The Number Warrior is hosting this month’s carnival! It’s up here, and I was impressed with the entries [both in number and quality]. Plus, there was even something on Bourbaki, though not the symbol I wrote on earlier.
I also really like the post right before the Carnival, about a possible case of cheating at the 2009 Philadelphia Inquirer Sudoku National Championship. You can find some of the contest Sudoku puzzles at this pdf.
Now I want to try some Sudoku: here’s one that’s a jigsaw puzzle (created by A.R. Nonenmacher and published under GNU-FDL)
Rock, Paper, Scissors is alive and well in elementary schools, at least from what I can hear [and hear I do, nearly every day]. I figured it was a simple game, but it turns out that it’s maybe not as straightforward as I thought.
The first sign was when we were watching Big Bang Theorey and Sheldon proposed a variation:
Got that? Here’s a diagram to help you out:
But TwoPi discovered that Rock, Paper, Scissors, Lizard, Spock didn’t originate with the show: it’s been around since at least 2005 according to The New York Times.
But back to the original game. Did you know that there was an official league? You did? Well then, did you know that back in 2005 — apparently a banner year for Rock, Paper, Scissors — Takashi Hashiyama was going to sell his $20,000,000 art collection and he had to choose between Christie’s and Sotheby’s to run the auction, so he made them play Rock, Paper, Scissors. He gave them some warning, and there’s some evidence (again according to The New York Times) that the Christie’s official conducted Actual Research, at least in the form of having a friend ask his daughters.
Mr. Maclean’s 11-year-old twins, Flora and Alice, turned out to be the experts Ms. Ishibashi was looking for. They play the game at school, Alice said, “practically every day.”
“Everybody knows you always start with scissors,” she added. “Rock is way too obvious, and scissors beats paper.” Flora piped in. “Since they were beginners, scissors was definitely the safest,” she said, adding that if the other side were also to choose scissors and another round was required, the correct play would be to stick to scissors – because, as Alice explained, “Everybody expects you to choose rock.”
Sotheby’s didn’t admit to any strategy. Bad choice, perhaps, because the Sotheby’s official picked rock paper, which was beaten by the Christie’s person’s scissors. Clearly 11-year olds know their game theory.
But wait, there’s more! The following year, a judge made two parties settle a dispute using RPS: (From CNNmoney.com):
This matter comes before the Court on Plaintiff’s Motion to designate location of a Rule 30(b)(6) deposition (Doc. 105). Upon consideration of the Motion – the latest in a series of Gordian knots that the parties have been unable to untangle without enlisting the assistance of the federal courts – it is
ORDERED that said Motion is DENIED. Instead, the Court will fashion a new form of alternative dispute resolution, to wit: at 4:00 P.M. on Friday, June 30, 2006, counsel shall convene at a neutral site agreeable to both parties. If counsel cannot agree on a neutral site, they shall meet on the front steps of the Sam M. Gibbons U.S. Courthouse, 801 North Florida Ave., Tampa, Florida 33602. Each lawyer shall be entitled to be accompanied by one paralegal who shall act as an attendant and witness. At that time and location, counsel shall engage in one (1) game of “rock, paper, scissors.” The winner of this engagement shall be entitled to select the location for the 30(b)(6) deposition to be held somewhere in Hillsborough County during the period July 11-12, 2006.
Humans aren’t the only ones with an eye towards the game. According to Wikipedia, generator of this entire post and the inspiration of a new Category, E-coli plays as well:
antibiotic-producers defeat antibiotic-sensitives; antibiotic-resisters multiply and withstand and out-compete the antibiotic-producers, letting antibiotic-sensitives multiply and out-compete others; until antibiotic-producers multiply again.
And so do lizards out in California (from this bio page)
As in the rock-paper-scissors game where rock beats scissors, paper beats rock, and scissors beats paper, three morphs of lizards cycle from the ultra-dominant polygynous orange-throated males, which best the more monogamous mate gaurding blues; the oranges are in turn bested by the sneaker strategy of yellow-throated males, and the sneaker strategy of yellows is in turn bested by the mate guarding strategy of blue-throated males.
So there you have it. Maybe not the simple game I thought it was after all.
These are not the only two things I don’t know, mind you, but they’re two things I want to know, that I’ve tried to find out, but which are failing to succumb to the magic of the Internet.
