# Easter Math

Friend: Easter seems early this year. Wonder why it is so early? Surely there's a spiritual reason for it!

Me: The reason is more religious than spiritual.

Friend: What do you mean?

Me: Well, Easter's date is determined by the Spring equinox, but since that date always changes, a church council determined March 21 as the "ecclesiastical equinox."

Friend: Why'd they do that?

Me: There was a debate in the early church about Easter. Some pastors did not celebrate it at all because "all days were to be treated the same." Some in the East celebrated Easter on the Sunday closest to Passover, or on Passover, they were the Quartodecimans. Others celebrated at time that replaced a popular Roman holiday (and I don't mean the 1953 Hepburn film).

Anyhow, the ecclesiastical equinox gave a fixed starting point for Easter.

Friend: So, why is it not just March 21st if that's the so-called ecclesiastical equinox.

Me: Oh, it gets more complicated than that! You need a math degree to determine the actual date of Easter. That's why I said it is more religious than spiritual!

Friend: What do you mean again?

Me: Well, here's the formula to determine the date of Easter. Bear with me.

You start with the year--so this year is 2024. The year number is y, and you use it to calculate the golden number, g: g = y mod 19 + 1.

But we need the date of what is called the "paschal full moon" which has two corrections that are incorporated into the formula: the solar correction, s, and the lunar correction, l.

s = (y - 1600) div 100 - (y - 1600) div 400

l = (((y - 1400) div 100) × 8 ) div 25

So we start with the uncorrected date for the paschal full moon, p'; then we apply a minor correction--to help with leap years and other things--to get the exact date, p, as the number of days after the 21st of March.

p' = (3 - 11g + s - l) mod 30

if (p' == 29) or (p' == 28 and g > 11) then

p = p' - 1

else p = p'

Now we need the date of the following Sunday starting with a 'dominical number', d:

d = (y + (y div 4) - (y div 100) + (y div 400)) mod 7

We calculate d': d' = (8 - d) mod 7

We already have p, the date of the paschal full moon; next we determine p'' the first date in the year which falls on the same day of the week as the paschal full moon. First we determine the 'day number' of p with respect to January 1. This is 31 + 28 + 21 + p = 80 + p.

p'' = (80 + p) mod 7

= (3 + p) mod 7

The difference between d' (the first Sunday in the year) and p'' gives us the number of days that must be added to p to get the date of the following Sunday, which is Easter Sunday. There is one further thing though. This number must lie in the range 1-7, rather than 0-6, since Easter is not allowed to fall on the same day as the paschal full moon. We first determine x', the difference between d' and p'':

x' = d' - p''

= (8 - d) mod 7 - (3 + p) mod 7

= (8 - d - (3 + p)) mod 7

= (5 - d - p)) mod 7

To force this to lie in the range 1-7, we calculate x as x = (x' - 1) mod 7 + 1 = (4 - d - p)) mod 7 + 1

We can now calculate e, the number of days Easter falls after the 21st March:

e = p + x

or

e = p + 1 + (4 - d - p) mod 7

In other words Easter Day is:

if e < 11 then

(e + 21) March

else (e - 10) April

Friend: What. Was. That?

Me: I know, right? Religious, not spiritual; and you need a math degree.

Friend: So Easter's early was my only point. Sheesh. One other question: ever wonder why we use eggs and lilies and bunnies for Easter?

Me: Don't get me started.