To Find out the Day of the Week of any Day
You need to remember
 a magic number per month, and
 a magic number per century
 (although for the 1900's the magic number is 0.)
Now add together
 the date,
 the year,
 the year divided by 4,
 the magic number for the month,
 the magic number for the century,
and there might be a correction.

Magic Month No. *
January  1  4  4  March 
April  0  2  5  June 
July  0  3  6  September 
October  1  4  6  December 

Magic Century No.**
1700  1799  4 
1800  1899  2 
1900  1999  0 
2000  2099  6 
2100  2199  4 

Example
Let's do 29th April, 1951.
Add together:
Day  29 
Year (last 2 digits)  51 
¼ of year (forget any remainder)  12 
Magic Month Number*  0 
Magic Century Number**  0 
(1 if Jan/Feb of Leap Year)  0 
TOTAL B =  92 
Now divide your total by 7, and keep the remainder.
Our total was 92, which is a multiple of 7 with 1 remainder.
Hence 29th April 1951 was a Sunday

Remainder  Day of the Week 
1  Sunday 
2  Monday 
3  Tuesday 
4  Wednesday 
5  Thursday 
6  Friday 
0  Saturday 

Notes:
 Using modulo arithmetic (mod 7) and a reasonable memory can make this method accessible to mental calculation in a reasonable time so as to perform this as a party trick.
 In modulo arithmetic (mod 7) adding 6 is the same as subtracting 1, which could be easier for mental calculation.
 Some conjurers/memory experts who use this as part of their performance will remember a magic year number for many of the last 100 years.
 Magic year number = [year + ¼ of year (forget any remainder)] mod 7
 For the purposes of this calculation the century starts in years ending in 00.
 e.g. 2000 is considered to be in the 21st Century
 Years ending in 00 are normally not leap years except those which are divisible by 400
 e.g. 1900 was not a leap year but 2000 was a leap year.
 Calculating the day of the week for England before 14th September 1752 is problematic because of the change from the Julian to the Gregorian calendar.
 Isaac Newton would have considered himself to have been born on Christmas Day of 1642 however most present day chronicles will quote his birthday as 4th January 1643.
This is one of the many
Enrichment tasks on this site.