Is it better to fill up a gas take once a week for $80, or put in a quarter tank 4 times a week for $20 each time? That question does have two reasonable answers, depending in no small part on whether you have access to $80 or just to $20 at a time, but what isn’t in doubt is that four quarter-fill-ups at $20 each isn’t actually cheaper overall than one full-fill-up at $80. Or at least, that shouldn’t be in doubt.

There’s an article about it here, but it doesn’t lessen the confusion at all.

(The reference to the question of which is closer, the West Cost or the Moon, is a reference to a discussion from a year ago.)

One of our alums (Thanks CJ!) sent a link this summer about how MATH was used in the design of the Alignment Optical Telescope in the Apollo Lunar Modules. I mean, yes, of course it was, but in particular an Archimedes Spiral — a spiral where the distance is increasing steadily, such as r = θ in polar coordinates — was used instead of a heavier piece of equipment. This is, I think, the first time I’ve seen a modern application of the Archimedes spiral.

The video is available below, and there is more info at NASA.

Our 9-year old recently introduced us to Kid Snippets, a series of short videos feature adults acting out children’s conversations. (The children are given a scenario and “act” it out vocally; the adults then do the actual acting to the children’s dialogue.) These are now sweeping through the department, since watching funny videos is way more fun than studying for math tests. It’s also possible that I’ve watched more than a few of these for a break myself, even when no kids were around.

As you’ll know if you’ve been on Google in the past twelve or so hours, today is Jim Henson’s 75th birthday, and in his honor we share with you a rousing rendition of the How Many Game, hosted by the wonderful Guy Smiley!

Follow up question: should they have won with the sheep?

According to the Muppet Wiki, Guy Smiley’s enthusiasm was rough on Jim Henson’s voicebox, so his portions were rerecorded so that it could be played over and over without Mr. Henson having to keep repeating his dialogue. The Muppet Wiki also says that there were two two-headed monsters: the one in the clip above was designed by Jim Henson, but a second, who was on Fanfare and the Mike Douglas Show rather than Sesame Street, was performed by Jim Henson.

I was thinking of all the things I meant to post in 2010, that I diligently saved, but that became less timely as time went on. D’oh! Crucial mistake, since it turned out the alternative was…blankness.

So I thought it might be fun, at least in a New Year Cleaning sort of way, to post them. And I thought I should call it Ten Things I Meant to Post, But Didn’t Get Around to. Except I’m not sure that there are 10, so this might be a Hitchhiker Trilogy kind of Ten.

Thing #1 was really Saturday’s post: The fact that the subtraction principle in Roman Numerals evolved gradually and (really, like almost everything I think) with some back and forth.

And Thing #2 is this Mandelbrot video, which was passed along by a colleague (Thanks Betsey!) in October, only a few days after Benoit Mandelbrot’s death. So in honor of the man and all that he did, here’s a tribute, prepared several years ago. I hope he saw it and enjoyed it.

From Youtube: “A music video for Jonathan Coulton’s song Mandelbrot Set by Pisut Wisessing made in Film 324: Cornell Summer Animation Workshop, taught by animator Lynn Tomlinson every summer for Cornell’s summer session, in the department of Theatre, Film & Dance.”

This Brian Regan video isn’t new, but I saw it recently for the first time and found it hilarious (Thanks for the link Michael!). And timely, given the holiday season.

Math Teachers at Play #5 is up at Let’s Play Math! It’s organized by topic, with pictures and neat quotations thrown in for good measure. And as usual, it is full of interesting posts!

Speaking of Carnivals, apparently the Carnival of Mathematics was just hibernating and it will reappear next week. At least, that’s the rumor, where “rumor” means I read it on jd2718.

And speaking of reading things, after reading Keith Raskin’s comment I headed over to Natural Blogarithms to look for Pythagorean Triple stuff, and I was immediately distracted by the following puzzle:

The four numbers A, B, A+B and A-B are all prime. The sum of these four numbers is

A) Even
B) Divisible by 3
C) Divisible by 5
D) Divisible by 7
E) Prime

(from the 2002 AMC 10/12B #15).

I quickly convinced myself of one answer, then decided there was a misprint and talked to Batman about it at work, and then we he realized that exactly one of the possible answers was correct. At that point the problem seemed a lot more interesting.

If you need something a little more lighthearted this Friday afternoon, here’s the 1-20 roll call from Sesame Street.

I love the show Top Gear. In particular, I am a big fan of the “challenges” they have. Recently (yesterday afternoon, when I should have been grading), there was an episode on BBC America from 2006 in which host Richard Hammond was at London’s ExCeL Centre for the British Motor Show, and before the exhibits were set up, he had The Stig (the show’s “tame racing driver”) test two vehicles to see how fast they could go in the 385-metre hall. The first was a Chevy Lacetti, their “Star in a Reasonably-Priced Car” car, which reached 70 mph. The second was a Toyota F1 car (the TF105, I think), which reached…

only 81 mph?

This led me to wonder what the top speed of a car could be on a 385-metre stretch. Let’s find out.

