Stories tagged Math

Sep
23
2008

On November 4, America will go to the polls and choose its next president. But we do not vote for the president directly. Rather, we vote for electors to represent our state in the Electoral College, and they ultimately choose the president.

By a strange quirk of math, voting in an indirect, divided election such as this actually gives vote4rs more power than if we voted in a direct election. The best way to explain is through an example:

  • In 2004, over 121 million people voted for the top two candidates. If we had held a direct election, the winner would need one more than half of those votes—60,500,001. (Roughly.)
  • But in our electoral system, a candidate doesn’t need a majority in the entire nation. Instead, they need to win a majority in the ten largest states. In 2004, those states had a combined voting population of over 66 million. So a candidate would need only 33,000,010 votes to win.

So, your favorite candidate needs only about half as many votes to win a divided election as they would to win a direct election. Which means your vote has the potential to be worth almost twice as much!

But what if you don’t live in one of the ten biggest states? That’s OK—those states almost always split between the major candidates, so that voters in other states also become crucial to winning the election.

It is true that under the Electoral College system, there are years when your vote doesn’t matter at all. But in the years that it does matter, it matters so much that, on average, you still come out ahead.

We recently put together a web exhibit to demonstrate this phenomenon. It includes an interactive calculator that allows you to change the voting populations of states and see how this affects voting power.

The exhibit is a simplified version of the work of Dr. Alan Natapoff, a physicist at MIT who first figured this out. You can find a nice summary of his work here,and an update here.

Sep
12
2008

Three black cats: So that's a triple negative... run! Run!
Three black cats: So that's a triple negative... run! Run!Courtesy heyjupiter
Evolutionary biologists and math wizards have put their minds together to summon a pulsating, glistening packet of truth from the void.

Biting into the fruit of this magnificent spell, the meta-scientists gained the following information: superstitions, it seems, are an evolutionary adaptation.

This isn’t an entirely new train of thought, even here on Science Buzz, but this research takes the notion a little further. It has already been proposed that superstitions—false connections between cause and effect—prepare us for “just in case” scenarios. That is to say, as Gene put it, it won’t actually rain on a particular day just because we forgot our umbrellas, but thinking that that’s true will encourage us to bring our umbrellas just in case. The scientists behind this new study are looking at that idea in a more mathematical way.

They started with a similar premise: that assuming a potentially false connection between cause and effect will sometimes be beneficial. For example, to a prehistoric man, rustling grass might sometimes mean that there’s a lion getting ready to pounce on you. Even though a lion isn’t the only thing that will make grass rustle, treating rustling grass as a sign of danger isn’t a bad idea in the long run; the caveman looses nothing by avoiding grass that is actually being disturbed by the wind, but gains everything by avoiding grass the few times that it actually hides a predator.

The scientists then decided that the theory could be tested mathematically. By weighing the losses of false associations (avoiding wind rustled grass) against the gains from when those associations turn out to be real (hungry lions hiding), we can see if that sort of behavior is beneficial to survival in the long run, and will therefore be selected for evolutionarily. The model gets more complicated when there are multiple potential causes to connect to an effect (is it the rustling grass, the full moon, or the random sneezing that means a lion is on its way?), but it seems that assuming false causes is, in general, a decent survival strategy. Fortune favors the timid, apparently.

In modern times, the scientists say, this behavior can manifest in things like attitudes toward alternative and homeopathic medicines; while most of them may be ineffective, the chance that some work is enough to get people to use them all.

Superstitions like avoiding black cats, paths under ladders, and opening umbrellas indoors, however, may have more to do with evolutionarily superstitious behavior getting mixed up with culture and “modern life.” These days, the researchers point out, superstitions are probably less beneficial than they used to be.

That’s a little bit of a copout, I’d say. Fortune, after all, favors the bold, so why not go out on a limb here?

You don’t want black cats crossing your path, obviously, because a much larger black cat could be chasing them—and you don’t want to mess around with huge black cats (especially if they’re being chased by an even larger cat).

Walking under ladders is an easy one. There’s always the chance that a bucket of paint could fall on your head, and once you’ve got a bucket stuck on your head any number of awful things can and will happen to you. Trust me.

Opening umbrellas indoors—if you’re in a very small house, you could seriously damage your umbrella.

Unlucky number thirteen? Thirteen of anything can’t be divided fairly between friends, leaving you with no other option than to kill one of your friends. That’s how blood feuds start.

I should be a scientist. Or a fortune teller.

