# Easter Calculator

Created by Davide Borchia
Last updated: Jul 08, 2022

Easter's Sunday jumps around like a rabbit: with our Easter calculator you will only have to insert the year: we will run the algorithm and swimmingly give you the answer

• What is Easter, and why does its date move;
• How to calculate Easter's date with the Moon;
• The calculation of the Easter date's formula, just with math!

Our other tool can help you calculating the day of the week, so be sure to check!

## What is Easter?

Eater is one of the most important Christian holiday: on this day, believers celebrate the resurrection of Christ, following the events of the Passion.

Over the years, Easter as a purely religious festivity has been slowly complemented by a set of more mundane celebrations: from eggs treasure hunts to the gifting of chocolate, the religious functions are not the only part of this holiday anymore.

## Which Sunday? How to calculate the Easter date

Easter is a moveable feast: to determine the date of Easter in a given year, we need to perform the traditional computus, or... look at the sky.

The definition of Easter's date is the first Sunday after the Paschal full moon (the first full moon after (or coinciding with) the 21st of March.

The 21st of March is not chosen randomly! It is a good, fixed approximation of the spring equinox: Easter is the only Christian festivity with a connection to the natural cycles.

What to do if you expect rainfall on Easter or any other day of the year? Use the volume of rainfall formula to estimate how much water you can collect!

## The Easter date calculation formula

To calculate Easter's date, we need to calculate first the date of the Paschal full moon: this is not an easy task, and usually, we use tables to find that date.

Since the repetition of the lunar phases' dates follows the Metonic cycle, with a length of $19$ years, we can use relatively short tables to find the corresponding day of the week. The Easter date calculation is straightforward from there.

And then again: what if you don't have the tables? And how to consider the discrepancies that the Metonic cycle accumulates?

Gauss, the XIX century's polymath comes in our help. He was the first one to devise an algorithm to calculate the date of Easter just by knowing the year. Apart from some minor mistakes, now corrected, the procedure is accurate. Let's see it:

1. Call the year $Y$.
2. Calculate the variable $A=\left\lfloor \tfrac{Y}{19}\right\rfloor$.
3. Find the values of the variables $B$ and $C$:
• $B=\left\lfloor \tfrac{Y}{100}\right\rfloor$; and
• $C=Y\ \text{mod}\ 100$.
4. Find the variables $D$ and $E$:
• $D=\left\lfloor \tfrac{B}{4}\right\rfloor$; and
• $E=B\ \text{mod}\ 4$.
5. Calculate $G=\left\lfloor \tfrac{8\cdot B +13}{25}\right\rfloor$.
6. Find $H$, the result of the expression $(19A + B -D -G +15)\ \text{mod}\ 30= H$.
7. Calculate the pair of variables $J$ and $K$ with:
• $J=\left\lfloor \tfrac{C}{4}\right\rfloor$; and
• $K=C\ \text{mod}\ 4$.
8. Calculate $M=\left\lfloor \tfrac{A+11H}{319}\right\rfloor$
9. Find the remainder $L$: $2E + 2J - K - H + M +32\ \text{mod}\ 7$.
10. Calculate the quotient $N=\left\lfloor \tfrac{H-M+L+90}{25}\right\rfloor$

We are in the endgame now. $N$ is the Easter month, with $N=3$ being March and $N=4$ April. To calculate the day, follow this equation:

$\footnotesize P=H-M+L+N+19\ \text{mod}\ 32$

Thus. Easter falls on the $P$th of $N$. All these calculations will run in our calculator: you will only have to insert the year in the apposite field. We'll tell you the date of Easter in the given year, plus the one in the closest years.

## An example of the Easter date calculation formula

Choose a year: $1991$ is good. Let's calculate all the variables!

\footnotesize \begin{align*} Y&=1991\\ \\ A&= 1991\ \text{mod}\ 19 = 15\\ \\ B&=\left\lfloor \frac{1991}{100}\right\rfloor=19\\ \\ C&=1991\ \text{mod}\ 100 = 91\\ \\ D&=\left\lfloor \frac{19}{4}\right\rfloor=4\\ \\ E&=19\ \text{mod}\ 4 = 3\\ \\ G &= \left\lfloor \frac{8\cdot 19+13}{25}\right\rfloor=6\\ \\ H&=(19\cdot15+19-4-6+15)\ \text{mod}\ 30=9\\ \\ J &= \left\lfloor \frac{91}{4}\right\rfloor=22\\ \\ K&= 91\ \text{mod}\ 4=3\\ \\ M&=\left\lfloor \frac{15+11\cdot 20}{319}\right\rfloor=0\\ \\ L&=(2\cdot3+2\cdot22-3-20+0+32)\ \text{mod}\ 7=0\\ \\ N&=\left\lfloor \frac{9-0+0+90}{25}\right\rfloor=3 \\ P&=(9-0+0+3+19)\ \text{mod}\ 32=31 \end{align*}

Easter in 1991 fell on the last day of March, the 31st.

If you are interested in other tools related to days, check our birthday paradox calculator!

Davide Borchia
Year
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