Archive for October, 2008

Acting Tips

October 16, 2008

I was recently cleaning watching Friends, “The One with the Race Car Bed” from Season 3, and lo and behold there was some math! in this episode Joey got a job teaching “Acting for Soap Operas”, and in one scene he shares some tricks of the trade: how to cry, how to convey evilness, and how to respond to bad news. The math is in that last one: “Let’s say I’ve just gotten bad news. Well all I do there is try and divide 232 by 13.”

How well does that work?  [The math part starts about 40 seconds into the scene.]

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One, Two, Three, Four, Six(?)

October 15, 2008

How many necklaces can you make if you have two different color beads (like purple and gold) at your disposal?

If you put 0 beads on your necklace, there’s only one way to do so and it’s mighty boring.

Suppose you put 1 bead on your necklace. It might be purple or it might be gold, giving you two possible necklaces.

If you want to put 2 beads on your necklace, you can do so in three ways. And if you want to use 3 beads, there are four possibilities.

What about with 4 beads? It looks like the answer should be five, right? Right? You know you want to say that. ALAS, there are actually six different ways to do it.

Once the pattern of consecutive integers is broken, it stays broken. There are eight ways to make a necklace out of 5 beads, and either thirteen or fourteen ways to make it out of 6 beads: thirteen if you allow reflexive symmetry — that is, the necklace is “free” and you can flip it over in addition to rotating it — and fourteen if the necklace is “fixed” so you can rotate it but can’t flip it. And then the numbers start to increase a little faster, into the thousands if you have fifteen beads (and, umm, enough time to make a whole bunch of necklaces).

Lego Math

October 14, 2008

I ran across an interesting article about how many different ways there were to stack 2×4 Lego bricks into towers. “Aha!” I thought, “This would make a great post! I could include lots of pictures and everything!”.

I thought that because this is what the floor of our living room looks like:

It’s like a giant I Spy puzzle. I figured I wouldn’t have any trouble finding bricks, but it turns out that among the75 million Lego pieces adorning our house, there were precisely five of the right size.

Shortly after I took this, one of them mysteriously disappeared.

In thinking about the number of ways to build a tower from six 2×4 blocks, it’s easiest to first analyze how many ways there are to build towers out of just two of the blocks. Here are two examples:

Here are two more ways to stack them:

But wait, there’s more! The orange top piece could cover more of the bottom piece, or go on the other side, or even be perpendicular to the bottom piece. If you think of the bottom piece as being fixed, there are 46 different ways that the top piece could be aligned (21 ways with the top piece being “parallel” to the bottom, and 25 ways with the top piece “perpendicular”). Except that the 46 ways aren’t really different: all but two of them could be morphed into a different one by rotating the tower 180°.

So there are really only 24 different ways to start. But then you can build higher and higher towers, and do you know how many different towers can be made from 6 legos? Millions and millions and millions. More precisely, there are 102,981,504 if you want them to be 6-legos high, and 915,103,765 if they can be shorter, with several legos on the same level.

That’s a lot of towers. The paper I read that explained this (including citations to an earlier quote by Lego that was off by 4 towers) is “A Lego Counting Problem” by Søren Eilers, Mikkel Abrahamsen, and Bergfinnur Durhuus. (I also did a search to see what “Lego” and “Group Theory” might net, and came across the LEGO Algebra Group which made me really happy until I discovered that it had nothing to do with the toys.)

91115/11116 !

October 13, 2008

One of my favorite encryption techniques isn’t really an encryption at all: it fails in the basic sense that if you know the method of encryption, you can easily decrypt the message. But it does reduce every message to a fraction, and I like that. I can envision entire conversations consisting of phrases like “73/9”. Here’s how it works. You start by replacing each letter with a number: say, A=1, B=2,…,Z=26, and perhaps (space)=27, plus whatever punctuation that you want. Then you write your message as a continued fraction (that is, a nested fraction where each numerator is 1). For example, to write “HELLO” you’d let H=8, E=5, L=12, L=12, and O=15 and write:

8+\frac{1}{5+\frac{1}{12+\frac{1}{12+\frac{1}{15}}}}.