The first is a spinoff of yesterday’s post, in which I quoted a paraphrase of Nixon’s from the Fall of 1972 in which he said that the rate of increase of inflation was decreasing [which, since inflation measures the change in prices, amounts to saying that a third derivative is negative]. Although this was just a postscript, it got me curious as to exactly how he’d phrased it. So I looked, figuring that Google would turn up something and…it did, but nothing about that speech. There is something sort of close in this speech from February 1, 1971 in the Annual Message to the Congress: The Economic Report of the President:
Fiscal policy should do its share in promoting economic expansion, and our proposed budget would do that. But fiscal policy cannot undertake the responsibility of doing by itself everything needed for economic expansion in the near future. To try to do that would drive taxes and expenditures off the course that is needed for the longer run. The task of economic stabilization must be accomplished by a concert of economic policies. The combined use of these policies, starting near the beginning of 1969, finally checked the accelerating inflation that had kept the economy overheated for years. [bold added]
See, it uses the word “acceleration”! Which is the main word that stood out, because I’m afraid that trying to sort through speeches by Nixon for references to inflation is a wee bit mind-numbing.
This 1971 speech occurred more than a year before the one referenced in the quote paraphrase paraphrased quote; the timing, however, seems to match the economic data (well, sort of). From Inflationdata, the average inflation rate was:
2.79% in 1967
4.27% in 1968
5.46% in 1969
5.84% in 1970 [And somehow I bet that fact that the increase was decreasing wasn't so reassuring at this point]
But in 1971 it started to go back down, so that in the Fall of 1972 it was back to 2-3% levels. That doesn’t quite match the claim that the increase was decreasing then — you could just say that inflation was decrease. But then again, it doesn’t look like inflation was accelerating per se either — it was mostly decelerating prior to his 1971 speech.
So the end result is that I have no idea which speech it was, nor am I sure that it was right in any case.
That’s the first thing. The second is a minor point. I was just reading a student paper about the secret society Bourbaki [the paper came with a short film she made re-enacting the start of Bourbaki!] and there was a reference to a curvy Z-like symbol that Bourbaki used to use, to signify “dangerous bends” in the road where it woudl be easy to get lost. I was curious as to what it would look like, but the closest I could find was the adaptation that Donald Knuth used: Which is all well and good, except that I’m curious as to whether the black curvy part is identical to Bourbaki, or merely inspired by “him”. Maybe a search through online books is the next way to go for that search.
The White House is talking about derivatives again! As in Calculus, though that’s not the word being thrown around. Christina Romer is the Chair of the Council of Economic Advisers, and a week ago she was quoted in an article in the Christian Science Monitor (from the October 22 JEC hearing) as saying:
Most analysts predict that the fiscal stimulus will have its greatest impact on growth in the second and third quarters of 2009… By mid-2010, fiscal stimulus will likely be contributing little to growth.
That article apparently caused some confusion, so she clarified the situation in The White House Blog:
As a teacher, I should have realized that many people have trouble with the distinction between growth rates and levels….When we go from no stimulus to substantial tax cuts, increases in government spending, and aid to state governments, this has a large effect on the growth rate of real GDP – just as when you press hard on your car’s accelerator and go from 0 to 60, you have a great change in your speed. This sense of acceleration is exactly what we have been experiencing since the start of the year. Fiscal stimulus has been steadily increasing, raising GDP growth by between 2 and 3 percentage points in the second quarter and between 3 and 4 percentage points in the third quarter….. We expect that stimulus will continue to have a positive effect on growth in the fourth quarter of 2009 and well into 2010, though, by design, not by as much as it did in the second and third quarters of 2009. As a result, we expect the largest effect of the stimulus on the levels of GDP and employment to occur well after the largest effects on growth rates.
At some point, the stimulus plateaus at a high level. That is important too. Such continued stimulus may not add much to growth, but it is keeping the levels of GDP and employment much higher than they otherwise would have been – just as keeping pressure on the accelerator keeps the car going at 60 mph.
So here’s another kind of situation to discuss in those calculus classes! And presumably the words “point of inflection” could also be brought into play, since that is apparently where Christina Romer thinks we are at right now.
*”again” referring to Hugo Rossi’s quote “In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.” from the October 1996 Notices of the AMS.
