The Gregorian Calendar, also the internationally accepted calendar of today, has the following scheme:

Taking all of the above into account, the anchor day for a given year is defined as:

$\mathrm{Anchor} \= 5 \cdot \left(\mathrm{floor}\left(\frac{\mathrm{year}}{100}\right) \mathbf{mod} 4\right) \+ \mathrm{Tuesday}$

Note: The $\mathrm{floor}$ of a number $z$, also denoted $\mathbf{ }⌊z⌋$, is the greatest integer smaller than $z$. In other words, the $\mathrm{floor}$ function rounds a number down to the preceding integer.

There are multiple methods to finding the doomsday of a year. All three methods mentioned use the common variables:

Method 1

$x \= \left(y \+ ⌊\frac{y}{4}⌋\right) \mathbf{mod} 7$

Method 2

1. Let $a$ to be the floor of $\frac{y}{12}$.

2. Let $b$ to be the modulus of $\frac{y}{12}$.

3. Let $c$ be the floor of $\frac{b}{4}$.

4. $x$ is the modulus of $\frac{a \+ b\+ c}{7}$.

$$\begin{gathered} a \= ⌊\frac{y}{12}⌋\; \\ b \= y \mathbf{mod} 12\; \\ c \= b \mathbf{mod} 4 \end{gathered}$$$$\begin{gathered} x \= \left(⌊\frac{y}{12}⌋ \+ y \mathbf{mod} 12 \+ ⌊\frac{y \mathbf{mod} 12}{4}⌋\right) \mathbf{mod} 7 \\ \= \left(a \+ b \+ c\right) \mathbf{mod} 7 \end{gathered}$$

Method 3

1. Let $x \= y$.

2. If $x$ is odd, add $11$ to it.

3. Divide $x$ by $2$.

4. If $x$ is odd again, add $11$ to it.

5. Subtract the modulus of $x$ by $7$ from $7$.

$x \= - \left(\frac{y \+ 11\left( y \mathbf{mod} 2\right)}{2} \+ 11\left(\frac{y \+ 11\left(y \mathbf{mod} 2\right)}{2}\right)\mathbf{mod} 2 \right)\mathbf{mod} 7$

Shift the anchor day by $x$ to obtain the doomsday.

Special Doomsday Dates

Date(s)

Mnemonic

4/4, 6/6, 8/8, 10/10, 12/12

Dates in the form n/n, where n is an even number between 4 and 12

5/9, 9/5, 7/11, 11/7

"I work from 9 to 5 at 7-11"

Feb 28/29 or 'March 0'

Last Day of February

March 14

Pi Day

October 31

Halloween