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## Doomsday Improvements

Published on Sunday, July 07, 2013 in , , , ,

Doomsday Improvements - I bet those are two words you never expected to see together!

Doomsday, in this context, refers to the Doomsday algorithm developed by John Conway, which helps determine the day of the week for any given date. Since it was originally published in 1973, there have been many variations and improvements, and we'll look at some clever ones in this post.

The classic Doomsday Algorithm starts by determining the day of the week in a given year on which the last day of February falls. From there, you're able to work out any date in the rest of the same year.

There are other approaches, of course. Another popular approach for calendar calculation returns results from 0 to 6, where 0 represents a Sunday, 1 represents a Monday, and so on all the way up to 6, which represents a Saturday. Effectively, these methods are based on working out how far a date in a given month and year are from the first Sunday. We can call this a “First Sunday” approach, to differentiate it from the Doomsday approach.

The great thing about the Doomsday approach, of course, is how easy it makes some of the calculations. The great thing about the First Sunday approach is the ease of interpreting the results. Back in 2009, Bob Goddard combined the best of both worlds with his First Sunday Doomsday (FSD) algorithm!

If you've learned the Doomsday approach, you can quickly adapt to this new approach. Let's take a close look at it, and maybe even improve on it.

YEAR CALCULATION: Step 1 determines a key for the year, based on the popular Odd + 11 approach, described on Grey Matters here, and on Wikipedia here. If you read carefully, however, there is an important difference.

The last step of the standard Odd + 11 approach requires you to subtract the result of the previous steps from 7. In the FSD doomsday approach, this step is skipped entirely! Naturally, the fewer the calculations, the quicker you can get to the result.

In 2008, Mike Walters published a quicker way to get the same result as Odd + 11 that would work well here. You can read it in detail at that link, but I'll describe the practical steps here.

Note that Mike Walters' approach also involved subtracting from 7 as the last step. Since it's not used in Bob Goddard's version of Odd + 11, we also won't be using that subtraction from 7 here, either. That way, the results remain consistent.

When given a year, ask yourself whether the last two digits are evenly divisibly by 4. If they aren't, keep adding 11 years at a time, until you get to a year that is evenly divisible by 4. You'll only ever have to add 11, 22, or 33 years, and you don't need to worry if the total goes over 100. Once you're dealing with a year which is evenly divisible by 4, divide that year by 2, and then subtract the nearest multiple of 7 which is equal to or greater than that number. You're done!

Let's use Mike Walters' example of 1953, so we'll only focus on the 53. 53 itself isn't evenly divisible by 4, so we add 11 and get 64, which works! 64 divided by 2 is 32, and 32 - 28 (the nearest multiple of 7 equal to or less than 32) = 4, we now know our key number for the year is 4!

Notice that in Mike Walters' write-up, he gets 3 instead of 4 for 1953? Again, that's because Bob Goddard's FSD approach differs from the standard Doomsday approach by removing the final subtraction from 7. Removing this final step from Mike Walters' approach ensures that we get the correct results required for the FSD approach.

In practice, any even number which is NOT evenly divisible by 4, such as 1962, just needs to have 22 added (62 + 22 = 84). Also, any odd numbered years, such as 1933 or 1947, will require either 11 or 33 to be added to get to a multiple of 4 (33 + 11 = 44, and 47 + 33 = 80).

CENTURY CALCULATION: This is where Bob Goddard's FSD approach really shines! As you may know, the Gregorian calendar repeats exactly every 400 years. Bob Goddard's approach uses the 1700s as a basic century, and the century keys adapt from there, using his ingenious “No tune for Friday” mnemonic.

The 1700s are used as the basic century for 2 reasons. First, because the Gregorian calendar wasn't used in Britain and its colonies (including America) until 1752. Second, it's easy to think of the 1700s as the century in which America got its start, so you can think of the Gregorian cycle as an American history cycle, too.

The astounding thing about the FSD algorithm, though, is that it handles Julian dates easily! Instead of using the 4 century pattern for Gregorian dates, you simply use the century number for Julian dates. For the 1400s, you add 14, and for the 900s, you add 9. It's surprisingly simple.

If you're familiar with how the traditional Doomsday approach handles months, you'll note the same mnemonics are used in the FSD approach, so that's an easy adaption.

If previous approaches have been giving you trouble, try Bob Goddard's FSD algorithm. You just may be surprised how quickly you can pick it up.

### 2 Response to Doomsday Improvements

5:46 AM

Hi Scott,

I really like the odd11 approach to calculating the year, i find it just as fast as calculating which of the 28 memorised years apply. Also, you can just memorise a few years such as current year and next year and use odd + 11 for all the other years.

I particularly like it for calculating the doomsday for years pre 2000

thanks,

Sea

Travis
3:55 PM

I agree with Sea Maguire. When I first learned the doomsday algorithm, I quickly forgot and abandoned it because of the difficulty. Then I learned the odd+11 method, and I now use the odd+11 doomsday algorithm very often.