This is another nice trick, due to John Horton Conway. It is also described in Winning Ways, Volume 2.
This method uses the fact that the phases of the Moon follow
a 19-year cycle, such that the phase of the Moon is repeated almost
exactly on the same date in the year as it had on that same date
19 years earlier. This gives a very slick way of estimating the
age (or phase) of the Moon, given any date in history. This 19-year
cycle has been known since ancient times, and is known as the
Metonic Cycle. It is also the basis of the calculation
of the dates of Easter and Passover.
In this century, for any year 20xx, do the following:
Divide xx by 19, getting a quotient (which you discard) and a
remainder R. If the remainder is greater than 9, subtract 19 from
it. Multiply the result by 11. Add to it the day of the month
and the number of the month (except for January and February use
3 and 4 respectively instead of 1 and 2).
Add the Century Number -8 to this result.
(In the 20th century, for years 19xx, the Century Number is
-4. See table below).
Add or subtract a multiple of 30 to get a number between 0 and
29. The result is the age of the Moon, in days (i.e., the number
of days since the Moon was new).
Example: What was the age of the Moon on D-Day, June 6, 1944?
(Military campaigns are frequently timed to the phase of the Moon
for reasons of tide and/or light).
44/19 = 2 (discard) remainder 6. This is not greater than 9, so
we leave it alone.
6 x 11 = 66.
66 + 6 + 6 - 4 = 74.
Subtract 2 x 30 in this case to get 14. The Moon was 14 days old--nearly
full--on D-Day.
The only thing that changes from century to century is the Century Number, which instead of being -4 is given in the following table:
Century Numbers for Gregorian Calendar
| 15xx | 16xx | 17xx | 18xx | 19xx | 20xx | 21xx | 22xx | 23xx | 24xx |
| 16.33 | 12 | 6.67 | 1.33 | -4 | -8.33 | -13.7 | -19 | -24.3 | -28.7 |
If you wish to know the phase of the moon for a date on the Julian calendar, you can use the following table (which cuts off at 17xx but can be extended by subtracting -4.33 for each century after that):
Century Numbers for Julian Calendar
| 8xx | 9xx | 10xx | 11xx | 12xx | 13xx | 14xx | 15xx | 16xx | 17xx |
| 27 | 22.67 | 18.33 | 14 | 9.67 | 5.33 | 1 | -3.33 | -7.67 | -12 |
This table is a little complicated, but if you are only interested in Gregorian centuries and only interested in centuries near our own, you only have a few numbers to memorize
Conway has suggested some refinements, which are described
in his book (Vol. 2, p. 799). They provide somewhat greater accuracy
and account better for leap years; but the simple rule is very
useful nevertheless.
For fun, you can check your calculation for today by looking at
the web page Moon
Phase Calculator, which will tell you how many days until
the next full moon. Remember that the age of the moon is counted
from new moon, and full moon is 15 days later. So, if you have
(for example) calculated that the age of the moon is 7 days, then
it will be about 8 days until the next full moon.
Here's another moon
phase calculator, which is a bit fancier (and also takes longer
to run). It will give you the phase and other information on the
Moon for any date in history. Please note that the page comes
from Italy, and the date is entered in the European style, first
day, then month, finally year.