(Science: day of the week) Zeller's congruence is an...

(Science: day of the week) Zeller's congruence is an algorithm developed by Christian Zeller to calculate the day of the week. The formula is \[h=\left(q+\frac{26(m+1)}{10}+k+\frac{k}{4}+\frac{j}{4}+5 j\right) \% 7\] where ? h is the day of the week (0: Saturday, 1: Sunday, 2: Monday, 3: Tuesday, 4: Wednesday, 5: Thursday, 6: Friday). ? q is the day of the month. ? m is the month (3: March, 4: April, . . ., 12: December). January and February are counted as months 13 and 14 of the previous year. ? j is the century (i.e., ). ? k is the year of the century (i.e., year % 100). Note that the division in the formula performs an integer division. Write a program that prompts the user to enter a year, month, and day of the month, and displays the name of the day of the week. Here are some sample runs: Enter year: $(e . g ., 2012): 2015$ Enter month: $1-12: 1$ Enter the day of the month: $1-31: 25$ Day of the week is Sunday Enter year: $(e .9 ., 2012): 2012$ Enter month: $1-12: 5$ Enter the day of the month: $1-31: 12$ Day of the week is Saturday (Hint: January and February are counted as 13 and 14 in the formula, so you need to convert the user input 1 to 13 and 2 to 14 for the month and change the year to the previous year.)

(Science: day of the week) Zeller's congruence is an algorithm developed by Christian Zeller to calculate the day of the week. The formula is
\[h=\left(q+\frac{26(m+1)}{10}+k+\frac{k}{4}+\frac{j}{4}+5 j\right) \% 7\]
where
? h is the day of the week (0: Saturday, 1: Sunday, 2: Monday, 3: Tuesday, 4:
Wednesday, 5: Thursday, 6: Friday).
? q is the day of the month.
? m is the month (3: March, 4: April, . . ., 12: December). January and February
are counted as months 13 and 14 of the previous year.
? j is the century (i.e., ).
? k is the year of the century (i.e., year % 100).
Note that the division in the formula performs an integer division. Write a program that prompts the user to enter a year, month, and day of the month, and
displays the name of the day of the week. Here are some sample runs:
Enter year: $(e . g ., 2012): 2015$
Enter month: $1-12: 1$
Enter the day of the month: $1-31: 25$
Day of the week is Sunday
Enter year: $(e .9 ., 2012): 2012$
Enter month: $1-12: 5$
Enter the day of the month: $1-31: 12$
Day of the week is Saturday
(Hint: January and February are counted as 13 and 14 in the formula, so you need to convert the user input 1 to 13 and 2 to 14 for the month and change the year to the previous year.)

Key Concepts

Calendar Date Conversion

Proper conversion and adjustment of calendar dates are essential when applying algorithms like Zeller's Congruence. For instance, the convention of treating January and February as months 13 and 14 of the previous year is crucial for the formula’s accuracy. This adjustment compensates for the differences in the calendar structure and ensures that calculations are consistent with the intended historical or standard calendar system.

Integer Division

Integer division, which discards any fractional part of the quotient, is central to the calculations in Zeller's Congruence. Since the formula requires precise operations without using floating-point arithmetic, integer division ensures that the computed values remain integral. This is important for maintaining the mathematical integrity of the steps involved in the algorithm, particularly when dealing with the factors and divisors in the formula.

Zeller's Congruence

Zeller's Congruence is an algorithm devised to compute the day of the week for any given date. It transforms the components of a calendar date—day, month, and year—into a numerical value corresponding to a day of the week using a formula that incorporates arithmetic operations such as addition, multiplication, and division. This method is particularly useful because it can systematically calculate weekdays even for dates far in the past or future.

Modular Arithmetic

Modular arithmetic involves calculating remainders and is a fundamental concept in many algorithms, including those used to determine the day of the week. In the context of Zeller's Congruence, the final result is obtained by taking the modulus of a computed value, ensuring the outcome is within the range of valid weekdays. This method of wrapping numbers into a limited set of values is crucial for mapping numeric results to specific days of the week.

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