Sep
04
2008

Once more into the breach: We few, we happy few, we band of kitties; for he to-day that sheds his blood with me shall be my puss-puss.
Once more into the breach: We few, we happy few, we band of kitties; for he to-day that sheds his blood with me shall be my puss-puss.Courtesy o205billege

The general holds the binoculars up to his eyes and surveys the battlefield. This will be the first test of the new Coordinated Autonomic Tactical force—C.A.T. for short—an army of robot warriors with electronic brains as complex and powerful as a small mammal’s.

The exercise begins, and all goes exactly to plan. The mechanized warriors sweep across the terrain in formation. Faced with unexpected obstacles, they improvise their own solutions. Soon, they are overwhelming the enemy positions.

Suddenly, a squirrel darts across the field. The entire right flank breaks rank to pursue. Corporal Whiskers beings licking himself. Sergeant Buttercup and Lieutenant Muffy begin hissing at each other. Private Snookums climbs a tree and can’t get down.

The general lowers his binoculars. Staring off into the middle distance, he says to his second-in-command, “We may not have thought this through thoroughly.”

Aug
11
2008

Ask not...: It tolls for thee, bro, it tolls for thee.
Ask not...: It tolls for thee, bro, it tolls for thee.Courtesy carl.jones
Just messin’, y’all!

Only some of us will die on September 10th! And that’s only because we were going to die anyway. There will be sudden heart attacks, tragic car accidents, hilarious full-body prolapses, and possibly some mysterious cases of spontaneous combustion, and none of that will have anything to do with the Large Hadron Collider turning on on the tenth of September.

That’s right, everyone, you can stop holding your breath, and start crossing your fingers, because the LHC now has a date for its first proton collision.

Some people have raised concerns that turning on the LHC could lead to the destruction of the earth in one of several very sciencey ways. Other people have shouted down these jokers, however, because they are very, very, very probably wrong.

And if the world doesn’t end, well, we’ll probably learn all sorts of rad things about the nature of the universe. We might even get some visitors from the future. But I might put a larger bet on the destruction of the solar system (but, you know, fingers crossed).

So, Buzzketeers, on September 10, do your best to protect yourself from the everyday dangers of existence. Wrap your head in packing foam, fill your tummy with starch-based peanuts, and keep yourself wet and/or naked to prevent sparks catching in your clothing and hair, because you probably won’t want to miss what’s coming out of the LHC.

Jul
15
2008

Some fingers: Some fingers down, more fingers up.
Some fingers: Some fingers down, more fingers up.Courtesy anna_t
Researchers from MIT have found that a tribe in remote northwestern Brazil has no words for specific numbers.

The language of the tribe, of which there are only about 300 members, seems to be unique in that it has no numbers. Counting was thought to have been innate in human cognition. Apparently that isn’t totally the case. Specific numbers weren’t useful to this culture, so they never developed them in their language.

Instead of specific numbers, the group, called the Piraha, has a couple of relative terms, translating to something like “some” and “more.” Piraha math classes, I assume, would be awesome.

Some + some = more (obviously)

Nothing + some = some (duh)

Nothing + more = some (interesting!)

More – some = some (probably)

Some – more = your mind blown (Whoa!)

Something very anthropologically and linguistically crazy is going on here. Something about how even though we think our thoughts shape language, language actually ends up shaping our thoughts. So if you come from a culture whose language has no concept of specific numbers, how does that shape your perception of the world?

Oh, if only I had been a better student.

Jul
10
2008

US missile test: There is nothing wrong with this picture.
US missile test: There is nothing wrong with this picture.Courtesy US Dept. of Defense (not Mark Ryan)
Click here and look at the photograph accompanying the story. Agence France-Presse claims the image was obtained from a website of the media arm of the Iranian Revolutionary Guard. The photo makes it look like the Iranians are flexing their military muscle during a recent missile test launch, but in reality they seem to be merely flexing their Clone Stamp Tool in their (probably illegal) copy of Adobe Photoshop.

Now look at the stock photo on the right. This is a minuteman test done by the US military over the Pacific Ocean. I swear to God I have not manipulated this image in any way whatsoever. Not at all. Not one single pixel has been changed in this original photograph. Really.

Well, okay, actually I may have enhanced it just a bit, but only to make a point.

Photo tampering has been around since the earliest days of photography. It was (and still is) a practice used often in advertising, propaganda, magazine covers, and even news (where it is gravely frowned upon). So this kind of thing is nothing new. But advances in digital photography and computer software that allows for pixel-level image manipulation has really created an atmosphere ripe for extreme skepticism of any kind of photograph you see out there nowadays. And the Internet is full of such “real photographs”; stuff like the guy who keeps his dead wife encased in a coffee-table, paratroopers coming in over a lake full of hungry alligators, or president Bush having a good time in hurricane-ravaged New Orleans. All lies!!

