Problem 49 - Discrete Mathematics with Applications

Question

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 April 5th.

Step 1: Calculate a, b, c, d, e, and r

We are given that y = 1996. Using the formulas provided: \(a = y \bmod 19\) \(b = y \bmod 4\) \(c = y \bmod 7\) \(d = (19a + 24) \bmod 30\) \(e = (2b + 4c + 6d + 5) \bmod 7\) \(r = (22 + d + e)\) We begin by calculating a, b, and c: \(a = 1996 \bmod 19 = 12\) \(b = 1996 \bmod 4 = 0\) \(c = 1996 \bmod 7 = 4\) Now we will find the values of d and e: \(d = (19 * 12 + 24) \bmod 30 = (228 + 24) \bmod 30 = 252 \bmod 30 = 12\) \(e = (2 * 0 + 4 * 4 + 6 * 12 + 5) \bmod 7 = (0 + 16 + 72 + 5) \bmod 7 = 93 \bmod 7 = 2\) Lastly, we compute the value of r: \(r = 22 + 12 + 2 = 36\)

Step 2: Determine the date for Easter Sunday

Now, we check if r ≤ 31. In this case, r = 36, so r > 31. This means Easter Sunday falls on April (r(mod 31)): April \((r \bmod 31) = 36 \bmod 31 = 5\) Easter Sunday in 1996 falls on April 5th.

Key Concepts

Modular Arithmetic

Modular Arithmetic is like clock arithmetic where numbers wrap around upon reaching a certain value, known as the modulus. It's a handy tool when dealing with cycles, like calculating dates. To illustrate, when we compute \(1996 \bmod 19\) and get 12, we find the remainder when 1996 is divided by 19. Similarly, \(1996 \bmod 4 = 0\) means 1996 is divisible by 4 without remainder.

• A modulus is the number at which we wrap around. For the year 1996 and mod 4, 4 is the modulus.
• Modular Arithmetic helps simplify complex calculations by reducing large numbers into manageable remainders.

The use of modular arithmetic is crucial in this exercise to sequentially break down the year into smaller cyclic components—namely, \(a\), \(b\), and \(c\)—before further calculations.

Algorithmic Problem Solving

This exercise involves systematic calculation to determine Easter's date through specific steps. Algorithmic problem solving breaks problems into smaller tasks, solved in sequence, often involving patterns of logic or computation techniques.

We start by calculating intermediate variables \(a\), \(b\), \(c\), followed by \(d\) and \(e\). The problem follows a determined sequence:

• Calculate modulo values \(a\), \(b\), \(c\)
• Use these to find \(d\) and \(e\)
• Finally, compute the value of \(r\)

Each step uses results from the previous step, illustrating a fundamental algorithm design—a flow from one calculated result to another. This methodical approach ensures accuracy and efficiency, minimizing errors.

Mathematical Formulas

The solution relies heavily upon mathematical formulas, which are predefined mathematical expressions offering solutions under given inputs. For determining Easter's date, formulas capture the relationship between the year and date computations. Consider:

• \(d = (19a + 24) \bmod 30\): calculates a key intermediary based on modular arithmetic of coefficients.
• \(e = (2b + 4c + 6d + 5) \bmod 7\): combines several factors to adjust the final day calculation.

These formulas are efficient for recurring calculations. They incorporate constants (like 19 and 24) that, through historical significance, accurately compute dates. Formulas like these are integral to the algorithm, offering precision and reducing manual error. Knowing when and how to apply these formulas is essential for mastering easter date calculations.