A math puzzle for you

If I don’t finish reading Othello today, this puzzle is why. Since I love passing on methods of procrastination to you, here you go. Even though it’s a math puzzle, the most advanced math you need to solve it is knowledge of divisibility rules, which aren’t too advanced.

Here’s your puzzle:

Arrange the digits 1-9 such that the number formed by the first n digits in order is divisible by n.

That is, the first digit is divisible by 1, the number formed by the first two digits is divisible by 2, and so on. You should be able to figure out the place of one of the digits very quickly. The rest will take a bit more work.

Have fun!

Gauss facts

You know about Chuck Norris facts if you’ve been on the Internet long enough. If you’ve done NaNoWriMo and you run in certain circles, you probably also know about Mattkinsi facts and that you should #lickmattkinsi to improve everything in your life.

But you might not know about Gauss facts, and that’s a crying shame. Carl Freidrich Gauss was one of those genius mathematicians who left no area of math untouched during his lifetime, and mathematicians today know it. You probably know it too, even if your math education ends at high school. Did you ever learn how to take the sum of n consecutive numbers? Your teacher might have told you a story about a mathematician who figured it out, probably involving a brutish schoolteacher who wanted to keep the kids busy only to find out that one kid figured out the solution instantly. That mathematician was probably Gauss, but whether he really did that is debated. There’s also an entire Wikipedia list of things named after Gauss.

With all that in mind, there’s definitely a good reason for Gauss to have his facts. These facts are hilarious and nerdy, as you’d expect. I laughed out loud while reading them since I understood almost all of them. The gist is the same as Chuck Norris facts (or any facts of these nature, really). Some of them are funny because they’re not possible in our world, mathematical or not. Others are funny because Gauss is suddenly the supermathematician from above that some have thought of him as. Others… aw, why am I explaining the joke? Go read some of them. Laugh. Look up a couple of the ones you don’t get. Learn something.

This is what I didn’t get about open and closed sets.

Hitler learns topology and explains one of the things that always confused me when I learned about open and closed sets. I’m with Hitler on this one. Open and closed sets? Neither open nor closed? It’s enough to make your head hurt if you think about it too hard. Too bad Downfall parodies weren’t a big thing when I was taking real analysis, or my professor would have heard references to this video every day (in a joking way, of course).

Pi’s not wrong: on the Tau Manifesto

A friend asked me via Twitter what I thought of this io9 article about why pi was wrong. That’s right, pi: the circumference of a circle divided by its diameter. In Euclidean geometry that’s equal to approximately 3.1415926535897… and is irrational. The person io9 interviewed for the article is a proponent for a new constant called tau (after the Greek letter) that is the circumference divided by the radius. In other words, a constant for twice pi.

There’s a website promoting the correctness of tau as the correct value, claiming pi to be wrong just as the article does. The website isn’t not talking about the mathematical incorrectness of pi, but the site and io9 are using headline trickery to pull readers in.

The Tau Manifesto is disappointing. All it shows is that the choice of pi vs. tau is arbitrary at best, and we as a society chose pi over history because pi was discovered first. Besides, writing 2*pi is nicer than tau/2. This tau movement isn’t going to catch on. It’s not even on Wikipedia!

Abstract algebra: my idea of a fun Friday night

It’s no secret that I’m a huge geek, but spending the better part of an evening reading an abstract algebra book may seal the deal. I started by comparing the two books that I learned algebra from. The first book was a book written by my professor just for the class; the second was a dense and well-known algebra book (Dummit and Foote’s Abstract Algebra, if you’re really curious) and the book I used for my special study in algebra. I knew there would be a lot of differences just because of the natures of the two books. One book contained humor; the other did not. One book left more exercises for the reader than the other. One book has a downright awful definition for a quotient group. Well, once you read all the context behind the definition it’s a little less awful, but the definition can really be condensed into much easier terms. Really, Dummit and Foote, you don’t need to use fibers to explain a quotient group.

I’ve had a lot of fun this evening going through my algebra books, though. If I weren’t pressured by the need to sleep I’d be staying up longer to keep reading and working on exercises. Yes, this is my Friday night: doing abstract algebra for fun. That’s what the weekend is for, though.

Mathematical emotions

I made my first-ever B in calculus. This shocks some people, especially when they find out that I have a degree in math and when they find out just how much I like the subject. We can blame derivatives. Derivatives, it turned out, were surprisingly tricky creatures. Rates of change and applied problems involving derivatives that went beyond maxes and mins flew over my head, and for the first time I could say that I was really lost in a math class. This wasn’t supposed to happen; in my precalculus class the year before I actually understood some of the leftover scribblings from AP Calculus the hour before, and that involved derivatives, so what was going on now?

That lost feeling only turned into a mind-blowing feeling as I took more math. Maybe that mind-blowing feeling is a lost feeling in disguise as they share a lot of the same characteristics. I never did decide that. Whatever the case, you’d think feeling would go away at some point. It doesn’t. If anything, it gets worse after discovering exactly how much math is out there and how little I actually know. Every math class I took has introduced something so mind-blowing that I have to stop and let it sink it for a minute, sometimes several minutes, sometimes several days. One of my math professors actually made us stop to let theorems sink in, especially if they were particularly crazy and nonintuitive. (This happens a lot in real analysis when you can’t cling to calculus anymore.)

