Problem 49 - 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. $$1996$$

Step-by-Step Solution

Answer

The date for Easter Sunday in 1996 is March 29.

Step 1: Find a, b, and c

First, we need to compute the values for \(a, b\), and \(c\) using modulo operations as follows: \(a = y \% 19\) \(b = y \% 4\) \(c = y \% 7\) For \(1996\), we have: \(a = 1996 \% 19 = 3\) \(b = 1996 \% 4 = 0\) \(c = 1996 \% 7 = 4\)

Step 2: Find d and e

Next, we need to find the values for \(d\) and \(e\) using the provided formulas: \(d = (19a + 24) \% 30\) \(e = (2b + 4c + 6d + 5) \% 7\) Let us substitute the values we calculated for \(a, b\), and \(c\): \(d = (19 \cdot 3 + 24) \% 30 = (57 + 24) \% 30 = 81 \% 30 = 1\) \(e = (2 \cdot 0 + 4 \cdot 4 + 6 \cdot 1 + 5) \% 7= (0 + 16 + 6 + 5) \% 7= 27 \% 7 = 6\)

Step 3: Find r

Now, we need to compute the value of \(r\), which determines the day of the Easter Sunday: \(r = 22 + d + e\) Let us substitute the values of \(d\) and \(e\): \(r = 22 + 1 + 6 = 29\)

Step 4: Obtain the date for Easter Sunday

Lastly, we need to determine whether Easter Sunday falls in March or April by checking if \(r \leq 31\). Since our calculated value of \(r = 29\), which is less than or equal to 31, we find that Easter Sunday falls in March, and its date is given by: March \(r =\) March 29 Therefore, the date for Easter Sunday in 1996 is March 29.

Key Concepts

Modulo Operations

Modulo operations play a crucial role in calculating dates, such as Easter Sunday. A modulo operation finds the remainder after division of one number by another. In mathematical terms, it is expressed as \[ a \mod n = r \] where \( a \) is the dividend, \( n \) is the divisor, and \( r \) is the remainder. In the context of the Easter date calculation for any year \( y \), modulo operations help determine periodic patterns by preparing numbers for further computation steps.

• For instance, we determine \( a = y \mod 19 \), which tells us the position of the year \( y \) within the 19-year cycle observed in determining Easter's date.
• Similarly, \( b = y \mod 4 \) and \( c = y \mod 7 \) figure out the year cycle moduli connected to the lunar calendar.

These calculations identify the basis for further steps in the algorithm, ultimately deciding whether Easter falls in March or April in any given year.

Algorithmic Problem Solving

When determining complex problems like finding the date of Easter, algorithmic problem solving is essential. This strategy involves breaking down a problem into smaller, manageable steps and then solving each one systematically. Here, the issue is composed into a sequence of calculations using defined formulas.
The steps include:

• Calculating intermediary values using given formulas.
• Using these values to make logical decisions moving forward.

For example, once we have the values \( a, b, c \), we proceed to calculate \( d \) and \( e \). These calculations: \[ d = (19a + 24) \mod 30 \] \[ e = (2b + 4c + 6d + 5) \mod 7 \] are crucial in determining the final outcome. What makes an algorithm effective is its ability to consistently yield the same outcome when applied correctly under the same conditions. In this instance, the repetition of a modular arithmetic procedure ensures the accurate prediction of Easter's date regardless of the year, making algorithmic problem solving a powerful tool in calendar computations.

Calendar Computation

Calendar computation involves translating the intricacies of our calendar system into mathematical formulas, as is the case in determining the date of Easter Sunday. This requires an understanding of both fixed periods (months) and cyclic events, such as lunar cycles.
The calendar is not straightforward due to its reliance on astronomical events, leading to varying Easter dates each year. Easter is determined by a mix of solar and lunar calendars, appearing after the first full moon following the vernal equinox. Hence, calculations must account for both the Solar and Lunar year nuances.
In the formula used:

• Values like \( d \) and \( e \) help pinpoint Easter's position in March or April.
• The final date is put together by checking whether \( r \leq 31 \), indicating Easter occurs in March (as in 1996 when it fell on March 29), or if the calculations result in an April date.

The combined use of modulo operations, algorithmic problem techniques, and precise mathematical logic ensures accuracy in such a multifaceted determination, demonstrating the complexity and need for detailed computations in calendar conversions.