From a recent Nature Valley ad in the London *Metro* newspaper:

Perhaps the second bar is twice as delicious as the first.

*Via Language Log. Photo from Spiderham.*

12 tables, 24 chairs, and plenty of chalk

From a recent Nature Valley ad in the London *Metro* newspaper:

Perhaps the second bar is twice as delicious as the first.

*Via Language Log. Photo from Spiderham.*

The folk from *Glee* paid unintended homage to the title of this week’s episode (“A Night of Neglect”) by showing Mr. Schuester forgetting his basic math skills. Actually that’s not entirely true; he does math in his head correctly as he explains his plan to use salt-water taffy to earn money to go to Nationals in New York:

When I was a student here we paid for our entire trip to Nationals selling this…. So, to make $5000 at 25 cents apiece, we need to sell 20,000 pieces of taffy.

So far, so good. But wait, what’s that equation in the background?

Poor Will…he didn’t even notice that the equation wasn’t quite right (and neither did the four members of the Academic Decathalon team). But don’t worry, we understand how busy this time of year is, what with all the projects and end of the year assignments coming due. So shall we just fix that up for you?

There, all better. Now you can go concentrate on raising that money. Just be sure to have someone else in charge of the ledger.

For no reason that I can think of, I decided to see how much Wolfram Alpha knew about probability, so I typed “probability of a full house” into the search box and got the following:

I thought that was pretty cool, especially since it includes the derivations, so I asked a few more questions, such as “probability of at least 2 red cards in a 5 card hand“:

Odd that it will count the numerator but not the (easier) denominator . At this point, I thought I’d try a standard probability question (balls in an urn) that might be harder to parse because of the additional statements: “probability of drawing a blue ball from an urn contaiing 5 blue balls and 7 red balls“. However, I missed the ‘n’ key when typing “containing” and got the following:

So, yeah, OK, Wolfram Alpha doesn’t provide “adult” content (why the quotes?), and I’m pretty sure I know what it’s reading as “adult”, but c’mon. Note that fixing the typo doesn’t alleviate the problem, but it does cause Alpha to hiccup and request more computing time. With variations on the wording, I’ve also had it return a picture of a blue ball along with the HTML code to generate it. Nice.

Godzilla is a well-known mind-reader, and in honor of final exams, which are coming up sooner than seems possible, he’d like to demonstrate his powers. Even over the internet, because his powers are MIGHTY. Like him.

Start with 3-digit number that is not a palindrome (so 360 is OK, but 363 is not). Then reverse the digits, and subtract the smaller number from the larger. You get a **NEW AND IMPROVED** number. So if you do start with 360, your **NEW AND IMPROVED** number will be 297 (which is 360 minus 063).

Treating your **NEW AND IMPROVED** number as a 3-digit number, reverse the digits. This means that if your **NEW AND IMPROVED** number appeared to only have two digits, or even one, then you have to tack on one or two leading zeros that you include in the reversal.

Now add your **NEW AND IMPROVED** number to its reverse. Godzilla will now tell you the sum, even over all the miles and electrons that separate you from this friendly beast…..

The Carnival of Mathematics is still going strong. This round – #76 – is hosted over at Walking Randomly and has, as usual, something for everyone, including a post from one my favorites: Language Log. (Yes, they use math there. Fairly often, in fact.) Go check it out, and while you’re at it, contact Mike if you’d like to host one.

We luv us some failblog (regular or decaf), particularly on a Friday. Lately they’ve had a bunch of math fails, where “lately” means “since the last time we posted from there” and “bunch” is closer in number to “I bought a bunch of bananas” than “I have a bunch of papers to grade”. So without further ado, here are some favorites.

There’s trouble with dates:

and trouble with money:

and lots of trouble with percents:

Apparently, as Barbie once said, math is hard.

The number 1729 has a right to be proud : it initially had only a small role on a taxicab in England but its super-power of being the sum of two positive cubes in not one but **two **ways (1^{3}+12^{3} and 9^{3}+10^{3}) led to a big break in a Feature Story starring GH Hardy and Srinivasa Ramanujan, with follow-up appearances for years to come on the likes of *Futurama *and *Proof*. So, you know, yay 1729.

But lest this Hardy-Ramanjuan number get too boastful, it’s not the only sequin at the Oscars. Its neighbor, that unassuming 1728, turns out to be an interesting character in its own right.

The origin of this is in the dozen. Although ten is a pretty natural base to use, in the sense that a lot of cultures break numbers up by tens in some form, it’s not the only possibility. We have not only a special word for 12 (dozen), but a special word for 12^{2} (gross), which suggests that our language carries hints of a Base 12 system. And that leads to the question: is there a special name for 12^{3}?

There is! The official name is a Great Gross. And while dozen and gross show up in egg cartons, it’s in measurement that the great gross really shines: there are a **dozen **inches in a foot, a **gross **square inches in a square foot, and a **great gross** cubic inches in a cubic foot.

But while the great gross is helping out with set design, there’s a rumor (which we’re apparently happy to help spread) that 1728 actually has a stage name. That’s because there’s a theorem about L-functions of elliptic curves called the Gross-Zagier Theorem, named after Benedict Gross and Don Zagier. So the natural extension of a gross is…a Zagier! Or at least that’s the name that 1728 goes by on the cocktail circuit according to Wikipedia, our local gossip rag. Which makes us wonder where this down-to-earth yet whimsical number will show up next.