Then you’d simplify it before sending it. For example, the fraction above simplifies to:

8+\frac{1}{5+\frac{1}{12+\frac{1}{\frac{181}{15}}}} and then 8+\frac{1}{5+\frac{1}{12+\frac{15}{181}}}

all the way to \frac{91115}{11116}.

To decrypt, use division to write the fraction as a continued fraction (keeping in mind that words that end in “A” will turn out to be ambiguous unless you develop a way around it). Want to have some fun? Try decrypting \frac{2358}{169} (or the longer \frac{536341314626}{38440072317}).

I don’t know if the guy in the painting by Lesser Ury is really a spy, but I kind of think he looks like one.

Fun with the Inclusive OR

October 12, 2008

I’m a little disappointed in header “SQUARE OR DIAMOND?” I think they could have added “OR RECTANGLE” for additional options.


more fail, owned and pwned pics and videos
[Note: Not always safe for work, although The Fail Blog now offers a G-rated option].

The 41st Carnival of Mathematics

October 10, 2008

Welcome, one and all, to the 41st Carnival of Mathematics! Step right up and marvel at the amazing, the astounding, the prime number 41! It is, of course, the 41st natural number, and is equal to the product of itself and 1!

Perhaps more interesting is the fact that 41 is a twin, supersingular, Germain, Eisenstein, Proth, and Newman-Shanks-Williams prime (which has to be some kind of record). It is also a centered square number.

We begin with a very accessible discussion of infinite sets, and the difference between countable and uncountable, by Carnival XL host Barry Leiba, in Countable and uncountable sets, part 1, at Starting at Empty Pages. (There’s also a part 2.)

How cool would it be to have a Mathematician for President? Denise tells us all about James Garfield and his proof of the Pythagorean Theorem over at Let’s Play Math.

The Central Limit Theorem is a well-known result in statistics, but in order to use it, one must assume a sufficiently large number of samples. John Cook, from The Endeavor, wants to know about quantifying the error in the central limit theorem, and how close an approximation we can really get. He also compares three methods of computing standard deviation – turns out they’re not all equally good.

Motivated by a confusing chapter in a book on game theory, Rod Carvalho decided to analyze the “Tragedy of the Commons” in Bandwidth-Sharing Games, over at Reasonable Deviations. In his words:

Suppose that n players would like to have part of a shared resource: each player wants to send information along a shared channel of known maximum capacity. I analyze this problem using a game-theoretic approach.

Barry Wright III, at 3 Style Life, goes in depth on elections in Facts about the Copeland Score (with PDF continuation), a way to generalize the Condorcet winner to elections that don’t have a Condorcet winner.

Music and math often go hand in hand, and David Stutz gives us a lot to think about in more musical Turing machines, at the synthesist. Inspired by Neal Stephenson’s new book Anathem, Stutz led a choral performance of a 3-state, 2-symbol Turing machine that performed binary addition. From there, things really take off!

Mike Croucher, 2-time Carnival host and owner of Walking Randomly, and a Mathematica power user, shows us how to Simulate Harmonographs. (A harmonograph is the result of letting a pendulum with a pencil attached to it swing over a piece of paper, like a Spirograph, but cooler.) Then he asks the question: NAG – The Ultimate MATLAB Toolbox? Read to find out if the Numerical Algorithm Group has hit a home run.

Looking for help with your math homework? Visit VideoJug’s math repository for a collection of advice videos on math.

Technology can be a great tool in teaching, and Maria, from Teaching College Math Technology Blog, demonstrates How Tablets Enhance the Math. Derivatives have never looked better.

Last, but certainly not least, how would you get a computer to use ordinal numbers? Mark Dominus tackles the question in Representing ordinal numbers in the computer and elsewhere, over at The Universe of Discourse.

An accidentally missed submission (sorry Jason!) comes from Jason Dyer. His entry, Visual Clarity in the Naming of Variables, examines how using similar letters for different variables can be confusing to students, and he offers several alternatives.

That’s it for #41, but tune in on October 24 for the next edition, hosted by The Endeavor.

Mail Goggles: Using math to save you from yourself

October 9, 2008

Earlier this week, one of the Official Gmail Engineers (Jon Perlow) announced the launch of a new lab: Mail Goggles. This is an optional feature that requires you to solve math problems before you can send out email.

