Problem 21 - Discrete Mathematics with Applications

Question

(Easter Sunday) Here is a second method" for determining Easter Sunday in a given year \(N .\) Let \(a=N\) mod \(19, b=N\) div \(100, c=N\) mod 100 \(d=b \operatorname{div} 4, e=b \bmod 4, f=(b+8) \operatorname{div} 25, g=(b-f+1) \operatorname{div} 3, h=(19 a+\) \(b-d-g+15) \bmod 30, i=c \operatorname{div} 4, j=c \bmod 4, k=(32+2 e+2 i-h-j)\) \(\bmod 7, \ell=(a+11 h+22 k) \operatorname{div} 451, m=(h+k-7 \ell+114) \operatorname{div} 31,\) and \(n=(h+k-7 \ell+114)\) mod \(31 .\) Then Easter Sunday falls on the \((n+1)\) st day of the \(m\) th month of the year. Compute the date for Easter Sunday in each year. $$2000$$

Step-by-Step Solution

Answer

In the year 2000, Easter Sunday fell on April 7th, using the given second method formula for determining the date of Easter Sunday.

Step 1: Calculate a, b and c

We are given that \(N = 2000\). Now, calculate the values of \(a, b\) and \(c\): \(a = N \pmod{19} = 2000 \pmod{19} = 5\) \(b = N \operatorname{div} 100 = \lfloor 2000 / 100 \rfloor = 20\) \(c = N \pmod{100} = 2000 \pmod{100}= 0\)

Step 2: Calculate d and e

Now, we'll find the values of \(d\) and \(e\): \(d = b \operatorname{div} 4 = \lfloor 20/4 \rfloor= 5\) \(e = b \bmod 4 = 20 \bmod 4= 0\)

Step 3: Calculate f and g

Next, calculate the values of \(f\) and \(g\): \(f = (b + 8) \operatorname{div} 25 = \lfloor (20 + 8) / 25 \rfloor = \lfloor 28 / 25 \rfloor = 1\) \(g = (b - f + 1) \operatorname{div} 3 = \lfloor (20 - 1 + 1) / 3 \rfloor = \lfloor 20 / 3 \rfloor = 6\)

Step 4: Calculate h, i and j

Now, we'll find the values of \(h, i\) and \(j\): \(h = (19a + b - d - g + 15) \bmod 30 = (19 \cdot 5 + 20 - 5 - 6 + 15) \bmod 30 = 138 \bmod 30 = 18\) \(i = c \operatorname{div} 4 = \lfloor 0 / 4 \rfloor = 0\) \(j = c \bmod 4 = 0 \bmod 4 = 0\)

Step 5: Calculate k and l

Now, calculate the values of \(k\) and \(\ell\): \(k = (32 + 2e + 2i - h - j) \bmod 7 = (32 + 2 \cdot 0 + 2 \cdot 0 - 18 - 0) \bmod 7 = 14 \bmod 7 = 0\) \(\ell = (a + 11h + 22k) \operatorname{div} 451 = (5 + 11 \cdot 18 + 22 \cdot 0) \operatorname{div} 451 = 203 \operatorname{div} 451 = 0\)

Step 6: Calculate m and n

Lastly, we'll find the values of \(m\) and \(n\): \(m = (h + k - 7\ell + 114) \operatorname{div} 31 = (18 + 0 - 7 \cdot 0 + 114) \operatorname{div} 31 = 132 \operatorname{div} 31 = 4\) \(n = (h + k - 7\ell + 114) \bmod 31 = (18 + 0 - 7 \cdot 0 + 114) \bmod 31 = 132 \bmod 31 = 6\)

Step 7: Find the date of Easter Sunday in 2000

Now that we have the values of \(m\) and \(n\), we can determine the date for Easter Sunday. Easter Sunday falls on the \((n+1)\)st day of the \(m\)th month. So, the date for Easter Sunday in the year 2000 is the 7th day (6+1) of the 4th month (April). Easter Sunday for the year 2000 was April 7th.

Key Concepts

Modular Arithmetic

Modular arithmetic, often called "clock arithmetic," simplifies the way we think about arithmetic operations by focusing on remainders. When dividing two numbers, the remainder obtained forms the basis of modular arithmetic. It's like wrapping around on a clock where numbers start again at zero after reaching the limit. This concept is crucial in calculating the Easter date, as it involves finding remainders after dividing by certain numbers, such as 19, 30, or 31.

For instance, in our exercise, we used modular arithmetic to determine "a" as 5 and "h" as 18 by mod 19 and mod 30 respectively. This strategy helps in breaking down complex problems into a series of manageable calculations that are easier to solve. By using modular arithmetic, we're able to efficiently handle cyclic patterns and grasp the essence of repeating sequences found in dates calculation algorithms.

Division in Mathematics

In mathematics, division is one of the basic operations, which determines how many times one number is contained within another. In the Easter date calculation, division is specifically applied to obtain whole numbers when dividing one number by another, removing the remainder. This process is denoted by the "div" operation in our problem.

For example, calculating "b" involved dividing 2000 by 100 to get 20. Here, the division helps identify century-related adjustments needed for determining Easter. It's important to note the differentiation from modular arithmetic, which focuses solely on remainders. Instead, division gives whole numbers or integers that are significant when precision is vital, like in identifying specific components of the algorithm used for date calculation.

Step-by-Step Problem Solving

Step-by-step problem solving is a methodical approach to tackling problems by breaking them down into smaller, more manageable steps. This technique is widely used in mathematics and algorithmic computations. By solving one part at a time, you gain a clearer understanding and reduce complexity.

During our Easter date calculation, which involved several variables such as "a," "b," "c," etc., tackling each computation one step at a time ensured accuracy. Through following sequential steps:

• Identify what needs to be calculated first.
• Complete each calculation in order.
• Use previous results to inform subsequent steps.
• Finally, compile all results to derive the final solution.

This ordered approach minimizes confusion and ensures one doesn't overlook key aspects of the solution.

Date Calculation Algorithm

A date calculation algorithm is a set of rules or a formula designed to determine specific calendar dates based on provided data. For Easter, the algorithm computes the date using specific constants and variables related to the Gregorian calendar. It's derived from mathematical computations that factor in Earth's rotations and calendar peculiarities.

In the exercise, we see this algorithm at work with calculations using variables like "a," "f," or "h," derived from modular arithmetic and division. The importance of such algorithms lies in their precision, allowing for accurate date identification, such as computing Easter Sunday, which doesn't fall on a fixed date each year.

The computation includes factoring in historical changes in calendars like leap years and ecclesiastical adjustments. Mastery of such algorithms is crucial for precise date-dependent tasks and understanding underlying mathematical structures in calendar-related problems.