### 1.0 Day of the Week

The Gregorian calendar was adopted in 1582. The formula to find the day of the week for any date after 1582 is,

d=N+ ⌊2.6M- 0.2⌋ +Y+ ⌊Y/4⌋ + ⌊C/4⌋ - 2C- (1 +L)⌊M/11⌋ (mod 7)

where,

*d* is the day of the week. *d* is 0 for Sunday, 1 for Monday, …, 6 for Saturday.

*N* is the day of the month.

*M* is the month. *M* = 1 for March, 2 for April, …, 10 for December, 11 for January and 12 for February.

*Y* is the last two digits of the year.

*C* is the first two digits of the year. For example, for the year 2023, *C* is 20 and *Y* is 23.

*L* is 1 for leap years, 0 otherwise. Any year divisible by 4 *and* not divisible by 100 is a leap year. However, the years divisible by 400 are also leap years. For example, the years 1992, 2000 and 2016 are leap years whereas 1800, 1900 and 2023 are not.

### 2.0 Example Program

An example program in C implementing the above formula is given below. We have a function, day_of_week (), which takes in day, month and year as integers, *dd*, *mm* and *yyyy*. The month argument is the usual 1 for January, 2 for February, …, and 12 for December.

// day_of_week.c: find the day of the week for a given date #include <stdio.h> #include <stdlib.h> #include <math.h> int day_of_week (int day, int month, int year) // returns 0 for Sunday, 1 for Monday ... { int N = day; int M = (month <= 2) ? month + 10 : month - 2; int C = year / 100; int Y = year % 100; int L = year % 4 == 0 && year % 100 != 0 || year % 400 == 0; int temp = N + floor (2.6 * M - 0.2) + Y + floor (Y / 4) + floor (C / 4) - 2 * C - (1 + L) * floor (M / 11); // if temp is negative, add a multiple of 7 to make it positive if (temp < 0) { int n = ceil ((double) (abs (temp)) / 7); temp += 7 * n; } return temp % 7; } #define BUFFER_SIZE 1024 char buffer [BUFFER_SIZE]; int main (int argc, char *argv []) { int day, month, year; printf ("Date (dd mm yyyy) : "); while (fgets (buffer, sizeof (buffer), stdin)) { sscanf (buffer, "%d %d %d", &day, &month, &year); int weekday = day_of_week (day, month, year); switch (weekday) { case 0: printf ("Sunday\n"); break; case 1: printf ("Monday\n"); break; case 2: printf ("Tuesday\n"); break; case 3: printf ("Wednesday\n"); break; case 4: printf ("Thursday\n"); break; case 5: printf ("Friday\n"); break; case 6: printf ("Saturday\n"); break; default: printf ("Error (%d)\n", weekday); } printf ("Date (dd mm yyyy) : "); } printf ("\n"); return 0; }

In most cases, the expression temp is positive and temp mod 7 is calculated as temp % 7, giving the day of the week. However, in some cases, especially for the dates at start of a century, like January 1, 1900, or January 1, 2000, the expression temp becomes negative. In these cases, we add a multiple of 7 to it to make it a positive number, and then the % operator can be used for finding the modulus.

We can compile and run the above program.

$ gcc day_of_week.c -o day_of_week -lm $ ./day_of_week Date (dd mm yyyy) : 11 11 2023 Saturday Date (dd mm yyyy) : 1 1 2000 Saturday Date (dd mm yyyy) : 29 02 1984 Wednesday Date (dd mm yyyy) : 1 1 2100 Friday Date (dd mm yyyy) : $

### 3.0 Reference

- Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory of Numbers, Fifth Edition, Wiley, 1991.