An increasingly important survival skill in the modern world is the ability to mentally

calculate on which day of the week a particular

date falls. Most of us have at some time been hospitalised, caused the death of loved ones or suffered the embarrassment of public nudity due to our inability to perform this simple task. But anyway, enough about me: let us consider how best for you to avoid a similar fate.

The technique described below is due to the famous mathematician John Horton Conway, who refers to it as the Doomsday algorithm. Conway delights in asking new acquaintances their birthdates and almost instantaneously telling them on which day of the week they were born; he claims (perhaps not unreasonably) to be the fastest human date-calculator in the world.

Firstly, a little memorisation is an essential investment for the budding Doomsday algorithmist. The heart of the technique is a list of what Conway calls "Doomsdays"--one date in each month--which conveniently all fall on the same day of the week each year. (For example, in 2000 each Doomsday was a Tuesday.) The dates are as follows:

January 31st (or 32nd in a leap year--this apparent absurdity is explained below)

February 28th (or 29th in a leap year)

March 0th (see below for explanation)

April 4th

May 9th

June 6th

July 11th

August 8th

September 5th

October 10th

November 7th

December 12th

Do

*not* give up in disgust at this point. With a couple of tricks, this list is very easy to memorise. For example, note that for all the even

months except February, the month number equals the day number: 4/4, 6/6, 8/8, 10/10, 12/12. As for May, July, September and November, Conway recommends using the

mnemonic "I work from 9 to 5 at the 7-11" which yields the dates 9/5, 5/9, 7/11 and 11/7. Next, remember that the last day of February is always a Doomsday--whether or not this happens to be the 28th or 29th. Since
this is one day before March 1st, we list it again as "March 0th", a
somewhat redundant entry but a convenient mental device when
calculating dates in March. A similar illegal date is given to the
January Doomsday in leap-years: "January 32nd" obviously refers to the
day after January 31st, even though this is not its proper name.

Having memorised this list, you're already able to calculate any date
in 2000, since we know that Doomsday 2000 is Tuesday. For example,
the 4th of July in 2000 is seven days before the Doomsday (Tuesday)
July 11, and is therefore also a Tuesday. January 30th 2000 comes two
days before a Doomsday (2000 was a leap year so "January 32nd" is a
Doomsday) and is therefore a Sunday.

The remainder of the technique involves calculating the Doomsday for
any given year. The procedure is
very simple and with practice will take a matter of seconds:

First, what century are we talking about? For the 2000s, as we have
seen, the first Doomsday of the century is a Tuesday. In 1900, it
was a Wednesday. In practice this is all you'll need to
remember--everyday usage rarely requires calculation of dates outside
1900-2099. (You can easily enough find the Doomsday for other
centuries yourself if necessary.)

Next comes the only bit of actual mathematics involved. Take the last
two digits of the year you're working with, and do the following:

Count the number of dozens in this number (i.e., divide by 12)

Add on the remainder left over when dividing by 12

Add on the number of 4s in this remainder (i.e., divide it by 4)

For example, if we're talking about 1995, then the last two digits are
95, into which twelve divides 7 times with a remainder of 11, into which four
divides 2 times. Thus the number calculated is 7+11+2=20.

This number you've just found represents the offset between that
year's Doomsday and the turn-of-the-century Doomsday. In the example
above, we've found that Doomsday 1995 is 20 days later than Doomsday
1900 (which was Wednesday). For our purposes multiples of seven
obviously don't make any difference, so we can throw away 14 days and
simply say that it's 6 days later than Wednesday, i.e., Tuesday. What
this means is that in 1995, all of the dates January 31, February 28,
April 4, etc., fell on Tuesdays. This makes it easy to find any other
date via simple arithmetic.

There's nothing more you need to know. Practice by finding on what
day of the week you were born. Here's one example
to clarify: let's say you were born on October 7th, 1968.
Remember that the Doomsday for 1900
was Wednesday, and we need to find by how much
1968 differs from that. Twelve divides 5 times into 68, with remainder
8. Four goes twice into this remainder. Add the three results:
5+8+2=15, which (throwing away a couple of multiples of seven, i.e.,
14 days) is an offset of 1 day from 1900's Doomsday. So Doomsday 1968
is Thursday, which in particular means the 10th of October was a
Thursday. Therefore the 7th of October, coming three days earlier,
was a Monday.

**Postscript:** Here's a poetic way to describe the Doomsday algorithm, possibly due to Kate Larson (at least, it appears on her homepage at `http://www.cs.wustl.edu/~ksl2/mathpoem.html`):

The last of Feb., or of Jan. will do

(Except that in Leap Years it's Jan. 32)

Then for even months use the month's own day,

And for odd ones add 4, or take it away^{*}

Now to work out your doomsday the orthodox way

Three things you should add to the century day

Dozens, remainder, and fours in the latter,

(If you alter by sevens of course it won't matter)

In Julian times, lackaday, lackaday

Zero was Sunday, centuries fell back a day

But Gregorian 4 hundreds are always Tues.

And now centuries extra take us back twos.

* According to length or simply remember,

you only subtract for September, or November.