If you've ever wondered what day of the week you were born, or when your birthday will fall this year, our day of the week calculator will come in handy!

In this article, you will discover how to calculate the day of the week without checking the calendar. If you are willing, you can learn how to do so in as little as a few seconds!

Instead, if you wonder how is Easter calculated, we have a tool designed precisely for that purpose. Don't miss it!

Days of the week

While the day and the year are quantities somehow rooted in the Natural world, corresponding to a rotation and a revolution of Earth, the subdivision of time into months and weeks is purely arbitrary. This led to many different calendars: from the one we know so well, the Gregorian or Julian calendar to the Chinese one or ancient calendars like the Mayan one.

What's more, you can even ask how old is my dog in human years. Is it a dog's calendar then :)

The Gregorian calendar: the easiest way to say what day it is.

It is possible to find the day of the week of each date, thanks to the regularity of the modern Gregorian calendar.

The Gregorian calendar, adopted in 1582 and bearing the name of Pope Gregory XIII, introduced a correction to the previously used Julian calendar, taking into account the correct duration of a day, leaving us with the current pattern of the duration of a year:

  • If the year is not divisible by 44 is a "normal year" and lasts 365365 days;
  • A leap year (with year number divisible by 44) has 366366 days;
  • A year divisible by 100100, but not by $400$$ is a regular year.

This way, the calendar accumulates only a small error due to the precession of equinoxes, equal to roughly one day every 77007700 years.

How to calculate the day of the week — without looking at the calendar

To know what day of the week is it for any date, you can either gather a lot of calendars or try to use some math to calculate what day it is in a few short passages.

🙋 This problem was covered many times in the past, maybe most noticeably by Lewis Carroll: the author of Alice in Wonderland. You can find his version of the calculations in this pretty old Nature's paper.

To calculate the day of the week, take a generic date:

d1d2/m1m2/y1y2y3y4\text{d}_1\text{d}_2/\text{m}_1\text{m}_2/\text{y}_1\text{y}_2\text{y}_3\text{y}_4

Follow these steps to calculate the day of the week:

  1. Take the last two digits of the year (y3y4\text{y}_3\text{y}_4), and divide by 44 keeping only the quotient:
    day=y3y4/4\text{day} = \lfloor\text{y}_3\text{y}_4/4\rfloor
  2. Add the day of the month:
    day=y3y4/4+d1d2\text{day} = \lfloor\text{y}_3\text{y}_4/4\rfloor +\text{d}_1\text{d}_2
  3. Add the month's key value mk\text{mk} according to the table below;
    day=y3y4/4+d1d2+mk\text{day} =\lfloor\text{y}_3\text{y}_4/4\rfloor +\text{d}_1\text{d}_2 + \text{mk}

🙋 Here is the table for the month's key value mk\text{mk}:
JFM AMJ  JAS  OND\text{J}\text{F}\text{M}\ \text{A}\text{M}\text{J}\ \ \text{J}\text{A}\text{S}\ \ \text{O}\text{N}\text{D}
1 4 4  0 2 5  0 3 6  1 4 6 1\ 4\ 4\ \ 0\ 2\ 5\ \ 0\ 3\ 6\ \ 1\ 4\ 6

  1. If you are dealing with a leap year, subtract 11 if the month is either January or February.
  2. Starting from the 1700's, add a factor given in the table below. For the other centuries, follow the pattern of subtracting/adding multiples of 400400 to the year and checking which one of the four centuries below the date falls.
for 1700’s add 4for 1800’s add 2for 1900’s add 0for 2000’s add 6\begin{align*} \text{for}\ 1700\text{'s}\rightarrow\ \text{add}\ 4\\ \text{for}\ 1800\text{'s}\rightarrow\ \text{add}\ 2\\ \text{for}\ 1900\text{'s}\rightarrow\ \text{add}\ 0\\ \text{for}\ 2000\text{'s}\rightarrow\ \text{add}\ 6\\ \end{align*}
  1. Add the last two digits of the year, y3y4\text{y}_3\text{y}_4.
  2. Finally, take the remainder of the division of the result by 77.

The resulting number corresponds to the day of the week, assuming that 11 is Sunday, 22 is Monday, and so on.

What day of the week was I born? Calculate the day of the week of your birthday.

Were you born on the 17th of July 1996? No? It's not important 😁 let's calculate!

Follow the steps laid down before:

  1. day=96/4=24\text{day} = \left\lfloor96/4\right\rfloor = 24.
  2. day=24+17=41\text{day} = 24+17=41.
  3. The month's key value for July is 00:
    day=41+0=41\text{day} = 41+0=41
  4. We don't subtract anything (though the year is a leap year).
  5. We add 00 since the year belong to the 1900’s1900\text{'s}.
    day=41+0=41\text{day}=41+0 = 41
  6. day=41+96=137\text{day}= 41+96 = 137.
  7. day=137 mod 7=4\text{day} = 137\ \text{mod}\ 7 = 4

Which corresponds to Wednesday. Use our birthday paradox calculator to find out the probability of sharing the birthday with someone else at a party!

Davide Borchia
Date
Day of the week: Sunday
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