The Doomsday Algorithm is a method for calculating the day of the week for any given date that’s easy enough it can be done mentally very quickly. When you get good enough at it, it really does look like a superpower and it’s a great trick for impressing friends and family. For example:

Me: Hey, would you believe me if I told you that I memorized the whole calendar?

You: There’s no way you could do that. When was my birthday, 12th of June 2002?

Me: that was a… Wednesday. The 2nd of the month actually!

You: 😮

The algorithm was designed by John Conway in 1973, who got so good at it that he could usually give the correct answer in under two seconds, and he even programmed his computer to quiz him with random dates every time he logged on - only he could so it was a great password!

The technique revolves around the fact that certain dates of the year will always fall on the same day of the week - we call this day doomsday. If we could calculate which day of the week this would be for a certain year (and remember one of these special days for each month), we could hypothetically easily find the day of the week for any day of the year. This is exactly what the doomsday algorithm accomplishes!

Useful links:

• The Doomsday Algorithm - Numberphile YouTube video
• Doomsday rule Wikipedia article
• doomsday CLI tool for explanations and testing
• John Conway himself describing the algorithm

Date math

Lets establish a common ‘language’ for doing maths with dates, which if you try to do with the names of the days (Monday, Tuesday, etc) is really difficult - for example, what’s 19 days after Thursday? Instead, we represent the days with numbers which allows for much easier calculations:

Now “19 days after Thursday” becomes 4 + 19 = 23. But what day of the week is 23? To simplify further notice how after exactly one week, or 7 days, you will always get the same day of the week. This means that 7 days after Thursday is still a Thursday, as well as 14 days after, 21 days after, 28 days after, etc. That means that 23 is really the same as 23 - 21 = 2. Finally, we’ve found that 19 days after Thursday is a Tuesday!

Finding the day of the week

As I said before, the technique uses the fact that every year has a special ‘doomsday’ which we can use to find the date for any date of the year. There are only 3 simple things that you need to calculate and add together:

1. The doomsmonth,
2. The doomsyear,
3. And the doomscentury

Doomsmonth

Here’s a fun fact: In any year, all of the following dates will be on the same day of the week:

• 3/1 or 4/1 on leap years
• 0/2 or 1/2 on leap years
• 0/3
• 4/4, 6/6, 8/8, 10/10, 12/12
• 5/9, 9/5
• 7/11, 11/7

Not only will they all be the same, they’re all a doomsday! That’s why, given any month, you should be able to quickly recite what day of that month is a doomsday. It looks daunting at first, but there are a few tricks you can use to make it easier:

1. The “0th” day of March
2. All the even months (after March) have doomsdays on the same number day
3. The mnemonic “I work from 9-to-5 at the 7-11” to remember the odd months

All you have to do to find the doomsmonth is to subtract the day of the month that’s a doomsday from the date you’re given. That’s it! What we’re really doing here is finding how far ahead the given day is from a doomsday, which we’ll soon learn how to find what day of the week it is.

Doomsyear

Finding how far away the given date is from the doomsday is well and all, but it’s not very useful if we don’t happen to know what the doomsday is for a certain year. That’s where the doomsyear and doomscentury come in! What we’ll focus on first is the doomsyear, which is based on the first two numbers of the year - i.e. the XX part of 20XX.

For every year that goes on, the doomsday increases by one. That’s except for leap years which add another day on as well. That’s really it! So if we knew the doomsday for the year 2000 (lets say x), it would just be the next day for 2001 (x + 1), the day after that for 2002 (x + 2), the next day for 2003 (x + 3), then it would skip an extra day for 2004 because it is a leap year (x + 5). So on and so forth!

As you can imagine, this would get tedious for higher years. There’s a trick though: 0, 28, 56 and 84 all have a doomsyear of 0. This means all we have to do is add how many years ahead we are from one of these 4 key years, and also add how many leap years there are which is just the difference divided by 4 (ignore any fraction part). This is the hardest part of the whole algorithm, but it gets easier with practice.

