The Doomsday rule: mental day-of-week calculations
The Doomsday rule is an algorithm for working out the day of the week of a given date. It’s based on John Conway’s observation that certain memorable dates called doomsdays (4/4, 6/6, 8/8, 10/10, 12/12, 9/5, 5/9, 7/11, 11/7, …) always occur on the same day of the week in any given year. This day is known as the year’s anchor day. To compute an arbitary day of week you work out the anchor day for the given year and the offset between the target date and an appropriate doomsday. It’s intended to be simple enough that you can do the computation in your head with a bit of practice.
A couple of recent Youtube videos have raised its profile, first one by James Grimes on Numberphile and then a second by Mike Boyd. In both videos they show that it’s possible to do the mental calculations quite quickly. I tried learning it some time ago and struggled to achieve that kind of speed on the algorithm as described in the literature. The first part of the computation involves counting leap years (i.e. dividing by 4) and the second part involves subtracting numbers mod 7 which means you have to get the sign right and sometimes have to deal with negative numbers. All definitely possible, but tricky to do quickly in your head.
I found that the effects of practice was basically to inline various bits of the strategy and commit the inlined version to memory, but the inlining wasn’t consistently applicable which meant that some dates ended up harder to compute than others. Since the point of practice seemed to be to commit various things to memory, I figured it’d be simpler to rework the algorithm to make use of memorisation directly.
There’s not actually that many things to remember, it only took me a couple of hours, and once I’d done the memorisation work I found I can run the algorithm pretty quickly even if distracted by other things such as continuing a conversation. I wonder if when other folks say they’re doing Conway’s method they’re really doing something more like this, or whether it’s just me that finds recalling memorised facts much easier than doing actual computations.
One handy feature of the method presented here is that you combine all the sub-computations together wih addition (mod 7) which is commutative and associative so you can process the components of the date in any order and reduce them to a single value at each step which saves on working memory slots. Here in the UK most folks say the date in day-month-year order; in the US month-day-year seems preferred, but really any order is reasonable.
Overview
Process the day-of-month, the month, the century and the year-of-century in any order you choose. Each component yields a value which you add to a running total, computed mod 7. The final running total after all four values are added (occasionally with a simple leap-year correction) indicates the day of the week of the target date, with 0 meaning Sunday, 1 meaning Monday and so on.
Day-of-month
When you hear the day of the month, add it to the running total, working mod 7.
Month
When you hear the month, convert it to a value according to the following table and add the value to your running total, working mod 7.
| Month | Value | Month | Value | Month | Value |
|---|---|---|---|---|---|
| January | 4 | May | 5 | September | 2 |
| February | 0 | June | 1 | October | 4 |
| March | 0 | July | 3 | November | 0 |
| April | 3 | August | 6 | December | 2 |
Just learn this mapping. There’s kind of a pattern but learning the values doesn’t take long and means you can process the month very quickly while listening for the rest of the date.
Century
When you hear the century—the first two digits of the year—convert it to a value according to the following table and add that value to your running total, working mod 7.
| Century | Value | Century | Value |
|---|---|---|---|
| 17xx | 0 | 21xx | 0 |
| 18xx | 5 | 22xx | 5 |
| 19xx | 3 | 23xx | 3 |
| 20xx | 2 | 24xx | 2 |
Again, just learn this mapping. There is a repeating pattern because the Gregorian calendar repeats every 400 years, but in practice you’ll mostly get one of these centuries so it’s simpler to remember them. You can reasonably reject years before 1700 because the UK and US didn’t start using the Gregorian calendar until 1752, and if you get a year after 2499 then it’s usually ok to take a bit longer to work out to which of 20xx, 21xx, 22xx or 23xx it’s equivalent.
Year-of-century
The year of the century—the last two digits of the year—is the trickiest bit to deal with. If it’s a multiple of 4 then it maps to a remainder mod 7 as follows which should be added to the running total:
| Year | Value | Year | Value | Year | Value | Year | Value | Year | Value |
|---|---|---|---|---|---|---|---|---|---|
| xx00 | 0 | xx20 | 4 | xx40 | 1 | xx60 | 5 | xx80 | 2 |
| xx04 | 5 | xx24 | 2 | xx44 | 6 | xx64 | 3 | xx84 | 0 |
| xx08 | 3 | xx28 | 0 | xx48 | 4 | xx68 | 1 | xx88 | 5 |
| xx12 | 1 | xx32 | 5 | xx52 | 2 | xx72 | 6 | xx92 | 3 |
| xx16 | 6 | xx36 | 3 | xx56 | 0 | xx76 | 4 | xx96 | 1 |
The values for the nine multiples of 12 are just the result of dividing by 12. The other 16 are all ±4 away from a multiple of twelve so you can in principle compute the nearest multiple of 12 and then add or subtract 2 according to the pattern, but it’s only 16 values so still not that hard to just memorise the mapping.
