The Formula
The formula is:

(Year Code + Month Code + Century Code + Date Number – Leap Year Code) mod 7

Here’s How it Works
I’ll run through an example with the date, 14 March 1897 — Einstein’s birthday.

The Year Code
To calculate the Year Code, use this formula:

(YY + (YY div 4)) mod 7

YY is the last two digits of the year. For the year 1897, it’s 97.

First, divide YY by 4 and discard the remainder: 97 div 4 = 24.

Then add 24 back into the YY number, which is 97 in this case, resulting in 121.

The next step is: 121 mod 7.

“Mod” means to divide the number and keep only the remainder. For 121 mod 7, start removing sevens:

10 x 7 is 70, leaving us with 51, because 121 – 70 = 51.
7 x 7 = 49, and 51 – 49 = 2.
We’ve removed all the sevens from 121 until we are left with a remainder of 2. That is the Year Code for 1897.

You could use a number shape image like a swan to hold that in memory while you calculate the items below.

The Month Code
This is easy — just memorize the number 033614625035:

January = 0
February = 3
March = 3
April = 6
May = 1
June = 4
July = 6
August = 2
September = 5
October = 0
November = 3
December = 5
Now you have the Month Code. For Einsteins birthday in March, it is 3.

Gregorian Dates

For the Gregorian Calendar, remember the number 4206420:

1700s = 4
1800s = 2
1900s = 0
2000s = 6
2100s = 4
2200s = 2
2300s = 0
If you are only doing this calendar trick with friends’ birthdays, you could probably leave this step out, because dates that fall in the 1900s get a Century Code of zero and don’t affect the outcome of the calculation.

Julian Dates
If you are looking at a Julian date, the formula is to take the century number and subtract it from 18 and then mod 7.

Example 1: if the year is 852 CE, take the century number, 8, and subtract it from 18, leaving 10. Then, 10 mod 7 =3.

Example 2: if the year is 1625 CE, take the century number, 16, and subtract it from 18, leaving 2. 2 mod 7 = 2, so in this case the Century Code is 2.

For Einsteins birthday in 1897, the Century Code is 2, because it’s a Gregorian date, and the chart above shows that dates in the 1800s get an adjustment of 2.

Leap Year Code
The other thing to take into account is whether you are dealing with a leap year. EDIT: If the date is in a January or February of a leap year, you have to subtract one from your total before the final step.

Gregorian Calendar
If you can divide a Gregorian year by 4, it’s a leap year, unless it’s divisible by 100. But it is a leap year if it’s divisible by 400.

1992 is a leap year because you can divide it by four.
1900 is not a leap year because you can divide it by 100.
2000 is a leap year because you can divide it by 400.

Julian Calendar
If you can divide a Julian year by 4, it’s a leap year.

Einstein’s birthday was in 1897 which was not a leap year (0), so it doesn’t affect the outcome.

Calculating the Day
Back to the original formula:

(Year Code + Month Code + Century Code + Date Number – Leap Year Code) mod 7

For 14 March 1897, here are the results:

Year Code: 2
Month Code: 3
Century Code: 2
Date Number: 14 (the 14th of the month)
Leap Year Code: 0
So:

(2 + 3 + 2 + 14) mod 7 = 21 mod 7 = 0

Match the resulting number in the list below, and you’ll have the day of the week:

0 = Sunday
1 = Monday
2 = Tuesday
3 = Wednesday
4 = Thursday
5 = Friday
6 = Saturday
Einstein was born on a Sunday.

More Examples
Here are three more examples from different centuries:

The Moon Landing
Humans set foot on the moon: 20 July 1969:

Take ’69 and divide by 4, discarding the remainder. That leaves 17. Add 69 to 17 to get 86. Then, 86 mod 7 = 2. The Year Code is 2.
The Month Code for July is 6.
The Century Code for the 1900s is zero.
The Date Number is 20, because it’s the 20th of July.
1969 wasn’t a leap year since it can’t be divided by 4.
2 + 6 + 0 + 20 = 28
28 mod 7 = 0
20 July 1969 was a Sunday.