Day of Week

How Day of Week Works

h = (q + ⌊13(m+1)/5⌋ + K + ⌊K/4⌋ + ⌊J/4⌋ − 2J) mod 7

h

Day of Week Index — The calculated day of the week result (0=Saturday, 1=Sunday, 2=Monday, 3=Tuesday, 4=Wednesday, 5=Thursday, 6=Friday).

q

Day of Month — The day of the month (1-31). For example, December 25 uses q = 25.

m

Adjusted Month — Adjusted month number: March=3, April=4, ..., December=12, January=13, February=14 of the previous year.

K

Year of the Century — The last two digits of the year (year mod 100). For 2024, K = 24.

J

Century — The zero-based century (year divided by 100, rounded down). For 2024, J = 20.

How to Use

1. Enter the month as a number (1-12), the day of the month (1-31), and the year as a positive integer.
2. Click calculate to find the day of the week for that date, like "Wednesday" or "Friday."
3. View the full day name, formatted date, and the complete Zeller's Congruence calculation broken down step by step.
4. See the month calendar grid highlighting the position of the target date within the month.
5. Check the "Days in Month" and "First Day of Month" for additional calendar context.

Understanding the Result

Frequently Asked Questions

Why are January and February treated differently in Zeller's Congruence?

In Zeller's Congruence, January and February are counted as months 13 and 14 of the previous year. This shift places the leap day (February 29) at the very end of the calculation cycle rather than near the beginning. Because the formula's month term ⌊13(m+1)/5⌋ encodes month lengths, having the irregular February at the end simplifies the arithmetic significantly. Without this adjustment, the formula would need additional conditional logic for leap years.

Does Zeller's Congruence work for dates before the Gregorian calendar?

Zeller's Congruence works for any valid date in the Gregorian calendar, which was introduced in 1582 by Pope Gregory XIII. For dates before 1582 (Julian calendar dates), the formula may not produce accurate results. Additionally, different countries adopted the Gregorian calendar at different times — Britain and its colonies adopted it in 1752, Russia in 1918, and Greece in 1923 — so historical dates may be ambiguous.

What do the results h=0 through h=6 mean?

The result h is a number from 0 to 6 that maps to the day of the week: 0=Saturday, 1=Sunday, 2=Monday, 3=Tuesday, 4=Wednesday, 5=Thursday, 6=Friday. Note that the cycle starts on Saturday rather than Monday or Sunday because of the historical convention Zeller used. The calculator automatically converts this number to the familiar day name.

How is Zeller's Congruence different from the Doomsday algorithm?

Zeller's Congruence is a direct formula that computes the day of the week from the date components using modular arithmetic and floor functions. The Doomsday algorithm, developed by John Conway, uses a different approach based on memorizing certain "anchor" dates (like 4/4, 6/6, 8/8, etc. that always fall on the same day of the week in a given year). Zeller's is easier to program into a calculator, while Doomsday is more practical for mental calculation.

Can this calculator handle dates in the Julian calendar?

This implementation uses Zeller's Congruence for the Gregorian calendar (the calendar system used by most of the world today). For Julian calendar dates, a slightly modified version of the formula exists, but it produces different results for the same date. To verify historical dates, check which calendar system was in use in the relevant country at that time.

Sources

• Wikipedia - Determination of the Day of the Week
• Wikipedia - Doomsday Rule