Here’s how it works: you preset certain days and times (e.g. weekend nights, the default) when you think you might be prone to sending out emails that you’ll later regret. You also select the difficulty of the math problems, on a scale of 1 to 5. If you try to send an email during a designated time, a window pops up with five math problems, which you have to solve in under a minute in order for your message to actually be sent.

Here’s a sample of the problem in Level 1, with the big red numbers counting down from 60 in order to let you know just how much (or little) time you have left to prove you’re thinking clearly:

Too easy to dissuade you when you really need stopping? Try Level 3.

And finally, here’s an example from Level 5 (which doesn’t seem so different from Level 3, and occasionally appears to be easier):

If you take too long to answer, you get the note, “Oops, looks like your reflexes are a little slow. Try again.” with five new problems. If you get any answers wrong, the program suggests water and bed. And if you do still manage to compose and deliver some news that you wish you hadn’t? At least you can say you really are too smart for your own good.

I initially learned about this from ars technica.

The Smarts of Slime Mold

October 8, 2008

When looking up the Ig Nobel prizes for Monday’s post, I was intrigued by the mention that slime molds can solve puzzles. As near as I can tell, here’s what was going on:

The Cast of Characters: Your everyday Physarum polycephalum, otherwise known as slime mold. Normally this mold likes to hand out in dark places, like under logs, but according to sites like this it can be useful to scientists because the cells are huge and easy to see. (“Cell” might be the wrong word; apparently lots of cells fuse together into one giant creeping amoeba-like thing.)

The Temptation: Slime mold gets hungry and wants to eat. Normally it eats things like rotten plants, but for illustrative purposes we’ll use a carrot and a cupcake.

The Problem: What if the slime mold is in a maze, and there are two pieces of food? If it were just one, then the mold would check out the entire maze and head over to the nutrients. With two different pieces of food, however, it runs into a problem: it can’t be in two places at once, and it’s not patient enough to eat one item and then move to the other.

The solution: It stretches between the two nutrients using the shortest path. It was for this recognition − that the slime mold was finding the shortest path − that Toshiyuki Nakagaki, Hiroyasu Yamadam, and Ágota Tóth won the Ig Nobel prize in Cognitive Science (for their September 2000 article “Intelligence: Maze-Solving by an Amoeboid Organism,” in Nature magazine). The article requires a subscription, but even without one you can look at a series of photos here. [From the photos, it looks to me like the slime is actually checking out a few different paths, but among them is the shortest.]

The follow-up: Almost three years later, the first two authors (Toshiyuki Nakagaki and Hiroyasu Yamadac) and a third (Masahiko Hara) pointed out that when the mold was trying to get to food in several different locations, it did the same trick of trying to be at each food source at once. Furthermore, “These findings indicate that the plasmodium can achieve a better solution to the problem of network configuration than is provided by the shortest connection of Steiner’s minimum tree.” (from the abstract to “Smart network solutions in an amoeboid organism,” which appeared in the magazine Biophysical Chemistry in January 2004).

But wait, there’s more! Klaus-Peter Zaune created a six-legged robot that was powered by this same kind of mold. Slime mold generally avoids the light; circuits underneath the slime could sense the stuff shying away, and that caused the robot to move away as well. The robot made its appearance at the Second International Workshop on Biologically Inspired Approaches to Advanced Information Technology in January 2006, according to New Scientist.

Finally, a January 2008 article back in Nature (“Cellular memory hints at the origins of intelligence” by Philip Ball) states that Slime Mold remembers the things it’s done:

When the amoeba Physarum polycephalum is subjected to a series of shocks at regular intervals, it learns the pattern and changes its behaviour in anticipation of the next one to come.

Poor slime mold, getting those shocks. I’m sure it would prefer the carrot.

Today in Pi

October 7, 2008

Happy Decimal Digits 23,913,007 through 23,913,015 of Pi day! Yes, at the 23,913,007th digit (not counting the 3) is the string 10/07/2008 [precisely, …95510072008255…]. Apparently being only 24 million digits in in a bit of a lucky stroke: tomorrow’s date of 10/08/2008 first appears 100 million digits later, at the 124,023,083th place. (See Pi-Search for more such fun.)