If the year isn’t a multiple of 4 then add its remainder mod 4 to your running
total, then round the year down to the nearest multiple of 4, grab the value
from the table above, and add that to the running total as well. There’s
various other ways you could achieve this, for instance you could memorise all
100 values, but representing the year as 4k+r and then processing 4k and
r separately works best for me.
Leap year correction
If the target is in January or February of a leap year then decrement the running total by one. This is kind of irritating to remember. However note that leap years are slightly simpler to compute since they’re always a multiple of 4 so there’s no remainder mod 4 to deal with at the year-of-century step.
Finishing up
The final total represents the day of the week of the chosen date, with 0 meaning Sunday, 1 meaning Monday, and so on.
Worked examples
19 July 1989
The day-of-month 19 becomes 5 which is the initial running total. July has value 3, and adding this to the running total gives 1. The century 19xx is also 3, which adds to the running total to give 4. The year-of-century xx89 is 1 greater than a multiple of 4, namely xx88, so we add the 1 to the running total to give 5. Finally xx88 has value 5 which is added to the running total to give 3, which means that 19 July 1989 was a Wednesday.
February 27, 2204
February has value 0 so the running total starts out as zero. The day-of-month 27 becomes 6 which replaces the running total of zero. The century 22xx has value 5 which adds to the running total to give 4 (recalling that adding 6 is the same as subtracting 1). The year-of-century xx04 is a multiple of 4 with value 5 which adds to the running total to give 2. But 2204 is a leap year and the date is in February so we must subtract 1, giving a final total of 1 which means that 2 February 2204 will be a Monday.
1900-01-15
The century 19xx has value 3 so that’s the initial running total. The year-of-century xx00 is a multiple of 4 with value 0 so the running total is unchanged. January has value 4 which is added to the running total to give 0. The day-of-month 15 becomes 1 mod 7 so this is the new running total. The month is January and the year is a multiple of 4 but note that 1900 was not a leap year so no further correction is needed. 1900-01-15 was therefore a Monday.
How it works
The day and month values added together (minus one for January and February in leap years) tells you the offset in days from the year’s anchor day to the target date. You can check that the memorable doomsdays all add up to a multiple of seven: 4 April = 4 + 3 = 7; 9 May = 9 + 5 = 14; …
The century and year-of-century values added together tells you the anchor day
for the year. The year-of-century table adds the number of years to the number
of earlier leap years in the century, i.e. it computes 5n/4 mod 7 for
multiples of 4.
Speed tips
Recalling a value from memory is enormously faster than working it out. I’d rather remember a table of 25 things instead of adding another computation step.
I spent some time actively memorising the remainders mod 7 of the numbers from 1 to 31, even though I can compute them fairly well, because you need to do this step quickly whilst distracted by listening for the rest of the date.
I also found it worth practicing doing sums of remainders mod 7. One of my very early maths teachers made us do speed exercises for adding small numbers, much like multiplication tables, which everyone hated at the time but I’ve since realised how powerful it is to be able to do these sums instantly rather than merely quickly. There’s not actually that many to learn in this case: the sums that don’t exceed 7 are just the regular ones; the sums that equal 7 are easy to map to zero. Sums involving zero are also easy, and adding 6 is easier to treat as subtracting 1. That just leaves 3+5, 4+4, 4+5 and 5+5 to learn.
I learned the year-of-century values in three batches: first the multiples of 12, then 04, 08, 16, 28, 32, 40, 76, 80, and finally 20, 44, 52, 56, 64, 68, 88, 92. Learning all 16 values at once didn’t work well for me, but working 8 at a time was very effective. The batches were pretty much randomly chosen except I made sure the split of values was as even as possible. There’s not much science behind that but it seemed to work.
I recommend learning the values for any years that you will use a lot, such as the current year and a couple of nearby ones. 2021’s is 0 and 2022’s will be 1.
If you’re coming at this cold then the only things you don’t already know are the values for the 12 months, 8 centuries, and the 16 years-of-century that aren’t multiples of 12, which is 36 individual items. To get faster you probably also need to practice the 31 remainders for days-of-the-month, 9 remainders for the multiple-of-12 years, and the 21 sums mod 7. That is 97 things in total, plus maybe the value for the current year ±1 to make it up to an even hundred. There’s loads of spaced-repetition flashcard apps out there that should help with this kind of thing. With a couple hours of up-front investment you can save literally seconds per month of looking up dates in a calendar.
