We were in AAA this weekend getting passport photos for the decennial renewal (though for kids it’s only pentennial)(or quincennial) (And yes, I did just have to look all those words up, thankyouverymuch.) Right by the front door TwoPi noticed a sign advertising the AAA discounts to various places.
OK, it’s hard to read, because TwoPi was being all subtle with his cell phone. Here’s a close up:
So you can go to Disney for 1 to 10 days, and “The more days — the less you pay!” Whoa — what a deal! Though in truth, I’m thinking that whoever made this sign needs a refresher course in the difference between functions and rates of change.
Hmmm — this stood out as a Fail, but in thinking more it’s a nice application of the sign of the second derivative as well.
Ray Bolger, who played the Scarecrow (THE Scarecrow) was born 106 years ago today. In his honor, here’s a clip from Youtube in which the Scarecrow gets a Doctorate of Thinkology and makes reference to the Pythagorean Theorem — sort of. (The exact words of the Scarecrow are, “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh, joy, oh, rapture. I’ve got a brain!”)
It seems (perhaps only to me) like it ought to factor nicely, because 20 is twice 10. But once you factor out that 10 and that 3 you’re just left with a prime, since 2010 is just 2·3·5·67. (Speaking of four primes, did you know that you can get four prime New York Steaks for $132.95 on Amazon? I was relieved to see that they were not available for Super Saver Shipping.)
Wolfram Alpha points out that 2010 itself is a factor of 296-1. And Number Gossip adds that it’s untouchable, which means that there aren’t any numbers whose proper divisors add up to 2010.
It can be written as 133122 in Base 4, which is kind of cool, and as 6, 3, 12 in Base 18; my favorite, however, is that it is 5, 10, 15 in Base 19.
Finally, it’s equal to:
669+670+671
400+401+402+403+404
127+128+…+141 and several others
[Hmmm...I can find a string for each of the 7 odd factors, but I'm not sure that exhausts all of the possibilities.]
While getting ready to post this, I noticed that MathNotations has a similar post from yesterday. Whoops!
It’s a new year, maybe a new decade (I’m a plebian), and the perfect day for a new Carnival! Though I can’t help but feel guilty that I didn’t mention the previous one, and not because it wasn’t good either — it was. So here’s a summary of the carnivals and a chance to start the New Year off on the right foot!
There was Math Teachers at Play #20 back on its original turf at Let’s Play Math, with it’s regular assortment of lots of fun math. Speaking of Let’s Play Math, Denise has her annual New Year’s post with the 2010 Mathematics Game; I remember thinking last year that I should get a head start on that, and didn’t, but we’ve got some mathy company coming this weekend so it’ll provide some good entertainment.
Then there was the Carnival of Mathematics #60 over at Σidiot’s Blog (heheh — I just got that). That carnival appeared back on December 4, and had some neat modular origami and puzzles and National Math Blog Writing month and a portrait of someone who isn’t a mathematician.
And now we’re at Carnival of Mathematics #61! It’s up at Walking Randomly today, with puzzles and statistics and many other good things. There’s a mention of calendars at the end, which reminded me of the printable dodecahedral one that TwoPi mentioned two years ago. Not that that is a calendar or anything, but at least it’s a dodecahedron.
(To the tune of “The Christmas Shoes”. Sorry about that.)
It was almost exam time
Sum. Solve. Is this a prime?
Trying to find a good shape or two
Not really in a polygon mood
Then appeared right in front of me
As I surfed somewhat anxiously
The creativity that designers do
But these were mathematical
Artsy pairs of shoes
And these shoes weren’t worn or old
They had V+F=E+2 from heel to toe
Fit for show more than for play
And I couldn’t believe, but they made me say
Yes I wanna show these shoes
To my classes please
There’s origami
Which adds quite a bit of style
I’d better hurry now
My datebook says there’s not much time
You see, classes have met for quite a while
And I know these shoes will make them smile
Cuz those shapes look kind of beautiful
While everyone is studying tonight.
These shoes, designed by Andreia Chaves of São Paulo, Brazil, were featured on yatzer (found via cnet). All photos are used with permission from the folk at yatzer (Thanks!)
I got a call this afternoon from Marc, a friend of mine from college in Minnesota who now lives back East and runs his own business. He was trying to find a model for a collection of functions, but couldn’t figure out what kind of function it would be.
The basic scenario was that the function should start at (0,0), increase rapidly to a point [say, (10,10)], and then slowly decrease. The x-axis could be a practical asymptote, although it didn’t really matter since this would only be looked at in finite time.
My first thought was surge function (something of the form ), and sure enough that works. But I was on Homework Patrol, so I handed it off to TwoPi, and he came up with and . Walphaing these shows that both work well:
This seemed to help. So here’s what I’m wondering — are there any other simple functions that fit the bill? Neither function is too complicated, but it would be fun to be able to share other examples of functions that rise quickly, but then that taper off after a while. [I think the tapering should be more gradual compared to the climb, though I believe that can be controlled with more constants.
It really wan’t my intent to post three math fails in a row — indeed, the only intent was to post something, since the last monthsemester six months have been rather sparce, post-wise. But then I saw an article on the Huffington Post which they got from Media Matters (which includes a great graphic and a video) and, well, another fail it is!
Here’s the scoop: Rasmussen Reports posted the results of a poll about climate change that included the following question and answers:
3* In order to support their own theories and beliefs about global warming, how likely is it that some scientists have falsified research data?
35% Very likely
24% Somewhat likely
21% Not very likely
5% Not at all likely
15% Not sure
If you were going to summarize this, how would you do it? Maybe combining categories, like this:
59% Very or somewhat likely
26% Not very or not at all likely
15% Not sure
But that’s a little awkward, isn’t it? So maybe you’d change the wording to emphasize the likely and not likely, and drop the people who are sure. that leaves:
59% Likely
26% Not very likely
Hmmmm. This is accurate enough, but to make it sound better, you might add back in the “Very likely” as a parallel to “Not very likely” [OK, you probably wouldn't do that. But let's pretend you would.]. That would give
59% Likely
35% Very likely
26% Not very likely
And that, my friends, is how Fox News you do a poll.
As a follow up, Producer Lauren Patterson claimed, “We were just talking about three interesting pieces of information from Rasmussen….We didn’t put on the screen that it added up to 100 percent.” Yeah, sure, that makes it all OK.
Finals ended today — woo hoo! [This is the earliest they've ever finished, and I can't say I'm upset.] In that spirit, here’s one of the latest from Fail Blog (similar to others I’ve seen, but this appears to be more recent).
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 decreasing. 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!
It seems (perhaps only to me) like it ought to factor nicely, because 20 is twice 10. But once you factor out that 10 and that 3 you’re just left with a prime, since 2010 is just 2·3·5·67. (Speaking of four primes, did you know that you can get 
), and sure enough that works. But I was on Homework Patrol, so I handed it off to TwoPi, and he came up with 
and 
. Walphaing these shows that both work well:
The Number Warrior
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 
