Problem 50 - Discrete Mathematics with Applications

Question

(Easter Sunday) The date for Easter Sunday in any year \(y\) can be computed as follows. Let \(a=y \bmod 19, b=y \bmod 4, c=y \bmod 7, d=(19 a+24)\) \(\bmod 30, e=(2 b+4 c+6 d+5) \bmod 7,\) and \(r=(22+d+e) .\) If \(r \leq 31,\) then Easter Sunday is March \(r ;\) otherwise, it is April \([r(\bmod 31)] .\) Compute the date for Easter Sunday in each year. $$2000$$

Step-by-Step Solution

Answer

Easter Sunday in the year 2000 falls on April 25th.

Step 1: Calculate a, b, and c

First, we need to calculate the values of \(a, b,\) and \(c\) using the year \(y = 2000\). $$a = y \bmod 19 = 2000 \bmod 19$$ $$b = y \bmod 4 = 2000 \bmod 4$$ $$c = y \bmod 7 = 2000 \bmod 7$$ Calculating these values: $$a = 5, \quad b = 0, \quad c = 1$$

Step 2: Calculate d and e

Now we will compute \(d\) and \(e\) using the values of \(a,\) \(b,\) and \(c\). $$d = (19a + 24) \bmod 30 = (19 \times 5 + 24) \bmod 30$$ $$e = (2b + 4c + 6d + 5) \bmod 7 = (2 \times 0 + 4 \times 1 + 6 \times d + 5) \bmod 7$$ Calculating these values with their known variables: $$d = 119 \bmod 30 = 29$$ $$e = (4 + 174 + 5) \bmod 7 = 183 \bmod 7 = 5$$

Step 3: Calculate r

Next, we calculate the value of \(r\) using \(d\) and \(e\). $$r = 22 + d + e$$ $$r = 22 + 29 + 5$$ Calculating the value: $$r = 56$$

Step 4: Determine the date of Easter Sunday

Since \(r = 56 > 31\), Easter Sunday falls in April. Therefore, we calculate the date in April as follows: $$r(\bmod 31) = 56 \bmod 31 = 25$$ So, Easter Sunday in the year 2000 falls on April 25th.

Key Concepts

Modular Arithmetic

Modular arithmetic is a critical concept when calculating the date of Easter, as it deals with the remainder of a division. It's like a clock where numbers wrap around after reaching a certain point.
When calculating Easter, you apply modular arithmetic to determine the remainders of divisions by different numbers like 19, 4, and 7.

• The operation \(a = y \bmod 19\) calculates the remainder of the year divided by 19.
• Similarly, \(b = y \bmod 4\) and \(c = y \bmod 7\) help break down the year into manageable parts.

These calculations reveal patterns or cycles in calendar events. In the Easter algorithm, modular arithmetic simplifies finding astronomically based dates by reducing larger numbers into repetitive, easier-to-manage values.
This makes it possible to compute specific calendar dates accurately, no matter how large the year number is.

Civic Calendar

The civic calendar is the everyday calendar used by most people worldwide, also known as the Gregorian calendar. It governs the months, weeks, and days we observe throughout the year. Easter Sunday is a moving feast, meaning its date changes each year but always falls between March 22 and April 25.
Understanding this aspect is key to recognizing why calculating Easter involves complex calculations. The reason it fluctuates is that Easter Sunday is determined by the lunar calendar, which doesn't line up perfectly with our civic calendar.
The civic calendar's structure influences how holidays are calculated based on both solar and lunar observations, an interesting blend of modern life and ancient tradition. This calculation means Easter is celebrated on the first Sunday after the full moon that occurs on or after the vernal equinox (March 21). The algorithm adjusts to align this lunar event with a Sunday in the civic calendar.

Algorithm for Date Determination

The algorithm for determining the date of Easter is a systematic process that converts celestial events into civic calendar dates. It considers factors such as phases of the moon and the spring equinox, meticulously traced through calculations.
The process begins with calculating intermediate values using modular arithmetic:

• Values \(a\), \(b\), and \(c\) come from the year being divided by 19, 4, and 7, respectively.
• The next step is calculating \(d = (19a + 24) \bmod 30\) and \(e = (2b + 4c + 6d + 5) \bmod 7\).

With these values, the Easter date is found using \(r = 22 + d + e\). If \(r ≤ 31\), Easter falls within March. Otherwise, it is in April.
This calculation results from centuries of adjustments to align closely with astronomical events and ensure the holiday is on a Sunday. It's a blend of tradition and precise calculation, allowing Easter to maintain cultural and religious significance across the globe.