Yet for some reason, that feeling is exactly why I pick up a math book to read and work it for fun. Sometimes you can’t fight that feeling.

God’s number is 20 and an algebra lesson

If you’ve ever played with a Rubik’s cube, you may have experienced frustration or excitement while trying to solve it. I know I have. You may not know that the Rubik’s cube forms a group–that is, a set with a binary operation (like addition or multiplication) that is associative and has inverses and an identity. The set of integers {…-2, -1, 0, 1, 2, …} is a group under addition because parenthesis don’t matter when you add, you can add -n to n to get zero, and of course the identity is zero. Add zero and nothing happens.

Hooray, you know some abstract algebra! Now for some more fun stuff. It turns out that the set of every single move you can make on a Rubik’s cube is a group. By looking at this group, a group of researchers (including a mathematician) figured out that any cube configuration can be solved in 20 moves or fewer. How’d they do this? Abstract algebra. Specifically, cosets. Not to be confused with closets, like Goodsearch did.

So what’s a coset, you ask? Before I tell you that, you need to know what a subgroup is. Let’s say G is a group. It can be any group. A set of elements H that are all in G is a subgroup if it’s a group under the same operations as G. Now for cosets. We’re going to look at some element g in G and the subgroup H. Then a left coset is gH = {gh : h an element of H }. That’s a set. We just take that element g and all the elements of the subgroup H and apply the operation. It may be a group. It might not be. Right cosets are defined similarly. Just switch the order of g, H, and h.

That’s great, but why do we care about cosets? Solving every single Rubik’s cube is a huge problem, so they made their work easier by choosing a subgroup and working with the cosets. Instead of solving 43,252,003,274,489,856,000 Rubik’s cubes, they solved 2,217,093,120. They got that number down to 55,882,296 thanks to symmetry. Cosets: difficult at first, but they make your life a lot easier.

Now that’s math in action. Thanks, Dr. Koch! I know my entire class cursed the existence of cosets when we learned them (and they were hard), but now the world knows they’re useful. I promise I’ll stop cursing modules when I understand them.

Math busking

I’ve seen many street performers in my time, mainly musicians and dancers. Take, for example, this accordion player I saw while in France two years ago.

An accordion player

Now replace that accordion player with someone performing math tricks. Yes, math tricks. This was not my idea. It is, however, an excellent one, especially given the general public’s simultaneous fear and amazement at math and those who are good at it. I just need to brush up on some math tricks to live to up to the title of mathemagician.

Math busking may not be on the level of writing proofs for food, and it may not even provide money for food, but it would be fun, right?

NaNoWriMo’s history and the function of a search term

As I’ve mentioned before, researching the culture of NaNoWriMo and Script Frenzy for Wikiwrimo is hard. Really hard. Blame the lack of Wayback Machine in recent years, blame the lack of site and forum archives in 2002 and 2004 and 2008, blame whatever you want. When a tradition starts early, it’s hard to dig through all those search results, and even if you wanted to dig through all of them, the search engine will cut you off eventually for site performance reasons.

So I started investigating the features of Google, Yahoo, and Bing. I had a hunch from the start that Google would come out as the winner, and as much as I love using Goodsearch for my searching needs, there are some things that the Yahoo-powered Goodsearch just can’t find. (Sorry.)

The primary item I investigated was searching within a timeline. If I knew that, say, Mr. Ian Woon came along before 2003 (and he must have since he’s mentioned in the 2003 forums quite a bit, though I can’t find a post where someone realized that Mr. Ian Woon is a nifty NaNoWriMo anagram), then being able to search the pre-2003 Internet would be a wonderful thing. There’s just one problem.

None of the major search engines can do that. Bing can’t do it at all from what I can tell. Yahoo and Google’s time-sensitive results can search only recent results, not exclude them. Yes, excluding recent results would likely lead to a much wider pool of results, but not if you can put a cap to when the results were created. It would be a great way to create a function of that search term. How many terms were being added in this interval, and how quickly were they being added? How many disappeared because the pages or relevant search terms were removed? This function would definitely be increasing with large derivatives when the term is getting discussed a lot. As an example, last week the iPhone would have a large derivative. Actually, the derivative would still be large and probably increasing since people have the things in their hands and rumors are buzzing even more loudly about the iPhone going to Verizon.

But what’s the long-term behavior of a given search term? Or more interestingly, how could such a function be useful?

The decimal makes all the difference

I’ve done a lot of math in my time and endured a lot of comments from people who not only aren’t great at math but who, whether intentionally or not, make obvious mathematical mistakes. Today at a Wendy’s drive-through I saw a sign that made me wonder if anyone proofreads the signs before hanging them.

Four sauces for a penny? Sure!
(Click for bigger)

Four sauces for a penny? I’ll take that. Better yet, I’ll take just one and watch you make change for this penny. Making change for higher denominations of currency would be cruel and unusual punishment. At this price I should buy four hundred for a dollar and sell the sauces for twenty-five cents each. I’d make ninety-nine bucks. Multiplying my money ninety-nine fold isn’t bad at all. Maybe I should look into this.