Happy Easter!

Did you know that determining the date for Easter is quite complex? Unlike Christmas, which is always on December 25th, Easter can fall between March 22 and April 25, making it difficult to predict the exact date in the future. In the 19th century, Carl Friedrich Gauss (1777 – 1855) developed an algorithm to calculate this date.

First, a brief history lesson: Easter is a Christian holiday that commemorates the resurrection of Jesus. According to the New Testament, the resurrection took place on the third day after the Crucifixion, which was on a Friday, so Easter must always fall on a Sunday. The Crucifixion, in turn, occurred during Jewish Passover, which is celebrated for eight days following the first full moon of spring. The pope later declared that spring always begins on March 21st.

This means that the date of Easter depends on the start of spring, which is tied to the solar year, and the full moon, which depends on a lunar cycle. A lunar month is about 29.53 days long, while a solar year is approximately 365.24 days long. It turns out that 235 lunar months are nearly equivalent to 19 solar years. This cycle of resynchronization between the moon and the sun is called the Metonic cycle, and it means that the moon can be in 19 different positions on a particular date.

To calculate the date of Easter for a given year Y, we need to take into account two steps. First, we need to calculate the number of days from the start of spring until the first full moon. Then, we need to find the first Sunday after that full moon. To perform these calculations, we use the modulo function, which returns the remainder when dividing one number by another.

To calculate the number of days from spring until the first full moon, we take A = Y mod 19 to determine the moon's position in the Metonic calendar for that year. However, there is a shift in the Metonic cycle of eight days every 2,500 years. We calculate the century Y is in using P, then calculate the shift using Q, adding an additional offset of 13 to realign the lunar cycle with the solar cycle.

P = floor(Y / 100)
Q = floor((13 + 8 * P) / 25)

We also have to account for the missed leap years in our calculations. Our calendar adds a leap year every four years to account for the 0.24 extra day every year. However, we don't have a leap year in years that are multiples of 100, with the exception of years that are multiples of 400. This means that every 400 years, we have 97 leap years, which accounts for the following expression for the three missed leap years: P – (P / 4).

We know that the first full moon after March 21st was 15 days into year 0, and we can account for the total shift with the following expression:

M = (15 – Q + P – (P / 4)) mod 30

Since a lunar cycle is about 30 days long, we can calculate the number of days from March 21st until the first full moon as follows:

D = (19*A + M) mod 30

To calculate the date of the next Sunday after March 21st plus D, we need to take into account various factors, such as the number of missed leap years and the days between March 21st and the next full moon. First, we calculate an offset based on the fact that January 1st in year 1 was a Friday.

N = (4 + P – (P / 4)) mod 7

Then, we add the fact that this day shifts one day every year and two days every leap year. This gives us the following equation to determine the days from the full moon to Easter:

B = Y mod 4, C = Y mod 7
E = (N + 2B + 4C + 6D) mod 7

Here, 6D is used for the days until Sunday without correcting for leap days. With the help of D and E, we can determine that Easter Sunday will be on March (22 + D + E). If this value exceeds 31, it will fall in April. However, the lunar month is slightly shorter than 30 days, so certain conditions must be observed to address this discrepancy. If D is 29 and E is 6, then the date must be April 19. When D is 28 and E is 6, the date is April 18.

All of the above calculations can be summarized in a simple Python code. By inserting any year, we can determine the date of Easter. For example, here are some future Easter dates:

• 2025-04-20
• 2100-03-28
• 3000-04-12

In conclusion, while the date of Easter may be a bit more complex than other holidays, it's fascinating to learn about the intricate calculations that go into determining it. This year, let's take a moment to appreciate the religious and cultural significance of the holiday and enjoy spending time with loved ones. Happy Easter to all!

Sources:

• Stewart, Ian. "Easter is a quasicrystal." Scientific American 284.3 (2001): 80-83.
• https://www.algorithm-archive.org/contents/computus/computus.html