John Horton Conway (the Princeton mathematician who is responsible for
"the Game of Life") wrote a book with Guy and Berlekamp, _Winning
Ways_, that describes in Volume 2 a number of useful calendrical
rules.  One of these is an easy "in your head" algorithm for
calculating the phase of the Moon, good to a day or better depending
on whether you use his refinements or not.

In the 20th century, calculate the remainder upon dividing the
last two digits of the year by 19; if greater than 9, subtract
19 from this to get a number between -9 and 9. 

Multiply the result by 11 and reduce modulo 30 to obtain a
number between -29 and +29.

Add the day of the month and the number of the month (except
for Jan and Feb use 3 and 4 for the month number instead of
1 and 2).

Subtract 4.

Reduce modulo 30 to get a number between 0 and 29. This is
the age of the Moon.

Example: What was the phase of the Moon on D-Day (June 6,
1944)?

Answer: 44/19=2 remainder 6.

6*11=66, reduce modulo 30 to get 6.

Add 6+6 to this and subtract 4: 6+6+6-4=14; the Moon was (nearly)
full. I understand that the planners of D-day did care about the phase
of the Moon, either because of illumination or because of tides. I
think that Don Olsen recently discussed this in _Sky and Telescope_
(within the past several years).

In the 21st century use -8.3 days instead of -4 for the last number.

Conway also gives refinements for the leap year cycle and also
for the slight variations in the lengths of months; what I have
given should be good to +/- a day or so.