Doomscentury

This is the last missing piece in our puzzle and by far the easiest part. The doomscentury tells what the doomsday was on the first year of every century, and all you need to do is remember these centuries:

An easy way to remember it is that “Y2K” has a doomscentury of 2. But what if you wanted to find for centuries beyond 1800 and 2100? Just subtract/add 400 years until they’re in that range. For example, the doomscentury for 2700 is the same as 2300 is the same as 1900, which is 3. Simple enough!

Bringing it all together

What remains to finding the day of the week? Well, we already know what day of the week doomsday is for the given year (the doomsyear + doomscentury), and we know how far away the given date is from a doomsday (the doomsmonth). What’s left to do is to add everything together! More often than not, adding everything together results in a number larger than 7 - don’t forget you can get rid of all the 7’s and get a number between 0 and 6. This is the day of the week, and the algorithm is complete!

Tips & Tricks

Here is a list of useful tips to make the whole process much easier:

1. Keep a running result instead of trying to remember multiple numbers. Although nice to wrap your head around the concept, doing it exactly as above can be difficult because you need to calculate then remember a couple different numbers, as well as the original date. This gets difficult really fast, and is very error prone! Instead, have only one number you constantly modify by adding/removing terms to with the following process:
   1. Start with the date you’re given.
   2. Subtract from this the day of the month that’s a doomsday - what you have now is the doomsmonth.
   3. Find and add onto this number the doomscentury.
   4. Find and add onto this number the doomsyear.
   5. Find what day of the week this is and you’re done!
2. If it makes the number easier, forget 7’s! If at the end of any step in the process I mentioned in the previous tip you have a multiple of 7, that means it is the same as 0 and you can essentially forget about it and start fresh on the next step. This makes it sooo much easier since most of the time you can just “discard” everything you’ve done till then and continue from 0. Plus, it’s really satisfying!
3. Don’t be afraid of negative numbers. Sometimes you might end on a negative number, such as -5. What day of the week is that? Just add 7 to make it positive; so for -5, It’s actually 2 which is a Tuesday. Embracing negative numbers can sometimes make your calculations easier and faster - for example, If you find at the end of step 2 of the process from the previous tip you get a doomsmonth of 5, sometimes it’s easier to instead of think of this as -2 and actually just subtract two instead of adding 5. It’s up to you!

Examples

This example had nothing too difficult, we just had to remember to account for leap years in the 2nd step because the month was February and the year was a leap year.

This example showed how using negative numbers can come in handy and end up simplifying steps. Lets just look at one more example that shows another trick to make the calculations easier:

What next?

Congratulations if you really made it this far! From here, it’s just practice practice practice - the ol’ adage that “Practice makes Perfect” absolutely applies, and you’ll only get faster at this by trying more and more dates. Try doing a couple a day for a minute and see the progress you make. That being said, there are also more intermediate/advanced concepts you can apply to get faster:

1. Start calculating as the date is being read out to you. Since most people will start with the day first, you can essentially start straight from step 1. When they get to the month, start working on step 2 before they even get time to think of a year. This is a neat little cheat that will make it much more impressive how quickly you can recall days!
2. When calculating the doomsyear we said that 0, 28, 56 and 84 all have a doomsyear of 0. This still leaves big gaps and is still sometimes difficult. Instead, just think about your twelves timestable. 12 has a doomsyear of 1, 24 has 2, 36 has 3, so on and so forth up to 84 having a doomsyear of 7 (I actually like to think of this as 0) and 96 having a doomsyear of 8 (or alternatively 1). This is the method I prefer as it is quick and easy to calculate, and always keeps the year close.
3. If even this doesn’t tickle your fancy, there’s been a recent discovery of a new algorithm which is short, dead simple to calculate and highly effective: The Odd+11 method. You can find the original paper that discussed this new method here, but this is the short version of the steps:
   1. If the year is odd, add 11
   2. Half the result
   3. If the result is odd, add 11
   4. Find how far the result is from the next highest multiple of 7

Of course, none of this would exist if it wasn’t for John Conway for originally creating the Doomsday Algorithm. Sadly, as of writing this post, he has recently passed away. But his contributions to the mathematical and computer science community were immense and his ideas will continue to be taught and learnt from for generations to come.