What day of the week was it?

Among the more esoteric forms of nerd-entertainment is the “day of the week” problem. Specifically, given a date from history, can you determine the day of the week on which it fell? For example, humans first landed on the moon on July 20, 1969, and unless you are old enough and have a great memory, you probably didn’t know that this was a Sunday. Of course you can always look it up somewhere (TimeAndDate.com is my personal favorite), but the challenge is to figure it out in your head.

In one extreme example, it’s said that the mathematician John H. Conway has a password-protection system on his computer that asks him to identify, in rapid succession, the days of the week for three randomly-generated dates. How does he (or anyone) do it?

Well, there are a few convenient and peculiar facts about our calendar that can help us do the math. First, there is a set of “reference dates” that all fall on the same day of the week every year. The first of these are 4/4, 6/6, 8/8, 10/10, and 12/12. For the odd-numbered months, we have some nice dates that also fall on the same day of the week as the first five: 5/9, 7/11, 9/5, and 11/7. (Check out this year’s calendar if you don’t believe me.) Notice that we don’t have any dates for January, February, or March. This is unfortunate, and is partly the result of having a leap day inserted every four years. But there are some ways around that problem. I use the dates 1/9 and 2/6, along with 3/0 (i.e., the last day of February) to do it. Be careful, though—we need to treat January and February of year x as technically being part of year x-1, but the 3/0 date as part of year x.

Thus, once you know the reference day (the day of the week on which all the reference dates will fall) for a given year, you can figure out the day of the week for any date that year by adding and subtracting weeks from a reference day. For example, July 20 is 7/20, which is 9 days after the date 7/11 (which was a Friday). So, 7/20 is one week and 2 days later, on a Sunday.

Next, we need to have a way of finding the reference day from a given year. As you might expect, that reference day moves each year, though this movement is complicated by that leap day every four years. An easy mnemonic is “12 years is but a day”. That is, if you move 12 years forward (say, from July 20 1969 to July 20 1981), the day of the week will move exactly one day forward (from Sunday to Monday). If you take into account that the day of the week skips one day forward in non-leap years and two days in leap years, you can determine the day of the week of a date in any different year. Of course, you can do all of this in reverse if you want to move backward in time.

And that’s it! We can now use the “reference day + “12 years is but a day” + “skip 1-2 days per year” rules to determine the day of the week for historical dates. For example, suppose we want to know what day of the week the stock market crashed in the U.S.—10/29/1929. Here are the steps:

  • The reference day for 2016 is Monday.
  • This means that 10/10/2016 is Monday, so that 10/29/2016 is 2 weeks and 5 days ahead—i.e., a Saturday.
  • Back up seven multiples of 12 from 2016 to 1932 (since 2016 – 7•12 = 1932), so the day of the week shifts 7 days back (to where you started)—still a Saturday.
  • Back up three more years, keeping in mind that you passed the leap day on Feb 29 1932, so you move back four days from Saturday—now on a Tuesday.
  • There you have it—Oct 29 1929 was a Tuesday (which is why it’s often referred to as Black Tuesday!).

As long as you know the reference day for the current year, you can apply this rule to go forward or backward to any date in the Gregorian calendar. In my case, my birthday happens to fall on that reference day of the week, but any of the reference dates will work. (And, for the calendar nerds out there: for historical events, you need to know whether the country in question was using the Gregorian or Julian calendars, which is a whole other can of worms.)

Thanks for reading! I’ll be back for more at an unspecified date in the future.

 

On the Fundamental Groups of Nations (Part II)

Hi everyone! I’ve been gone for a long while. Basically, we had a baby, then I got busy with full time work, and life changed. I won’t promise regular postings on this site, but when something interests me I will try to drop it here for you to enjoy.

To follow up on my last pre-blackout-period post, the weird boundaries and enclave/exclave situations with borders around the world led to some interesting math, but it looks like India and Bangladesh got there before I could tell you about it! Specifically, they signed a treaty last year to swap various enclaves and exclaves to simplify the border. So we say goodbye to the world’s only third-order exclave and look forward to less strife and confusion at the border.

That’s where I’ll leave this story. I hope to continue with other math topics moving forward—stay tuned for some peculiar math in the Gregorian calendar!