Easter Sunday calculation

Easter Sunday falls on a different day each year.  It can be as early as March 22 (the last one was in 1818; the next in 2285), or as late as April 25 (the last in 1943; the next in 2038).  This is because the date of Easter Sunday is tied to the lunar calendar.

UPDATE: Thanks to Bob Wright for spotting an error in one of the instructions. This is the corrected versiom.

The official definition is

‘Easter is the first Sunday
after the first ecclesiastical full moon
that occurs on or after March 21’.

Why March 21? That’s the first day of spring, or the Vernal Equinox. Here’s a detailed article on the date for Easter.

In 1800, the mathematician Gauss worked out the details for finding the date of Easter for any given year (after about 325). His algorithm accounts for lunar cycles and leap years,  among other things.

Since this is close to tax time – and may of you have probably been filling out the many and varied IRS forms – I thought it would make sense to present Gauss’ algorithm in a tax-form format.

Each line involves only one calculation, to reduce the chance of error.  I’ve included the results for 2013. (Easter is March 31).

If you print out the page, there’s space on the right to go through the calculations for any other year. (Start with 2013 to make sure you get the same results.)

You can check your results with tables of Easter dates.

1 Write the year 2013
2 Divide Line 1 by 19 and write the remainder [see Footnote] 18
3 Divide Line 1 by 4 and write the remainder 1
4 Divide Line 1 by 7 and write the remainder 4
5 write the first two digits of Line 1 20
6 Multiply Line 5 by 8 160
7 Add 13 to Line 6 173
8 Divide Line 7 by 25 and write the integer part 6
9 Divide Line 5 by 4 and write the integer part 5
10 Subtract Line 8 from 15 9
11 Add Line 10 to Line 5 29
12 Subtract Line 9 from Line 11 24
13 Divide Line 12 by 30 and write the remainder 24
14 Add 4 to Line 5 24
15 Subtract Line 9 from Line 14 19
16 Divide Line 15 by 7 and write the remainder 5
17 Multiply Line 2 by 19 342
18 Add Line 13 to Line 17 366
19 Divide Line 18 by 30 and write the remainder 6
20 Multiply Line 3 by 2 2
21 Multiply Line 4 by 4 16
22 Multiply Line 20 by 6 36
23 Add Line 20 to Line 21 18
24 Add line 22 to Line 23 54
25 Add Line 16 to Line 24 59
26 Divide Line 24 by 7 and write the remainder 3
27 Add 22 to Line 19 28
28 Add Line 26 to Line 27 31
29 If Line 28 is between 1 and 31, this is the date of Easter Sunday in March. Otherwise, continue to Line 30
30 Add Line 19 to Line 26
31 Subtract 9 from Line 30
32 This is the date of Easter Sunday in April

Footnote about remainders: If you’re doing all this with pencil and paper, you’ll get the remainder as a result of the division. But pencil and paper can be error-prone, so it’s probably better to use a calculator. Here’s how to get the remainder using a calculator:

Take Line 2, for example: “Divide Line 1 by 19 and write the remainder” Line 1 is the year, so we want the remainder of 2013 divided by 19. (For mathematicians and programmers, “mod(2013,9)”.)

Enter 2013.
2013
[divide by 19]
/
19
=
Subtract the integer part:

105
[multiply by the divisor]
*
19
=
See 18, the remainder.

Also, if the dividend (the number) is smaller than the divisor, the remainder is the number. For example, the remainder of 23 divided by 30 is 23.

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