We have a mystery on our hands. Greater than the mystery of what Frankenstein has to do with polygons. Greater even than the mystery of how to come up with a good Math Halloween Costume. The mystery is: Do pumpkins have some strange connection with the number 3?
It all started when we were getting ready to carve Jack O’Lanterns.
And we noticed this odd thing at the bottom of the pumpkin:
See it down there?
What the heck is that? The hole in the center goes all the way through to the outside, but the perfect 120° angle markers are only on the inside of the pumpkin. Did this come from a metal spike or something, and it’s just artificial? Or do pumpkins somehow naturally divide into three parts, like a banana. Hey kids — you can try this at home! Take a banana, break it in half, and then stick your finger down the middle. It will naturally split into 3 pieces lengthwise. I learned this in college [in the dining halls, not a classroom].
Apparently, I just now discovered via The Sneeze, this happens to bananas because the ones sold in the supermarket are triploid organisms, which means that they have 3 sets of each chromosome instead of 2. That’s a little weird, no? A little unnatural? Actually, unnatural is exactly what it is: triploid organisms occur (exclusively? This part I’m not sure on) when a biploid organism is crossed with a tetraploid, giving the average of 2 and 4 sets of chromosomes. This is bad for the resulting triploid banana, which is sterile, but good for the person who eats it, because the sterile banana contains no seeds. [No, not even those black spots, which according to this official sounding page are "the remains of aborted ovules that did not mature into seeds" and EEEWWWW who else is suddenly grossed out by bananas?]
But pumpkins don’t seem to be triploid organisms. At least, they aren’t on the list of examples I found on Wikipedia, though watermelons are. So the mystery remains: is this figure naturally occurring, or artificial?
Happy Halloween!
The Fibonacci connection: I’ve seen the fact that bananas split into 3 pieces used as an example of how Fibonacci numbers appear in nature. That our bananas might be a hybrid of those that have 2 and 4 sets of chromosomes, though, and that presumably split into 2 or 4 pieces seems to discount the whole Fibonacci relationship for bananas since 4 isn’t a Fibonacci number.
In celebration of the month of October Wait, you mean it’s October already? When did that happen? , here’s some belated Carnival News:
[Hey, it's the new PicApps! I'm trying to decide if I like it -- more pictures versus the less control thing. And that little film strip. Hmmm.]
Math Teachers at Play #16 appeared on October 3 over at I Want to Teach Forever. One of my favorite submissions was the Brain Games from mental_floss Blog, but there’s plenty of other good stuff. Then, two weeks later, there was Math Teachers at Play #17 over at mathrecrecreation. (who has a post on origami today!) with yet more interesting posts. And now we jump ahead to Math Teachers at Play #19, over at Math Mama Writes [What happened to #18, you ask? You'll have to check it out and see!]. It’s got some cool stuff, including a post about using math to solve a murder case [but can they really neglect air resistance? Wouldn't that make a difference, and maybe make it possible to travel further in the x-direction? HEY -- it's a project question for when I teach Diff Eq in the Spring!]
So there’s the way too late update! Stay tuned for the Carnival of Mathematics next week over at The Number Warrior (who also has a cool problem-solving/communication post currently up on the game Slitherlink). The Carnival of Mathematics will now be appearing the first Friday of each month, with Math Teachers at Play moving to the third Friday of the month. More details can be found here at Walking Randomly, who has taken over organzing the CoM.
Stay tuned for tomorrow: the Mystery of the Fibonacci Pumpkin!
I’m a sucker for a gimmick, particularly food gimmicks. I’ve tried just about every flavor of Mountain Dew, as well as every variety of Reese’s peanut butter whatevers. So when Starbucks released their new VIA instant coffee, I lined up with everyone else to take the taste test. (It’s not bad, if you like Starbucks coffee.)
As a reward for tasting the coffee, I got a coupon for $1.00 off any VIA purchase. I then heard the barista explaining to another customer that with the coupon, the 3-pack was only $2 (actually $1.95), but the 12-pack was a better deal. So I looked at the price of the 12-pack: $9.95, or $8.95 with the coupon. And then I did a quick calculation in my head, and discovered that no, the 12-pack isn’t a better deal after all. With the coupon. Of course, it is a better deal without the coupon, and I’m sure that’s what she meant, so maybe this isn’t a FAIL, but it’s still a fail.