Then I stared wondering about e. Sadly, being the less famous relation of π has its drawbacks and e doesn’t have its own search page. There is, however, a list of the first two million digits of e, which includes 1007 for 10/07, but not 10072008. Bummer we weren’t writing this a year ago: the string 10/7/2007 manages to squeeze in towards the end.

(The string 360 is much more popular: it starts at the 285th decimal digit in the expansion of pi, and occurs almost immediately in e: 2.7182818284590452353602874…)

Suppose you want to look for a name instead? Dr. Mike has translated 31,415,929 digits of pi and 27,182,818 digits of e into Base 27, to represent the 26 letters and punctuation. The laws of probability make long names hard to find, and even with all those digits GODZILLA doesn’t appear in the translated numbers (nor does MOTHRA), though a lot of other words do, as he explains at the bottom of each page. Interestingly, Dr. Mike used \sum \frac{1}{n!} in base 27 to estimate the digits of e. Hooray for Calc II and series convergence!

The Ig Nobels

October 6, 2008

The 2008 Ig Nobel prizes have been awarded!!! In the words of the official site:

The Ig Nobel Prizes honor achievements that first make people laugh, and then make them think. The prizes are intended to celebrate the unusual, honor the imaginative — and spur people’s interest in science, medicine, and technology.

So who were the winners this year? The complete list (including linked references) is here, but as a quick summary they are:

  • NUTRITION PRIZE. Massimiliano Zampini of the University of Trento, Italy and Charles Spence of Oxford University, UK, for electronically modifying the sound of a potato chip to make the person chewing the chip believe it to be crisper and fresher than it really is.
  • PEACE PRIZE. The Swiss Federal Ethics Committee on Non-Human Biotechnology (ECNH) and the citizens of Switzerland for adopting the legal principle that plants have dignity.
  • ARCHAEOLOGY PRIZE. Astolfo G. Mello Araujo and José Carlos Marcelino of Universidade de São Paulo, Brazil, for measuring how the course of history, or at least the contents of an archaeological dig site, can be scrambled by the actions of a live armadillo.
  • BIOLOGY PRIZE. Marie-Christine Cadiergues, Christel Joubert, and Michel Franc of Ecole Nationale Veterinaire de Toulouse, France for discovering that the fleas that live on a dog can jump higher than the fleas that live on a cat.
  • MEDICINE PRIZE. Dan Ariely of Duke University (USA), Rebecca L. Waber of MIT (USA), Baba Shiv of Stanford University (USA), and Ziv Carmon of INSEAD (Singapore) for demonstrating that high-priced fake medicine is more effective than low-priced fake medicine.
  • COGNITIVE SCIENCE PRIZE. Toshiyuki Nakagaki of Hokkaido University, Japan, Hiroyasu Yamada of Nagoya, Japan, Ryo Kobayashi of Hiroshima University, Atsushi Tero of Presto JST, Akio Ishiguro of Tohoku University, and Ágotá Tóth of the University of Szeged, Hungary, for discovering that slime molds can solve puzzles. [This may warrant its very own post in the near future!]
  • ECONOMICS PRIZE. Geoffrey Miller, Joshua Tybur and Brent Jordan of the University of New Mexico, USA, for discovering that a professional lap dancer’s ovulatory cycle affects her tip earnings.
  • PHYSICS PRIZE. Dorian Raymer of the Ocean Observatories Initiative at Scripps Institution of Oceanography, USA, and Douglas Smith of the University of California, San Diego, USA, for proving mathematically that heaps of string or hair or almost anything else will inevitably tangle themselves up in knots. [But of course you already read about that here last December!]
  • CHEMISTRY PRIZE. Sharee A. Umpierre of the University of Puerto Rico, Joseph A. Hill of The Fertility Centers of New England (USA), Deborah J. Anderson of Boston University School of Medicine and Harvard Medical School (USA), for discovering that Coca-Cola is an effective spermicide, and to Chuang-Ye Hong of Taipei Medical University (Taiwan), C.C. Shieh, P. Wu, and B.N. Chiang (all of Taiwan) for discovering that it is not.
  • LITERATURE PRIZE. David Sims of Cass Business School. London, UK, for his lovingly written study “You Bastard: A Narrative Exploration of the Experience of Indignation within Organizations.”

