On this blog, you can learn any number of feats, including determining the day of the week for any date, and more recently, finding the moon phase for any date.

In a recent comment, regular Grey Matters reader Jay suggested mixing the two, and teaching Conway's method for finding Easter in a given year. Just as you requested, Jay, here it is!

WHEN IS EASTER? Easter is held on the first Sunday following the first full moon following the vernal equinox. It's this detailed definition that makes finding the date of Easter in your head for a given year so impressive. There seems to be an impossible amount of information to know off the top of your head in order to determine the correct date.

If you've practiced both the day for any date feat and the moon phase for any date feat, you should have little trouble performing this feat. The only other thing you need to know is your multiples of 19 from 0 to 190 from memory.

A NOTE ON PRESENTATION: Since you'll be effectively combining 2 other feats, speed shouldn't be the focus of your presentation. If you've ever watched Dr. Arthur Benjamin's TED performance, you've seen the speed with which he squares 2-, 3-, and 4-digit numbers. Notice that, when he squares 5-digit numbers, he doesn't focus on speed, but rather the process itself. Even if aspects of the process aren't clear, he makes it fun just to see the process in action.

That's also a good idea for the Easter date feat. It will allow you do seemingly nonsensical math out loud, yet still entertain the audience and get the right result.

THE PROCESS: The process can be broken down into 2 large steps. First, you find the date of the paschal full moon (the first full moon after the vernal equinox). The next step simply involves determining the day of the week for that date, and then working out the date of the following Sunday.

The only thing you need to start is a year. In this tutorial, I'll assume you're given a year in the 1900s or 2000s. Other centuries will be discussed later. We'll use 1980 as our example year, in order to make the tutorial clearer.

STEP 1: To start, subtract 1900 from the given year, whether that year is in the 1900s or 2000s. If the remaining digits are 19 or more, subtract the nearest multiple of 19 that is equal to or less than those digits. In our 1980 example, 1980 - 1900 = 80, and the nearest multiple of 19 is 76, so we figure 80 - 76 = 4.

It's important to note that we're NOT using the special positive/negative year key taught in the moon phase tutorial.

The next step is to add 1, so our example becomes 4 + 1 = 5. After that, multiply by 11. As it happens, this is easy in this case, as 5 × 11 = 55. You want to be very sure you are comfortable quickly multiplying by 11. If you don't already know how to do this, watch the video below and then try the practice exercises here:



At this point, starting with 1980 has given us 55, but we've only taken the last 2 digits into consideration. To compensate for the century, subtract 6, regardless of whether the year is in the 1900s or the 2000s. In this case, 55 - 6 = 49.

If the number is 30 or more, you can subtract the nearest multiple of 30 equal to or less than the number you have. In this case, the closest multiple of 30 that is equal to or less than 49 is 30, so we work out 49 - 30 = 19.

This final number is how many days the paschal full moon is before April 19th, which you may also need to consider as being “March 50th.” Since we have 19, we could try subtracting that from April 19th, but April 19th - 19 days = April “0th”, which doesn't make any sense. Instead, we do March “50th” minus 19 and get March 31st.

So, in our example, we've determined that March 31st was the date of the paschal full moon in 1980. Verbally, this might sound something like, “1980? Let's see that's 4...5...55 minus 6 is 49...19 days before April 19th, which is March 31st.” Note that you don't have to explain each step, just run through the calculations and let your audience wonder about the numbers. The idea is to make the seemingly nonsensical calculations fun for your audience.

Quick review:
1 - Subtract 1900 from the year, then subtract the largest multiple of 19 that is equal to or less than the last 2 digits of the year: X = (year - 1900) mod 19
2 - Add 1: X + 1
3 - Multiply by 11: 11(X + 1)
4 - Subtract 6 to compensate for the century: (11(X + 1)) - 6
5 - Subtract the nearest multiple of 30 equal to or less than the current number: ((11(X + 1)) - 6) mod 30
6 - Subtract the number you now have from either April 19th or March 50th to get the date of the paschal full moon: (April 19th/March 50) - (((11(X + 1)) - 6) mod 30)

Now, you're ready for the next step.

STEP 2: At this point, you simply use your preferred version of the day of the week for any date feat to work out the day of the week for this date. Personally, I use Day One, but any version will work. At this point, you should have told your audience that March 31, 1980 is a Monday.

From there, work out the date of the following Sunday, and that will be Easter! In our example, it's a little tricky since we;re on the border between 2 months. In this case, the easiest solution is to go 1 day back from Monday (to Sunday, March 30th) and then ahead a week (March “37th” = 37 - 31 = April 6th).

You can have them type Easter 1980 into Wolfram|Alpha to verify for themselves that April 6th, 1980 was indeed the correct date for Easter that year.

ADDITIONAL NOTES: If you know you're always going to be dealing with dates in the 1900s and 2000s, and therefore always subtracting 6, you can simplify from (((11(X + 1)) - 6) mod 30) to the much simpler ((11X + 5) mod 30), thus saving a few steps.

The century adjustment for both the 1900s and 2000s as discussed above, is -6. The proper adjustment for other centuries can be found using this formula in Wolfram|Alpha.

For the 2100s, you'd set h=21, for the 2200s you'd set h=22, etc., and c will be the century adjustment. When working with other centuries, you'll also want to find an easy multiple of 19 to subtract to make things easier. For example, for dates in the 2100s, you could subtract 2090 from the year, since 1900 + 190 = 2090.

To find an easy way to deal with a given century, you can use this Wolfram|Alpha calculator. For the 2100s, you'd set h=21 (just as in the previous calculator), and the calculator returns two new numbers, c=-2100 and d=10. This means that, for years in the 2100s, all you have to do is subtract 2100 and then add 10 to adjust for the proper part of the 19-year cycle.

Since the year gets reduced to multiples of 19, you shouldn't be surprised to discover that there's a 19-year cycle of dates for the paschal full moon. This table gives the dates for the paschal full moon for the years 2014-2032. Each number on the chart that's less than 20 refers to that date in April (8 on the chart, for example, means April 8th). Each number on the chart that's greater than 20 refers to a date in March (23 on the chart, for example, refers to March 23rd). The 0 on the chart refers to “April 0th”, which is really March 31st.

If you're comfortable with memory systems, you could just memorize the paschal full moon dates associated with each year in the 19-year cycle, simplifying the process even more!

What happens if the paschal full moon falls on a Sunday? In that case, Easter wil be on the following Sunday.

If the formula tells you that April 19th was the date of the paschal full moon, step back one day to April 18th. Similarly, if the formula returns April 18th, and your year calculation, after reducing it and adding 1, is 12 or more, step back 1 day to April 17th. Otherwise, always stay on the final date you get.

Try calculating the Easter date for random years, and taking all the rules we've talked about into account, and you'll develop this skill quicker than you ever thought possible!