How to Calculate Easter’s Date

IMG_5742

Dyed eggs await further decoration.

Easter is so early this year that it got me thinking about how the higher ups know when to put it on the calendar.  Easter occurs annually the first Sunday after the first full moon after the Vernal Equinox.  Did you catch all that?  The Vernal Equinox feels like it was yesterday; in fact, I wrote a blog post about it.  And then a few days later, we had a spectacular full moon, which means that this Sunday is Easter!  Early Christians folded their holidays into preexisting Pagan celebrations in an effort to more easily spread their religion.  But this blog is not about religion or history — it is about math.

So what’s math got to do with Easter?  The German mathematician Carl Friedrich Gauss came up with an algorithm for calculating Easter.  Before we work through the calculation for 2016, let’s make sure you understand “mod” and “floor”.

Mod – the remainder.  27 mod 8 is 3, since 3 is the reminder when you divide 27 by 8.  100 mod 20 is 0, since 20 is a factor of 100.  24 mod 35 is 24, since 24 is less than 35.

Floor – the largest integer less than or equal to a given number.  So floor 12.7 is 12.  Floor 12.99 is 12.  Floor 12 is 12.

Here is a screenshot (from wikipedia) of Gauss’ Algorithm for Calculating Easter:

Screen Shot 2016-03-25 at 6.03.20 PM

Let’s work it through for 2016.  You will need a scratch sheet of paper and a writing implement (and maybe a simple calculator) if you want to calculate it with me.

  1. a = 2016 mod 19… 2016/19 is 106.105…  this means that the remainder when you divide 2016 by 19 is:  2016 – 106(19) which is 2.  So a = 2.
  2. b = 2016 mod 4… 2016/4 is 504… this mans the remainder is 0.  So b = 0.
  3. c = 2016 mod 7… 2016/7 is 288, therefore c also equals 0.  So c = 0.
  4. k = floor(2016/100)… 2016/100 is 20.16; floor 20.16 is 20.  So k = 20.
  5. p = floor((13+8k)/25)… (13 + 8(20))/25) is 6.92; floor 6.92 is 6.  So p = 6.
  6. q = floor(k/4)… 20/4 is 5 and floor 5 is 5.  So q = 5.
  7. M = (15 – p + k – q) mod 30… 15 – 6 + 20 – 5 is 24.  24 mod 30 is 24.  So M = 24.
  8. N = (4 + k – q) mod 7… 4 + 20 – 5 is 19.  19 mod 7 is 5.  So N = 5.
  9. d = (19a + M) mod 30… 19(2) + 24 is 62.  62 mod 30 is 2.  S0 d = 2.
  10. e = (2b + 4c + 6d + N) mod 7.  2(0) + 4(0) + 6(2) + 5 is 17.  17 mod 7 is 3.  So e = 3.
  11. Easter is 22 + d + e March.  22 + 2 + 3 is 27.  So Easter is March 27, 2016.

I leave the calculation of Easter 2017 to you.

 

 

One thought on “How to Calculate Easter’s Date

  1. Pingback: Tax Day | Math, Teaching, and Teaching Math

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