For simplicity, I will assume constant acceleration a (I said simplicity, not accuracy), and constant deceleration from braking (again, not particularly realistic). Let be the top speed. Then the distance covered in accelerating to top speed is

and the braking distance is

where μ is the coefficient of friction between the tires and the floor, and g is gravity. We then have the following equation:

and thus

At this point, I plead ignorance. I tried (not very hard) to find a reasonable coefficient of friction for racing tires, and to find a 0-60 time for a Formula 1 car (the McLaren F1 does it in 3.9 seconds). In the end, I made a spreadsheet for μ between 0.4 and 0.7, and a between 6 and 7 m/s^{2}. Given an ideal setup—starting at one end of the hall and stopping perfectly at the other end—the top speed is somewhere between 95.6 mph (μ=0.4, a=6) and 115.5 mph (μ=0.7, a=7).

What does this mean? I don’t know, but it makes The Stig’s 81 mph sound pretty good, given the initial burnout, the nonconstant acceleration and braking, driver reaction time, and an interest in personal safety (cf. Hammond’s “Formula 1 car-shaped hole” comment).

The 2008 William Lowell Putnam Mathematical Competition officially took place this weekend! Yup, six hours of grueling math problems. We had a record number of students take it this year: 21! [That’s just 21, not 21 factorial. That would be an impressive number.]

Here’s one of the videos that our students watched while they were gathered around my dining room table Friday night eating dinner (because we totally bribe our students with food: dinner the night before at my house, bagels for breakfast, and lunch at the local pub in between the two three-hour sessions). It’s “I Will Derive” and I know it’s made its way around the internet, but I still think it’s fabulous:

And here’s the problem (A2) that caused the most discussion over lunch:

Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?

Here’s my favorite problem (A1) because I was able to solve it right away and do you know how often that happens? Not very.

Let f : R^{2} -> R be a function such that f(x,y)+f(y,z)+f(z,x)=0 for all real numbers x,y, and z. Prove that there exists a function g : R->R such that f(x,y) = g(x)-g(y) for all real numbers x and y.

You can see the rest of the problems here, and pretty soon you should be able to find the answers here. Edited Monday 12/8 to add: yes, the answers are posted!

No, of course Godzilla didn’t really use a calculator on an exam. He’s a stickler for following the rules.

Classes are finishing up, projects are graded, and finals start next week. To celebrate the end of the semester, one of our majors brought in a mix of math songs. Only most of them weren’t initially about math, they just sounded that way after her editing.

In honor of this week’s busyness and for math songs everywhere, here’s a video of one of the songs that Jill used. It’s “U + Me = Us (Calculus)” from 2ge+her:

[And if you’re just feeling depressed about the mathematical subject, you can find the parody lyrics “Don’t Know My (Calculus)” here. Although a parody of a satire is kind of a funny phenomenon.]

They Might Be Giants has been doing videos and podcasts for kids. One of their recent releases (from earlier this year) is a video all about one of our favorite polygons: the nonagon! Several other polygons make guest appearances as well.

You can see the video below. The first minute or so is introduction, followed by the nonagon song, one about the letter O, and then a good-bye (5:32 total).

This news story made the rounds many months ago, but I didn’t read of it until I was paring down my Inbox this week (1388 messages. It was getting a little overwhelming) and found it in a news digest. In late March, the Japan Aerospace Exploration Agency accepted a proposal of a project led by Shinji Suzuki to make origami spacecrafts and launch them from the International Space Station. How cool is that?

One worry is that they would burn up because friction from entering the atmosphere tends to make things rather hot, but it’s possible/likely that they won’t both because of their shape and because they will be traveling so slowly through the atmosphere (plus the paper, made from sugar canes, is heat resistant). When I first read this I envisioned Giant Origami Planes, but they’re actually small: the shuttles will only be 8 inches by 4 inches after folding, and weigh just over an ounce.

Another worry is that there is no way of controlling where they land, or even how to track them. This is a much bigger deal, and perhaps one reason they’re not going with the Giant Origami I’d envisioned (can you imagine if one of those swooped down onto your lawn?). But don’t go looking too soon: the grant they received is for 3 years of feasibility studies.

While you’re waiting, you can learn how to make an Origami Rocket.

I read this story in many places, but got most of the info for here from Discovery News.

Now is the time on Sprockets 360 when we plagiarize share some of the fine things going on around the Internet.

Are you looking for some fun math problems to do? You can try out the bi-weekly Monday Math Madness. MMM #10 at Blinkdagger is a probability problem with a twist: if you know the probability of a group winning the lottery over 25 years, what is the probability of someone in the group winning the lottery at least once in a 5-year period? Solutions are due by Monday night/Tuesday morning at midnight. MMM #11 will appear the following Monday on Wild About Math.

There’s also a new regular math problem being posted. Walking Randomly has started posting an Integral of the Week. Integral #1 is . (Incidentally, if you google “Walking Randomly” then the program suggests that you mean “Working Randomly”, which feels rather like my summer.)

Finally, why was 6 afraid of 7? Because 7 8 9. (Thank you, thank you, I’ll be here all week.) The Barenaked Ladies said it much better on this YouTube video, which I actually found here at Wiskundermeisjes, a site created by Ionica Smeets and Jeanine Daems. (This isn’t the first time I’ve stolen borrowed from them, either — I originally found the idea for Godzilla’s Sierpinski Cookies from Evil Mad Scientist Laboratories on their site. Indeed, their site really makes me wish I could speak Dutch.)