Next up, however, is definitely a FAIL. The following was spotted at my local Target store, where Halloween candy is on sale right next to the Christmas decorations:
Can anyone figure out where this number came from? The box weighs something like 38 ounces (so it’s about $4.21 per pound). If it really were $159.84 per pound, a Butterfinger (2 oz) would cost $20. I think I’d frame it instead of eating it.
We recently discovered a series of math comics because first a commenter and then Mike himself linked to us on Spiked Math. Yay — more math comics! So as the weekend approaches, you can treat yourself to:
or
or
or this one which is, of course, near and dear to my heart:
All of these are copyrighted, but available for non-commercial use.
Recently, Batman mentioned a comic he’d seen about A4 paper and the golden ratio Lichtenberg Ratio (see the comments below). Thanks to the wonders of the Internet, we were able to track down the comic, which makes me laugh every time I read it. We were also able to find the author, who graciously gave us permission to post the panels here. If you click on them you’ll be directed to the Flickr site, where you can read them in a larger size.
The 2009 Ig Nobels were announced last week. These are awarded “For achievements that first make people LAUGH then make them THINK”. This year’s mathematics award is probably more of the think than laugh variety, however:
Gideon Gono, governor of Zimbabwe’s Reserve Bank, for giving people a simple, everyday way to cope with a wide range of numbers — from very small to very big — by having his bank print bank notes with denominations ranging from one cent ($.01) to one hundred trillion dollars ($100,000,000,000,000).
Most of the research is more amusing. For example, the Peace Prize was given to Stephan Bolliger, Steffen Ross, Lars Oesterhelweg, Michael Thali and Beat Kneubuehl for their analysis of whether full beer bottles or empty ones caused more damage. From their abstract:
Beer bottles are often used in physical disputes. If the bottles break, they may give rise to sharp trauma. However, if the bottles remain intact, they may cause blunt injuries. In order to investigate whether full or empty standard half-litre beer bottles are sturdier and if the necessary breaking energy surpasses the minimum fracture-threshold of the human skull, we tested the fracture properties of such beer bottles in a drop-tower.
(Short answer: empties broke at a higher impact, which I think means they’re more dangerous. I wouldn’t want to be hit by either one, though.)
But wait, there’s more. Cows who give more milk when called by name. A man who cracked the knuckles of his left hand — but not his right — every day for 60 years. Giant panda poop as garbage disposal. A bra that can turn into a pair of face masks in an emergency. And my favorite, the Chemistry Prize, which went to Javier Morales, Miguel Apátiga, and Victor M. Castaño for creating diamonds from Tequila.
I’m reading The Memorist by M. J. Rose, a book which is in the same 100+ chapter genre as The DaVinci Code but which doesn’t sport its own diet.
Early on, some criminal-guy leads some journalist-terrorist-wanna-be to an underground area beneath Vienna. After showing him the area beneath the main concert hall, they head back towards the outside world only to discover slashes in one of their rafts — a raft that might have looked a lot like the one in the picture above except that it was in a cave. And was inflatable.
But at any rate, they needed those rafts because there was an underground lake that was really hot. Now here’s the confusing part:
The water was thirty percent hotter than the human body’s temperature thanks to the geothermal heat under the lake’s bed. If you tried to swim across you’d be boiled to death.
What, exactly, does 30% hotter than the human body’s temp mean? My first thought was that the reference point should be absolute zero, or -459.7°F. This would make the water (1.3)(98.6+459.7)-459.7, which simplifies to about 266°F. This matches the line about scalding, but doesn’t quite fit later on when journalist-guy pulls the remaining raft from criminal-guy and dunks him into the water:
For a second David wondered if Wassong could somehow make it out. No, he knew that was impossible. He knew, because Wassong had warned him — no one survived the firewater. Wassong was splashing wildly, displacing a circle of water around him. He continued thrashing for fifteen seconds, thirty seconds, forty, and then all movement ceased. Hans Wassong lay still, floating facedown in the boiling lake, his glasses bobbing beside him.
Despite the reference to boiling, I’m pretty sure that splashing would be kept to a minimum if the water were really 266°F. So I don’t think that’s it. Maybe, since we’re in Vienna, we’re supposed to use Celsius for our reference point. The body’s temperature is 37°C, so the lake would then be (1.3)(37°C)=48.1°C, or about 118.6°F. That’s hot, but not really hot enough to kill so quickly — the Honeywell Burn Chart says an adult could swim for 10 minutes before getting 3rd degree burns. So that’s not it.