When I published a composite photo in a magazine some years ago, the publisher credited it as a “photo illustration” rather than photograph. And I had no problem with that. I’ve also sold (as photographs) images that were extensively manipulated by the addition and removal of elements to enhance the composition. Since I wasn’t trying to make any kind of editorial statement, I have no problem doing that. I look at it more as painting with pixels than tampering with photography. But it does raise the issue of photo ethics. Evidently, it’s okay when used in some ways (such as advertising where everybody expects everything to be a lie), but not okay in other ways (such as news photos).

If done correctly, and with a good deal of thought and meticulous attention to detail, a remarkable “photograph” can be created that even the experts will have difficulty determining whether it’s been doctored or not. Such as my fine illustrative example above. If I hadn’t told you otherwise, I’m sure you would have thought it was an actual photograph of multiple launches. People can be so gullible.

So, perhaps you want to join the Photo Tampering Bandwagon and learn the finer points of image manipulation, but you just don’t have the time to invest in reading the manual that came with your copy of Photoshop. Who can blame you? The thing is massive! I don’t even like reading it. But now, fortunately, there’s a wonderful series on YouTube called “You Suck at Photoshop”, which makes learning the ins and outs of what truly is a complicated program both fun and educational (especially if your current relationship is on shaky ground).

And, lastly, for those of you insisting on some sort of “science” angle to these posts, go here for that.

LINKS
More on the ethics of photo manipulation
Snopes Fauxtography site

Jun
24
2008

Opps! There's one!: So I guess it's 17 now.
Opps! There's one!: So I guess it's 17 now.Courtesy Minnete Layne
Well, if you were feeling anxious about there being no more undiscovered sea monsters, chill out. There are still some out there. About 18, to be specific.

See, ever since Science’s parents (Magic and Critical Thought) stopped putting Science’s stuff up on the fridge, Science has really been going out of its way to make sure we all know how special it is.

We get it, Science, you’re great. Take it easy.

As if.

Science, in its latest flailing and pathetic play for attention, has announced that there are indeed more huge, unknown sea creatures out there, and it knows that there are 18 of them.

Okay, Science, whatever you say. Act like you know.

But, no, Science goes on to explain, here’s my reasoning: If we first decide that a body length exceeding 1.8 meters defines a large sea creature (which, by the way, makes JGordon a large sea creature by 3 cm when he goes swimming), we can then look at the rate at which large sea creatures have been discovered in the last 180 years or so. The rate of discovery for large sea creatures remains pretty strong, and if you consider the places large sea creatures could be hiding, deep in the oceans, or under polar ice, say, it’s very likely that there are quite a few of them left to find. Using some flashy statistical modeling, Science predicts that there could be as many as 18 of these large sea creatures still undiscovered.

Science goes on to emphasize that there probably aren’t any monsters hiding out in lakes and lochs, and that accounts of sea serpents and their ilk can probably all be explained by known creatures, like colossal squid, and 30 plus-foot oarfish. Ah, thanks for that, Science.

Still, Science doesn’t hold all the cards. It may know that there are 18 monsters still hiding out there, but I know exactly what they are. Deal with it, Science.

Anguirus
Baragon
Destroyah
Ebirah
Gabara
Ganime
Gigan
Gorosaurus
Kamoebas
King Caesar
King Ghidorah
Kumonga
Megalon
Minilla
Mothra
Rodan
Urkel
Varan

Jun
22
2008

My Geo Metro: 47 miles per gallon
My Geo Metro: 47 miles per gallonCourtesy Art Oglesby
Guess the answer to this word problem before doing the math.

  • Car A (a compact) gets 34 mpg
  • Car B (a hybrid) gets 54 mpg
  • Car C (an S.U.V.) gets 18 mpg
  • Car D (a sedan) gets 28 mpg

Which would save more gasoline?

  • (a)replacing Car A with Car B
  • (b)replacing Car C with Car D
  • (c)both would save the same

Can you do the math?

I drive my car about 10,000 miles each year. One way to look at this problem would be to calculate how many gallons of gas each of the four cars would use to go 10,000 miles. Can you do the math? If gas costs $4 per gallon what is the cost for each car to go the 10,000 miles?

