The day for any date by mental calculation

Scope
Here is a method for quickly calculating the day of the week for any date by mental arithmetic and mnemonics. The method works for any date since the adoption of the modern Gregorian calendar, which happened in different parts of the world at different times from 1582 onward.

For dates within the current year, the day can be calculated within 2 or 3 seconds, as described lower down on this page.

The basic method
We add four numbers, as follows.

• The century number
• The year number
• The month number
• The date

I explain below how we obtain those numbers, but first, here is an example, 30 October 2008.

The century number for the 21st century is 0.

The year number for 2008 is 3.

The month number for October is 6.

The date in which we are interested is 30.

Adding these four numbers gives 39.

Then we find the remainder that is given when we divide by 7.

The remainder is 4. That remainder tells us the day for the date, with 0 meaning Sunday, 1 meaning Monday, 2 meaning Tuesday, etc.

Therefore 30 October 2008 falls on a Thursday.

The century number
Here are some century numbers.

1800 to 1899: 3
1900 to 1999: 1
2000 to 2099: 0

There is a more complete list of century numbers lower down on this page.

Finding the year number by arithmetic
The following method is probably too error-prone for mental calculation, but is fine if all that we require is a pencil and paper method of calculation.

I will illustrate the method with the calculation of the year number for the year 1985.

We take the last two digits of the year, 85.

We divide by 4, discarding any remainder, obtaining 21.

We add the 21 to the 85, obtaining 106.

We then find the remainder that is given when we divide 106 by 7.

The remainder is 1. Therefore the year number for 1985 is 1.

By similar calculations, the year number for 2008 is 3, and the year number for 1800 is 0.

The month number
The month numbers are given by the following table. But I don't remember them just as numbers. I use some mnemonics.

Month	Month number
January	6 in non leap years 5 in leap years
February	2 in non leap years 1 in leap years
March	2
April	5
May	0
June	3
July	5
August	1
September	4
October	6
November	2
December	4

Here are the mnemonics by which I remember those numbers (the non leap year ones — one must remember that for leap years, the numbers for January and February must be reduced by one).

I visualise for each month the first so many letters of its name. The number of letters gives the month number.

Month	Month number	What I visualise	Memory aid
January	6	Januar	The german word for January
February	2	Fe	The chemical symbol for iron
March	2	Ma	My ma (mother) was born in March
April	5	April	The whole word
May	0		Just an empty space (but see below)
June	3	Jun	One less than one would expect
July	5	July+	One more than one would expect
August	1	A	A lonely A
September	4	Sept	The usual abbreviation
October	6	Octobe	Rhymes with Obe Wan Knobe
November	2	No	Oh No! I can't remember that month number. Er.. oh yes I can, the clue is what I just said, "No".
December	4	Dece	Like "dessicated coconut", as abbreviated by someone with poor spelling (ok, this one is lame, but it's so lame that it's unforgettable).

For May, I can alternatively visualise a motorway where Fred Flintstone is driving his wood and stone car. Since he lived so long ago, it can't be the M1 (the first motorway built in the UK), so it must be the M0. So the month number is 0.

Ok, those are horribly lame, but they work for me. I'm sure you will be able to invent ones that work for you.

Which years are leap years?
In the Gregorian calendar, leap years are the ones that are divisible by 4, except that there is a special rule for years that end in 00. Years that end in 00 are leap years only if they are divisible by 400. Therefore 1900 was not a leap year but 1600 and 2000 were.

More century numbers
1500 to 1599: 1
1600 to 1699: 0
1700 to 1799: 5
1800 to 1899: 3
1900 to 1999: 1
2000 to 2099: 0
2100 to 2199: 5
2200 to 2299: 3
etc.

The pattern repeats endlessly.

An easy way to find the remainder given when we divide by 7
We simply subtract multiples of seven until we can no longer do that. For mental arithmetic, that seems easier than doing a division calculation.

The method is called casting out sevens.

So for the year number calculation higher on this page, we proceed as follows. From the 106, we can subtract 70, leaving 36. Then we can subtract 35, leaving 1. Since we can no longer cast out any sevens, the remainder is 1.

