s:

The answer you get will be in the form of x = some number and y = some number. x is the first two digits of the century, such as 20 for the 2000s, and y is the offset you'll need to add for years in the given century. For example, if you select the 2300s, you'll get the answer x=23, y=3. This simply means that for years beginning with 23 (x), you'll need to add 3 (y).

Naturally, you shouldn't forget to adjust your day of the week for any date feat calculations for the same centuries.

### Orthodox Easter Dates: 1500s through 3000s

Determining how far the moon is into its Metonic cycle is done just as in the previous section. When working in the 1900s and 2000s (and 2100s!), you normally need to add 9 to adjust for the Orthodox date (and subtract 30, if needed).

For other centuries, you may need different numbers. The widget below can calculate the required offsets for any given century:

Similar to the above widget, x represents the first 2 digits of the request century, y is the adjustment for the Julian date, and z is the number of days you need to move ahead to get the Gregorian date.

If you select the 2000s, you'll get 3 very familiar numbers: x=20, y=9, z=13. If you've practiced the Orthodox feat, you'll recall that, for years beginning with 20 (x), you need to add 9 (y) more days (subtracting 30, if needed) back from the Roman Catholic full moon to get the Julian date for the Orthodox full moon. You then need to go 13 (z) days forward to get the equivalent Gregorian calendar date.

If you want to work with the 2300s instead, just select 2300s in the pop-up menu, click the

*Submit* button, and you'll get a result of x=23, y=11, z=16. This means that for any year beginning with 23 (x), you'll need to move back 11 (y) more days from April 19th/March 50th to adjust for the Julian calendar date (remembering to subtract 30, if needed). 16 (z) is the number of days you'll need to move forward from the Julian date to get the corresponding Gregorian date.

You can even calculate the Gregorian offset (z) in your head, if need be. To learn how to do that, read

my *Changing Calendars Mentally* post.

Practice this feat, and you'll have an unusual mental feat ready to astound almost any audience! If you have any questions or comments, please include them in the comments section below.