Well then, maybe we should use Fahrenheit, which would lead to (1.3)(98.6°F), or about 128°F. Now we’re getting somewhere: the 40 seconds corresponds pretty much exactly to how long before criminal-guy gets 3rd degree burns all over his body, and that’s going to make it tough for him to escape.
I confess, I’ve been waiting to see if this guy is really dead or if he’s going to appear at the last minute. Nothing says, “See ya later!” like a claim that no one could have survived.
First off, there was a Math Teachers at Play (the second #15) just about two weeks ago. It’s hosted by Maria Droujkova of the Math 2.0 Interest Group. There’s the usual assortment of great stuff, including a computer game that I really want to play, and some stuff on Wolfram Alpha (which I know is totally old hat, but I gave two of my classes the homework assignment to play on W|A for 20 minutes and email me the neatest thing they found, and I think that’s been their favorite assignment so far. Plus one of my Math for Liberal Arts groups cited it in a project when a pattern of numbers got too big for their calculator. I love Wolfram Alpha) and sites for special needs students.
So that’s one carnival.
And the next is the Carnival of Mathematics #58 at Walking Randomly, up this past weekend, with near integers and binary in baby toys and maps. (And now the 9-year old is wondering why I’m listening to Men at Work). Even though it’s not in the Carnival, also check out the post about (-1)*(-1)=+1.]
This has nothing to do with the Carnivals, but do you want to know what I learned how to do on Excel this weekend? Conditional formatting! I use Excel for my gradebook and I usually highlight if a student gets below 70% on exams or projects. In the past I’ve done this by hand — which isn’t too time consuming because we’re not talking loads and loads of people — but it turns out that you can highlight a group of cells, go to Format, and then Conditional Formatting, and then set it up. I even got lazy and for a 32-point assignment I set it to highlight scores less than “=70%*32″. I expect this isn’t news to many people, but since the first person I mentioned it to hadn’t heard of it, I figure that’s good enough reason to bring it up. Thanks for showing me this Nicole!
There’s a neat post over at Language Log on determining whether or not someone is a manager if they say, “at the end of the day.” It is the latest in a recent thread (parts 1, 2, 3) about irritating phrases being associated (often incorrectly) with irritating people (see here for an earlier discussion).
As an avid juggler, I have a rather large supply of props to juggle, most of which are balls or beanbags. (Also on the shelves: clubs, rings, scarves, devil sticks, and a diabolo.) As anyone who’s ever bought juggling equipment can tell you, this stuff isn’t cheap: decent beanbags can run $10-15 apiece, and rings and clubs are much more expensive. So when I discovered these instructions to make your own beanbags, I was understandably excited. (Of course, I’ll have to ask Batwoman to sew the pieces together for me. Needles have a tendency to end up stuck in me instead of the fabric.)
Then I started clicking around the IJDb, and I found these. Marylis Ramos has clearly spent a lot of time thinking about tiling a sphere. Certainly any Platonic or Archimedean solid can be adapted to a sewing pattern to approximate a sphere, but a great deal of experimentation among jugglers and sewers has led to only a few becoming popular: the tetrahedron, cube, dodecahedron, icosahedron (one of the best, but really hard to sew), truncated tetrahedron, cuboctahedron, and the “lemon” (with 3, 4, 5, 6, or 8 panels).
The pictures are, of course, idealized beanbags with perfect 1-dimensional seams that cannot be achieved by terrestrial sewing machines (at least not the Singer in my basement). Or maybe they’re just from Wikipedia’s spherical polyhedron page.
I’ll be making some 4-panel lemons, and I’ll post a follow-up to discuss their sphericity (or lack thereof).
The Number Warrior is hosting this month’s carnival! It’s up here, and I was impressed with the entries [both in number and quality]. Plus, there was even something on Bourbaki, though not the symbol I wrote on earlier.
These are not the only two things I don’t know, mind you, but they’re two things I want to know, that I’ve tried to find out, but which are failing to succumb to the magic of the Internet.
The White House is talking about derivatives again! As in Calculus, though that’s not the word being thrown around. Christina Romer is the Chair of the Council of Economic Advisers, and a week ago she was quoted in 
Recently, Batman mentioned a comic he’d seen about A4 paper and the 