Doing the calculation within seconds.
In most cases that arise in everyday life, the calculation is for a date within the current year.

So we can simply remember the year number for the current year. Then for dates within that year, you can calculate the day within 2 or 3 seconds.

For example, at the time of writing this text it is 2009. I simply remember that for 2009, the year number is 4.

For use in everyday life, the above is probably all that we need. We can just remember the current year number and for other years calculate it by the arithmetic method, perhaps using pencil and paper to avoid error, depending on how reliable is our mental arithmetic.

To be able to answer within seconds for dates within any year, without recourse to pencil and paper, we can use the following mnemonic method to find the year number.

Finding the year number by mnemonics (optional)
The following method is faster and less prone to error than the arithmetic method.

The mnemonics are for the year numbers for all of the two digit years that are divisible by 4. The year numbers are as follows.

Year	Year Number
00	0
04	5
08	3
12	1
16	6
20	4
24	2
28	0
32	5
36	3
40	1
44	6
48	4
52	2
56	0
60	5
64	3
68	1
72	6
76	4
80	2
84	0
88	5
92	3
96	1

The mnemonic system is based on the following number/letter associations.

0 O
1 A
2 B
3 C
4 D
5 E
6 S
7 G
8 H
9 N

These associations must be memorised so that we have them at our fingertips instantly.

For each entry to be remembered, we associate something memorable, e.g. a mental image, word or phrase.

Here are some suggestions.

Year & Year No.	Letters	Memorable entity
00 0	OOO	Mnemonic not required - use the arithmetic method.
04 5	ODE	Mnemonic not required - use the arithmetic method.
08 3	OHC	"Oh! Crikey", Oliver Hardy eating Cake, or Overhead Camshaft
12 1	ABA	Abacus
16 6	ASS	Arnold Schwarzenegger with a snake around his neck, or simply the word ASS
20 4	BOD	Bod, Bodkin or Bodensee
24 2	BDB	Beady Bee (a bee ornament made of beads)
28 0	BHO	Black Hole - it's round like an O
32 5	CBE	CB Radio - Extra communications channel, or Commander of the British Empire
36 3	CSC	C S Lewis - Christian
40 1	DOA	Dead On Arrival
44 6	DDS	A Double Diamond (beer) Spillage or Donald Duck has a Squeaky voice
48 4	DHD	David Harris - Downunder (author of the Pegasus email client - he lives in New Zealand)
52 2	EBB	Ebbw Vale
56 0	ESO	Esoteric
60 5	SOE	So Easy, or School Of Engineering
64 3	SDC	SD Card (a storage device used in digital cameras)
68 1	SHA	A well-known hashing algorithm
72 6	GBS	Great Britain sinking into the Sea
76 4	GSD	Gilbert and Sullivan - Delightful
80 2	HOB	Ho ho ho - Santa turns up at the wrong party - a Birthday party
84 0	HD0	Hard Disc - it's round like an O
88 5	HHE	Howard Hughes - Eccentric
92 3	NBC	A US television network
96 1	NSA	National Security Agency

You will want to replace some of the above with mnemonics of your own, I am sure. E.g. I doubt whether many of you have heard of David Harris.

Some of the mnemonics that I use are not in the above table since they relate to people or events in my personal life.

To find the year number for 1991, we drop down from 91 to the next lowest multiple of 4, 88. That gives HH, from which we recall that Howard Hughes was eccentric and so remember HHE. The E gives the year number for 1988. It is 5. 91 is 3 years on from 88, so we add 3 to the 5 to give a year number for 1991. It is 8. Then we can cast out a seven to give a more convenient year number for 1991. It is 1.

Avoiding errors by being methodical
It is all too easy to make errors in cases such as 1 August 1997 where the same number (in this case 1) features several times in the calculation.

Therefore, always process the numbers in exactly the same order for every case.

• Phase 1, consisting of
  1. The rounded down year
  2. The extra to be added on
  3. The century number
• Phase 2 - the month number
• Phase 3 - the date

An alternative method for calculating the year number
As an alternative to the mnemonic method for finding the year number, those who are good at mental arithmetic may prefer the more efficient version of the arithmetic method that is described in an older version of this page, here.

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