Equally amusing is the online program from this past Saturday’s ceremony, which includes the 24/7 Lectures (three experts each speak for 24 seconds on their subject, then summarize it in 7 words), the Win-a-Date-With-a-Nobel-Laureate Contest (reported by The Boston Globe to be Benoît Mandelbrot himself!), and previous winners, including the creator of the pink plastic flamingo.

Godzilla flies a kite!

October 5, 2008

Today is the last day of the Niagara International Kite Festival, celebrating all things kite (including the fact that they had to use a kite back in 1848 to start building the first bridge over Niagara Falls. Seriously — there was a Kite Contest and it took 8 days to get a kite to fly from the Canadian side to the US side; once the kite had made it across a stronger string was attached, and then a cable, and from there they had a connection and could send bridge stuff across.)

Godzilla wasn’t able to make it to the festival itself, but he did celebrate by making a kite. He’s pretty crafty so he could have made a really complicated one, but he decided to go simple with this round and use the pattern from Big Wind Kite Factory in Moloka’i, Hawai’i. [This is designed as a classroom project for younger kids, and if you pre-cut the string and use a hole-punch you can avoid scissors altogether. The original instructions have more detail, but no giant lizards.]

To start, Godzilla took an ordinary piece of paper. The original design was for an 8½”×11″ piece of paper, but he used a slightly heavier construction paper because it had subtle glitter built in. Godzilla may be a monster, but he appreciates a good glitter paper.

He sat down and folded it in half width-wise. Apparently this is called a hamburger fold (as opposed to a hot dog fold, which is when you fold it so that it becomes long and skinny, like a hot dog bun).

The fold is on the right-hand side of the paper (on Godzilla’s left). He folded a slight diagonal near that side of the paper. This formed the spine of the kite.

Then Godzilla unfolded it and admired the result.

He taped the fold together, because taping is fun.

The directions at this point say to tape an 8″ dowel across the top. But cooking skewers are a lot cheaper, and he already had those in the cupboard because Godzilla is a bit of a gourmet chef.

This would cause a problem with kids — not just the length, but the fact that it’s a pointy object that would become a weapon in about 3 seconds if given to (our) kids. If this were done with children, I’d recommend chopping the end off to size before starting the whole kite-making. Godzilla, however, had a more unique way of making sure the stick was the right length:

The next step was to attach the string (unless you want to have a paper airplane. Our 8-year old did test it at this point, and it made fancy moves but didn’t really fly very far….). You can use scissors, a hole punch, or your talons to make a hole in the spine to tie the string.

Now, in theory, you attach a tail. Godzilla skipped the tail part because he was afraid if he took any more time this blog post would never get written, and he went straight to the flying. Fortunately there was a nice stiff breeze in the living room — just look how well his kite flies!

Edited 10/6 to add: I meant to mention that I used this kite design in an Inquiry workshop for teachers this summer.   Since you can vary it (or other kite designs) pretty easily — using different weights of paper, changing the fold, adding weights — it’s a great opportunity for kids to create their own experiments to find out which kite flies the longest, the highest, etc.  That’s Godzilla:  helping science students everywhere!

Fancy Fields

October 2, 2008

Michael, over at God Plays Dice, posted a neat article yesterday about the geometry in Baseball fields and the history of mowing the grass in interesting patterns (like these stripes at Petco Park):

There’s a lot more detail in the New York Times article, “Groundskeepers Display Artistry on the Diamond”.

I was immediately reminded of Crop Circles. There’s a bunch of geometry there, too, though it’s more about circles than stripes. Like this one:

Or this one:

Or this little guy (by Perfectblue97) which looks simpler but might have been just as hard to make:

There are some more designs at CoolMath4Kids, and more about how to physically make the circles on WikiHow. But if what you want is a step by step description of particular designs, your best bet is to head over to Zef Damen’s page, which has lots and lots of designs (updated frequently, too!). I think I’m totally going to assign this as a creative assignment in Geometry when we’re using a straightedge and compass.

All of which is to say that groundskeepers aren’t the only ones who get to make neat designs.