From the workplace · Learning

How to work out the day of the week from any date (takes 10 minutes)

This is a test of my ability to teach, and my students’ ability, and willingness, to learn.

You can work out the day of the week for any date in history if you can do the following 3 things: add 4 numbers together; divide numbers by 4 and by 7; learn a 12-digit sequence. If you practise it enough you will be able to do it in your head. (And people might call you “Rain Man”.)

I have used it on training courses to change the mood, to get the delegates to use different parts of their brain, and to test out the most effective way to teach the steps. These steps work best when taught but if you give this page 10 minutes of your attention you will have all the information you need to work out the day of the week from any date.

Overview

You will need to add 4 numbers together, then divide them by 7, and the “remainder”, as we called it at primary school, will give you the day of the week.

Adding 4 numbers together

The 4 numbers that you add together are:

  1. the day (between 1 and 31);
  2. the year (as a 2-digit number);
  3. the year (as a 2-digit number), divided by 4, ignoring any “remainder”;
  4. a 1-digit number (between 0 and 6) corresponding to the month.

Step 4. is the most difficult. For most of us it involves learning something new: you need to know a numerical sequence that corresponds with the 12 months of the year. Rather than write it out in long-form (“January = 1, February = 4, March = 4, and so on) this is how I write the sequence:

JFMAMJJASOND (for the months January, February, March and so on)
144025036146

Having practised it enough I know that the code for September or December is 6, for example, and the code for April or July is 0.

I remember it as 3 4-digit numbers: 1440 / 2503 / 6146. You might prefer to remember it as 4 3-digit numbers (144 / 025 / 036 / 146). Either way if the month is May you add 2, if it’s June you add 5, and so on.

Dividing by 7 to find the day of the week

  1. When you have added the 4 numbers above together, divide it by 7 and “keep the remainder”.
  2. Finally, the number that you end up with in step 5 corresponds with a day of the week: 0 = Saturday, 1 = Sunday, 2 = Monday and so on, through to 6 = Friday. Or, to use a similar shorthand to the months, above, here is the sequence starting with Saturday:

SSMTWTF
0123456

Some examples

I’ll do 3 examples, then run through the two exceptions (January or February in a Leap Year, and allowing for different centuries).

 Example 1 (30 July 1966)

30 July 1966, the day that England beat West Germany to win the World Cup:

The numbers you add are 30 (day), 66 (year), 16 (year divided by 4, ignoring any remainder), 0 (the month code for July)

30 + 66 + 16 + 0 = 112

Divide 112 by 7. It’s exactly 16 (no “remainder”), so the day of the week is Saturday.

Example 2 (6 July 1972)

6 July 1972, the day that David Bowie first performed “Starman” on “Top of the Pops”.

The numbers you add are 6 (day), 72 (year), 18 (year divided by 4, ignoring any remainder), 0 (the month code for July)

6 + 72 + 18 + 0 = 96

96 / 7 = 13 remainder 5

5 corresponds with Thursday.

(As anyone from my generation knows, “Top of the Pops” was always on a Thursday.)

Example 3 (6 June 1944)

6 June 1944, D-Day

The numbers you add are 6 (day), 44 (year), 11 (year divided by 4, ignoring any remainder), 5 (the month code for June)

6 + 44 + 11 + 5 = 66

66/7 = 9 remainder 3

3 corresponds to Tuesday: D-Day was a Tuesday.

Two Exceptions (January or February in a Leap Year; Different Centuries)

  1. If the date is in January or February in a Leap Year you need to subtract 1 from your total before dividing by 7.

(Ignore these bracketed sentences if you know what a Leap Year is. Some students, in my experience, don’t know. Broadly speaking, it’s any year that’s divisible by 4. For sports fans, that means any Olympic year. But, to complicate things, years ending in “00” are only Leap Years if they’re divisible by 400. So, 1900 and 2100 are not Leap Years but 2000 is. There is a bug in Excel, deriving from Lotus 1-2-3, which makes 1900 a Leap Year. It wasn’t.)

  1. The above steps work for all dates in the 20th Century; no adjustment is needed. For dates in the 21st Century add 6 (or subtract 1 – the end result, when dividing by 7 and looking for the remainder, is the same). For dates in the 18th Century add 4. For dates in the 19th Century add 2. The sequence, stretching ahead indefinitely, starting with the 20th Century is:

0, +6, +4, +2

That is 20th Century (no adjustment), 21st Century (add 6), 22nd Century (add 4), 23rd Century (add 2), 24th Century (no adjustment), 25th Century (add 6), and so on.

Example 4, using both exceptions (19 January 2016)

I’m posting this on Tuesday 19 January 2016 (January, in a Leap Year, in the 21st Century), so I know what the correct answer should be.

Using the first 4 steps above:

Day (19) + Year (16) + Year divided by 4 (4) + Month Code (1) = 40

40 divided by 7 = 5, remainder 5, giving the incorrect day Thursday.

However, the 2 exceptions state that you subtract 1 (January or February in a Leap Year) and add 6 (for a date in the 21st Century) before dividing by 7.

40 – 1 + 6 = 45

45/7 = 6 remainder 3

3 corresponds to a Tuesday, the correct day of the week.

Summary

The instructions above can be summarized on a single sheet of paper. Here’s what I usually write, as a reminder:

Month Codes JFMAMJJASOND = 144025036146

Add: Day + Year + Year Divided by 4 (ignore any remainder) + Month Code

Divide Total by 7

Keep the Remainder: 0 = Saturday, 1 = Sunday, 2 = Monday, and so on.

Exceptions:

January or February in a Leap Year: subtract 1 from the Total before dividing by 7

Different Centuries: Add the following to the Total before dividing by 7 if the date is not in the 20th Century: 19th Century +2, 21st Century +6, 22nd Century +4

Now, practise for a while before trying the shortcuts below

If you practise these steps for 10 minutes a day, twice a day, for 2 weeks, you shouldn’t need any reminders of the steps. You will get quicker. And then you’ll be ready for the shortcuts below.

Shortcuts (for later, when you’ve practised the above steps a few times)

If you practise this for a while you might work out that you can take shortcuts. For example, if the day of the week is 28 you don’t need to add 28 to your total: it’s a multiple of 7 so it won’t affect the end result, when you divide by 7 and look for the remainder. Similarly if the day of the week is 29 you could just add 1 to your total rather than 29.

And if the year is 70 you don’t need to add that to your overall total either, it won’t affect the end result. (You still need to divide it by 4 and ignore any remainder, for Step 3 above, but it’s easier to add 17 to the overall total than to add 70 + 17.)

Have fun.

 

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