Category: math

In the process of looking up John Conway’s “Doomsday Algorithm” for figuring out what day of the week “any” date falls on, I came across Chamberlain Fong’s description of a simplification of Conway’s algorithm that he and Michael Walters came up with, which they describe in a paper on arxiv. (“Any” is in quotes for reasons explained in Part 2.)

What does the Fong/Walters algorithm do differently? Let’s note what’s the same. Conway’s scheme of “Doomsdays” is still in place, and you still need to know the Doomsday for the start of the century. The difference is in computing the offset. There is no dividing by 12, remembering the quotient, dividing the remainder by 4, etc. (Apparently that part of the method was due to Lewis Carroll, as explained in Part 2 or in Fong’s Scientific American blog post.)

How does the alternate method , which they call “odd+11” work? If you love flowcharts, look here. But it’s pretty simple:

1. Take the last two digits of the year.
2. Is it odd? If so, add 11.
3. Divide your number (which now must be even) by 2.
4. Is your new number odd? If so, add 11 again.
5. Reduce your number modulo 7.
6. Subtract your number from 7 (“take the 7s complement”).

Continue reading “Doomsday Calendar, Part 3”

This internet thing, though. It seems to be all rabbit holes, all the way down. In my previous post, I wrote about John Conway’s “Doomsday Algorithm” for figuring out what day of the way “any” day falls on. It seems that there are quite a few people who have spent a lot more time thinking about calendars in general, and mental calendars in particular, than I have. Most of what’s in this post is based on this excellent piece by Chamberlain Fong.

The first helpful thing Fong does is to give some history. Conway’s method has two parts: the identification of particular days of the year as “Doomsdays,” which all fall on the same day of the week. Once you know what day of the week any give year’s Doomsdays fall on, it is easy to figure out the day of the week of an arbitrary date. The other part is to figure out what day of the week the given year’s Doomsdays fall on.

The first part of the algorithm seems to be entirely due to Conway. The second part, interestingly, was apparently worked out by Lewis Carroll—one of the few people in the orbit of mathematicians who may have been more playful that Conway. Martin Gardner, the legendary author of Scientific American’s Mathematical Games column, was a serious student of Carroll, and when he came across Carroll’s work on perpetual calendars, challenged Conway to come up with something simpler.

Continue reading “Doomsday Calendar, Part 2”

[This wound up being Part 1 of 3. You can find Part 2 here, and Part 3 here. You don’t need to read all three parts, but Part 3 does actually describe a significantly simpler algorithm than the one Conway used.]

John Horton Conway, a great mathematician of unmatched playfulness, died recently at age 82 of Covid-19. His biographer, Siobhan Roberts, wrote an excellent New York Times obituary. Conway was something of a showman, in a good way, so I should give a spoiler alert up front, especially these days, that this has nothing to do with Doomsday as in the end of the world.

I never met Conway, but in the mid-70s my math camp buddy Rob Indik had a summer in which he commuted to the UI Chicago campus “with Vera [Pless] driving, Conway in front, and me riding in the back and trying to ride along with Conway’s free flow of ideas.” I remember Rob regretting that he couldn’t remember a mental “perpetual calendar” that Conway had taught him. In the computer age we have no call for perpetual calendars, but back when dinosaurs roamed the earth they were novelty items: a small object with a few dials, which would let you figure out what day of the week any date falls on. A mental one sounded like fun, and I occasionally thought—not very hard—about how you might go about it. I never got very far. [I see that this is not the definition of “perpetual calendar” favored by Wikipedia, but that the American Heritage Dictionary has me covered.]

With Conway’s death, I decided to look it up, and found that it’s around the web in several places, but without a very clear explanation of why it works. So I thought I would explain both how it works and why it works.

Continue reading “John Conway’s “Doomsday Calendar””

I read a couple of books fairly recently that discuss the cubic equation: Barry Mazur’s Imagining Numbers: (particularly the square root of minus fifteen) and William Dunham’s Journey through Genius. They got me interested in this formula, which had always been an afterthought (to the extent to which it had been a thought at all) in my mathematics education.

The cubic and quartic formulas were perhaps the greatest accomplishments of Renaissance mathematics, and they were a key spur to Evariste Galois’ beautiful study of the symmetry inherent in polynomial equations. Yet I got a PhD in mathematics without knowing much more than the bare fact that such formulas exist. It should certainly be possible to understand these formulas in light of Galois theory, but such explanations are not easy to find.

I wrote this paper to pass on what I learned about the cubic equation, and how it can be derived in the context of Galois theory.

Continue reading “The Cubic Formula and Galois Theory”

This article, “A User-Friendly Introduction to the Theory of Determinants” was inspired by a number of discussions of the best way to teach determinants on Facebook. It turns out to be a topic that inspires a lot of opinions from mathematicians. Continue reading “Determinants!”

Lots of proofs are based on simple ideas, but get bogged down in notation or exposition that doesn’t bring out the salient points, as if the soloists and all the members of the choir were singing at equal volume.

Sometimes some warm-ups can help people understand what is essential in a proof and what is extra. They can help the simplicity of an idea shine through.

So the first warm-up is a purely geometric proof that the golden ratio is irrational. A number of years ago, I saw such a proof… but the diagram that went with it was laid out on a single line, and it got bogged down in a bunch of notation, and I kind of got it but it certainly didn’t excite me.

Then one day I was looking at my business card and I realized that the proof was right there.  When I was designing my business card, I tried to figure out a good logo, and I eventually settled on a golden rectangle and golden spiral:

Continue reading “A Purely Geometric Proof that the Square Root of Two is Irrational—With a Couple of Bonus Side-Trips”