*In an amusing turn of events, it turns out that Gross and Zagier won the Frank Nelson Cole prize in Number Theory in 1987 from the American Mathematical Society for their paper “Heegner points and derivatives of L-series” which contained the above theorem. The other winner that year for a different paper was Dorian M. Goldfeld who, the following year, published a paper with M. Anshel entitled “Applications of the Hardy-Ramanujan partition theory to linear diophantine problems,” bringing it all back full-circles to the people who made 1729 famous. It’s like one giant family reunion. *

See Mini-G look at this fine piece of stripey art:

Isn’t that interesting, full of nuance? NO — it looks totally boring. But Mini-G is actually looking at it at an angle, which turns out to be a completely different story.

No more simple stripes! And while it’s no Mona Lisa*, it’s pretty cool to see the shapes appear just as you start to walk away in search of something less vertical to look at. Even better, it’s simple knitting. REALLY simple knitting, just knits and purls, where using stockinette stitch makes a color fade into the background when viewed from the side, and using garter stitch makes a color stand out. There’s a great explanation here, where “great”=“uses legos”.

This comes from Woolly Thoughts (“In pursuit of Crafty Mathematics”) and their newish illusion site. It’s a free pattern — Woo hoo! — and not that I’m suggesting that you knit during meetings or anything, but if you DID knit during meetings this particular pattern is simple enough that you can do it without being distracted from the Important Conversations and Presentations, and then you can feel good at the end of two hours that you made quite a bit of progress on your knitting whether the meeting led to a resolution or not, plus you get to point out that you’re really doing mathematics if anyone asks what you’re knitting. Win-win!

* though there is a pattern for that.

Happy 3/3 everyone!

I just graded a bunch of proofs that √3 is irrational. The proofs had a lot of holes in them. This didn’t surprise me too much, in large part because the students weren’t math majors; rather, it was for a liberal arts math class taken largely as a gen ed requirement, and the whole proof by contradiction thing is really pretty scary and abstract for most people the first time around under the best of circumstances.

But actually, even when I’ve assigned this to math majors, they struggle. They can have the proof that √2 is irrational right in front of them, be instructed that instead of even numbers they want to look at multiples of 3, and despite my Find and Replace instructions, they still don’t understand what to do. The most common mistake is to replace “even” with “odd”.

In some ways this doesn’t surprise me, but in some ways it does. Why is it such a conceptual leap to go from 2 to 3? It’s a HUGE leap for many people. And so I was pondering this while grading, and Batman suggested it might be because we have a special word for “divisible by 2″ but don’t for “divisible by 3″. So you get, what, 10 years of reinforcement that there is just this one special way to divide the integers, and it doesn’t generalize.

What we need is a new word for these numbers.

And fortunately we have one: threeven. So 0, ±3, ±6, ±9, … are all threeven, and the rest are…umm, not. (Maybe we need two new words). This word isn’t mine or even Batman’s; it actually was suggested by one of his students in response to this exact same problem.

As a bonus, it generalizes: there’s fourven, fiven, sixen, seven-en (sev-en? )…as far as you want. Which, admittedly, might not be very far but it still makes for a smoother sounding proof.

Happy threeven day!

It was a crazy January (with an inadvertently extended sabbatical, thanks to the ice storm down south at the time of the Joint Math Meetings!) and now February is coming in like, well, February. Rochester is in the middle of a winter storm, and though it doesn’t quite seem to be the **WINTER STORM **that the forecasters predicted, there’s still a respectable amount of snow and ice. Leading to conversations like these:

**Last night, looking at the closings online:**

**Person 1 : ** Wow, they’ve even closed all the Curves gyms in the area, except for one that’s on a delay. They list them all separately — that’s weird.

**Person 2: **Isn’t that a complicated what of describing it? If they just made one announcement it would be Simple Closed Curves.

*************************

**And then this morning…**

**Person 1: ** Can you take the kids to school tomorrow? I’m giving an exam and want to allow plenty of time to drive slowly if the roads are still icy.

**Person 2: ** Is that a Margin of Terror?

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.”*

Happy New Year! And since the New Year is all about numbers (especially if you have come to look forward to Denise’s annual January 1 post on Let’s Play Math: form all the integers from 1 to 100 using (exactly) the digits 2, 0, 1, 1 and common mathematical symbols), here’s a picture of a number that I meant to post in ~~October~~ ~~November~~ December.

Recognize this number? Even though it’s not written as LIV? This is from the 54th entryway to the Colosseum in Rome, which was built almost 2000 years ago when Roman numerals didn’t always use the subtraction property that we’re taught, where 4 is written as IV instead of IIII.

I found that to be interesting in and of itself, since I’d heard that the subtraction was a later addition but never witnessed it. But what’s weird? It wasn’t a sudden change. Here’s the forty-fifth gate:

The subtraction principle was used with 40, just not with 4. Which leads to a natural question: what about gate 44?

I’m bummed that we didn’t get a better picture of this, but you can kind of see all four Is after the L. Apparently, according to our usual font of knowledge, the reluctance to use IV is because that was the standard abbreviation for Jupiter’s name in Rome (IVPPTER), and this mixture of sometimes using four symbols in a row continued for more than a thousand years: in the 1390 English cookbook ** The Forme of Cury** (here on Project Gutenberg) the author still uses IIII [as in the Table of Contents, where Section IIII is

And even 100 years ago [last year, in 1910], the Admiralty Arch in London uses MDCCCCX instead of MCMX in the inscription

ANNO : DECIMO : EDWARDI : SEPTIMI : REGIS :

: VICTORIÆ : REGINÆ : CIVES : GRATISSIMI : MDCCCCX :

(*In the tenth year of King Edward VII, to Queen Victoria, from most grateful citizens, 1910).*

So what does all this mean? Nothing much, except that Roman Numeral Rules were maybe not quite as hard and fast as I once believed.