Show me your answers in the comments

I will do the math for my Geo Metro as an example. It now has over 100,000 miles on it. Until recently it got 50 miles per gallon. Two gallons would take me 100 miles, 20 gallons would take 1000 miles. 100,000 miles would take 2000 gallons. With $4 gas that 2000 gallons would cost $8000.

Save the world's gas

I once owned a Ford pickup truck. If it got 20 mpg and if I drove it 100,000 miles I would need 5000 gallons which would cost me $20,000. By replacing my pickup with the Metro I use less than half the gas and save over $1000 a year. I used to commute to work and put on 30,000 miles per year. That figures out to a $36,000 saving over 10 years.

Jun
19
2008

A different crop circle: This one was created by confused cattle. Amazing.
A different crop circle: This one was created by confused cattle. Amazing.Courtesy Alan L. Baughman
Want to blow some minds, Buzzketeers? You’ve got a couple of options.

The first and most obvious route to some serious brain-blasting is to become a motorcycle stunt jumper. I don’t care if you aren’t into engines and broken bones; if you see a man in a cape riding a dirtbike fly over 30 flaming school buses, your brain will ooze out your ear holes (in the most awesome way).

The other option is to learn some math. It doesn’t have to be too much math—a lot of people couldn’t tell trig if it bit them in the calc, and so a little math can go a long way. And if you can combine that math with another skill…minds will be blown.

Take, for example, the latest, greatest crop circle. Now, we all know that crop circles are made by aliens, right? Duh. It’s a case of Occam’s razor—the simplest explanation is the best. So we have unusual patterns battered into fields of crops. What’s the explanation with the fewest assumptions? That beings we have never encountered traveled from a place we know nothing about, and use their very likely highly advanced minds and inter-stellar travel technology to draw circles and things in our food for reasons we can’t fathom.

For the sake of argument and education, however, let’s pretend that crop circles have a much more complex origin—that they come from dudes (and dudettes, undoubtedly) with an artistic bent, and too much time on their hands.

So, back to this particularly mind blowing circle. It appeared on a field near Barbury Castle (which, I’m afraid, isn’t much of a castle), and consists of a ten layered, jagged-looking spiral, with a few circles and dots and things. It looks pretty cool—check out the photograph—but it means nothing to me. Then again, I majored in the liberal arts. When the circle was examined by an astrophysicist (or a “professional cleverboots,” as they are sometimes known), however, something remarkable jumped out of the shape: it’s a mathematical code.

And what secret equation or figure is hidden in this alien thought bubble?

Pi.

“The code is based on 10 angular segments with the radial jumps being the indicator of each segment,” says the astrophysicist on-call. “Starting at the centre and counting the number of one-tenth segments in each section contained by the change in radius clearly shows the values of the first 10 digits in the value of pi (3.141592654). The tenth digit has even been correctly rounded up. The little dot near the centre is the decimal point.”

How about that? I wouldn’t have noticed, but now that I’ve been told, my mind is hissing and steaming out of my tear ducts. Or are those just tears of happiness?

Very clever, crop circle-person, very clever. Consider all minds blown. And you couldn’t have done it without your old pals math and geometry.

Check out this page for ten of the most impressive crop circles to be seen on Google Earth (the new one isn’t on there yet

Jun
16
2008

Sizing up the competition: Capuchin monkeys can be trained to use money, and react to it in surprisingly humanlike ways.
Sizing up the competition: Capuchin monkeys can be trained to use money, and react to it in surprisingly humanlike ways.Courtesy Arriving at the horizon

Researchers in Italy have taught capuchin monkeys how to use money to buy food in lab experiments. The monkeys were observed making simple economic decisions.

The study is not yet available on-line, but it sounds an awful lot like this research from Yale, published last year. There, the monkeys not only learned to use money, but to respond to price fluctuations, and even to steal. However, they could not budget and they gambled irrationally – a lot like their hominid cousins.

There was also an incident of flinging feces at fellow monkeys who did not play by the rules – perhaps the most dismal experiment in the history of the dismal science.

Gone fishin': A long-tailed macaque in Thailand demonstrates its unorthodox technique.
Gone fishin': A long-tailed macaque in Thailand demonstrates its unorthodox technique.Courtesy nittynorn

Meanwhile in Indonesia, industrious long-tailed macaque monkeys aren't waiting for handouts. They are supporting themselves by observed fishing, scooping fish from the water with their bare hands and eating them.

This is the first documented evidence of fishing in this species, though other primates – including baboons, chimps and orangutans – are known to fish. Researchers believe this behavior in macaques is an indication of their flexibility and ability to adapt to new conditions.