How to Calculate the Day of the Week by Hand

Zeller's Congruence

Christian Zeller was a German mathematician who in 1882 published a formula for computing the day of the week for any date in the Julian or Gregorian calendars. His formula for the Gregorian calendar is:

h = (q + ⌊13(m+1)/5⌋ + K + ⌊K/4⌋ + ⌊J/4⌋ − 2J) mod 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, ..., 14 = February — January and February are counted as months 13 and 14 of the previous year)
• K is the year within the century (year mod 100)
• J is the zero-based century (⌊year / 100⌋)
• ⌊x⌋ means floor — round down to the nearest integer

The January/February rule is critical: those months are treated as months 13 and 14 of the preceding year. So January 15, 1900 is treated as month 13, year 1899.

Worked Example: July 4, 1776

Let's verify what day the Declaration of Independence was signed. The date is July 4, 1776.

Step 1: Identify the components.

• Day: q = 4
• Month: July = m = 7 (no adjustment needed — not January or February)
• Year: 1776, so K = 76, J = 17

Step 2: Calculate each term.

• q = 4
• ⌊13(m+1)/5⌋ = ⌊13 × 8 / 5⌋ = ⌊104/5⌋ = ⌊20.8⌋ = 20
• K = 76
• ⌊K/4⌋ = ⌊76/4⌋ = ⌊19⌋ = 19
• ⌊J/4⌋ = ⌊17/4⌋ = ⌊4.25⌋ = 4
• 2J = 2 × 17 = 34

Step 3: Sum the terms.

h = (4 + 20 + 76 + 19 + 4 − 34) mod 7
h = 89 mod 7
h = 89 − (7 × 12) = 89 − 84 = 5

h = 5 corresponds to Thursday. July 4, 1776 was indeed a Thursday — the Declaration of Independence was adopted on a Thursday. (It was officially signed by most delegates on August 2, 1776 — also a Thursday, as it happens.)

The January/February Adjustment in Practice

Let's try February 14, 2000 (Valentine's Day, year 2000).

Because February is month 14 in Zeller's system, we shift the year back by one: we use year 1999 instead of 2000.

• q = 14
• m = 14 (February)
• Year = 1999, so K = 99, J = 19

Calculate:

• ⌊13(14+1)/5⌋ = ⌊13 × 15 / 5⌋ = ⌊195/5⌋ = ⌊39⌋ = 39
• ⌊K/4⌋ = ⌊99/4⌋ = ⌊24.75⌋ = 24
• ⌊J/4⌋ = ⌊19/4⌋ = ⌊4.75⌋ = 4
• 2J = 38

h = (14 + 39 + 99 + 24 + 4 − 38) mod 7
h = 142 mod 7
h = 142 − (7 × 20) = 142 − 140 = 2

h = 2 corresponds to Monday. February 14, 2000 was indeed a Monday.

Month Code Table

The ⌊13(m+1)/5⌋ term is the heart of the formula — it accounts for the varying lengths of months. Here is what it evaluates to for each month, which you can memorize to speed up mental calculation:

Day Code Output Table

Handling Negative Results

The formula can produce a negative number before the mod operation, particularly for dates in the 1800s and earlier (because −2J can become a large negative number). In modular arithmetic, −1 mod 7 = 6, not −1. The safe approach: if your sum before mod 7 is negative, add a multiple of 7 large enough to make it positive before dividing.

For example, if your sum before mod is −5: −5 + 7 = 2. h = 2 → Monday.

Julian Calendar Variant

Zeller also published a version for the Julian calendar (used before October 1582 in most of Europe). The Julian variant replaces the −2J term with +5 − J. The January/February treatment remains the same. For dates before October 15, 1582 in Western Europe, use the Julian variant.

If you would rather not do the arithmetic yourself: try the Day of Week Calculator →