Lands’ End catalog

Under the category “Weird but True”: In today’s junk mail, I learned that Lands’ End has an entire line of clothing $f(x)$ , apparently targeted toward mathematicians.

Cheating

“[Students] cheat… because they’ve imbibed the message — from parents, from peers, from schools — that looking successful is more important than being honest. They cheat because they have been taught, however unwittingly, that it is worth it.”

From How We Teach Students to Cheat.

Epsilon

Years ago, when I taught calculus, I’d usually include the following extra credit question on the first exam: “In the small box, write a good value for $\varepsilon$ . A valid answer gets 4 points; the smallest answer in the class will get 5 points.” It was basically free extra credit… any positive number would work, but it was a (hopefully) fun way for students to be a little competitive in coming up with small positive numbers, which is the intuitive meaning of $\varepsilon$ in mathematics. (I still remember when my high school math teacher was giving me directions to a restaurant, concluding “You’ll know you’re within $\varepsilon$ of the restaurant when you see the signs for Such-and-Such Mall.”)

Most students volunteered something like $0.0000001$ or $10^{-9999999999999999}$ . Except for one particularly gutsy student who wrote, “The probability that Dr. Q gets a date on Friday night.” For sheer nerve, he got the 5 points that year.

Also getting 5 points that year was the best answer of the class: “Let $x$ be the smallest answer that anyone else wrote. Then $\varepsilon = x/2$ .” That was especially clever from a calculus student, as that’s the essence of a fairly common technique when writing proofs in real analysis.

A good clean joke

Two algebra teachers are on a plane. Shortly after reaching cruising altitude, one of the engines conks out. However, the flight attendant announces that the plane has three other engines. However, instead of needing 3 hours to fly to their destination on 4 engines, it will now take 4 hours to fly on 3 engines.

A little while later, another engine goes out. Never fear, says the flight attendant: the plane can fly on two engines. Unfortunately, the length of the flight has now increased to 6 hours.

Later still, a third engine fails. Not to worry, says the flight attendant. The plane can fly on only one engine. But the flight will now last 12 hours.

So one algebra teacher says to the other, “I really hope that last engine doesn’t go out, or else we’ll be up here forever!”

Interdisciplinary studies (part 2)

A provost complains to the physics faculty about how much money it costs for labs, lasers, technical staff, and other associated costs of doing their work. “Why can’t you be more like the Math Department?” asks the provost. “All they need is money for pencils, paper, and a wastebasket. Or better still, you could be like the Philosophy Department. All they need is money for pencils and paper.”

The Hitchhiker’s Guide to the Galaxy

In 1979, Douglas Adams envisioned the iPad at a time when the Apple II was the state of the art. From now on, I’ll always think someone’s reading the Hitchhiker’s Guide to the Galaxy when they’re fiddling with their tablet (especially if they also have a towel nearby).

“[Ford Prefect] also had a device that looked rather like a largish electronic calculator. This had about a hundred tiny flat press buttons and a screen about four inches square on which any one of a million pages could be summoned at a moment’s notice. It looked insanely complicated, and this was one of the reasons why the snug plastic cover it fitted into had the words DON’T PANIC printed on it in large friendly letters. The other reason was that this device was in fact that most remarkable of all books ever to come out of the great publishing corporations of Ursa Minor — The Hitchhiker’s Guide to the Galaxy. The reason why it was published in the form of a micro sub meson electronic component is that if it were published in normal book form, an interstellar hitchhiker would require several inconveniently large buildings to carry it around in.”

From Chapter 3 of “The Hitchhiker’s Guide to the Galaxy.”

World’s largest model solar system

I think this is really cool: the Department of Astronomy at Stockholm University (Sweden) have devised the world’s largest model solar system, at a scale of $1:20,000,000$ . The inner planets are in the vicinity of Stockholm, while the outer planets, dwarf planets, and distant objects are spread throughout the country.

Triangles of pennies

Useless fact of the day: Make a triangle of arbitrary size with pennies, like in the picture below. What is the least number of pennies that have to be moved to make an upside-down triangle?

Turns out the answer is the number of pennies divided by 3, ignoring the remainder. So for the 10-penny triangle above, the answer is 3 moves (since $10/3 = 3 + 1/3$ ). A good formal write-up for why this works, with specific discussion about implementing this in a middle-school classroom, can be found here: http://132.68.98.62/Courses/Algebra_206/Algebra%20-%202005/Articles/2-MTMS-Inquiry%20and%20triangle%20array-2004-9-6.pdf

Image credit: http://www.coolmath4kids.com/math_puzzles/p4-pennytriangle.html

Probabilities in Monopoly

Somebody had to figure this out: Probabilities in Monopoly, including long term probabilities for each square, expected income per roll for each square, payback times for purchases, and mortgage payback and loss times.

Bug in TI-83 Plus?

On my TI-83 plus emulator, I input the following in the statistical command 2-SampTInt:

x1 = 51.71
s1 = 0.79
n1 = 10
x2 = 136.14
s2 = 3.59
n2 = 10
C-Level: .95
Pooled: No

The calculator returns 9.896 degrees of freedom; it should be more like 12 degrees of freedom according to Welch’s formula (which is also implemented with T.TEST in Microsoft Excel).

I’m assuming the TI has some non-standard way of computing the number of degrees of freedom, but I haven’t the faintest idea what it could be. What’s odd is that the number of degrees of freedom appears to be computed with Welch’s formula when using the parallel command 2-SampTTest.