January 29, 2005
Can Math Be Fun?
How do you convince adolescents that math is interesting and useful? It's a big problem. Here's a webpage where I have some things that might help--at least for me...:
Brad DeLong's Collaborative Website: OneHundredInterestingMathCalculations: How do you convince adolescents that there is a big long-run payoff from math? Teaching them (mine at least) that there is a huge short-run payoff from reading and a huge medium-run payoff to writing is easy. But math is harder.
- [World War II Bomber Pilot Survival Odds]
- [How Many Extraterrestrial Civilizations Are There?]
- [Gravity and "Weighing the Earth"]
- [Economic Growth Since 1500]
- [Exponential Growth and Human Populations]
- [How Much Blood Is There in the World?]
- [Julius Caesar's Last Breath]
- [The Birthday Fact]
- [The False-Positive Problem]
- [The Grass-Is-Greener Paradox]
- [The All-Knowing Alien Paradox]
- [Repeating Decimals]
- [Introduction to Compound Growth]
- [Elementary Ballistics: The Kinematics of Falling Bodies]
- [Elementary Ballistics: What Goes Up Must Come Down]
- [How Rich Is Fitzwilliam Darcy?]
- [The Clock Hands Problem]
- [Sunscreen, or the Freak Mutant Near-Albino Problem]
- [The Distributive Law, or the Get-Out-of-the-Way Problem]
- [The Federal Reserve Problem]
- [The Ancestor Problem]
- [Strategy Secrets of ENRON]
- [The Muddy Parent Problem]
- [The Kissing Problem]
- [Understanding "Risk Arbitrage"]
- [Orbiting the Earth]
And here are some suggestions from others for problems it would be interesting to write up:
Brad DeLong's Collaborative Website: SuggestionsForEntries:
- < How long can Moore's Law go on? Starting from the average distance between atoms in a silicon crystal, find the time when chip features will be (supposedly) one atom wide.
- Intro to counting and combinatorics. Suppose there were 14 (or 12) cards in one suit. Suppose there were 5 (or 3) suits in a standard deck of playing cards. How would the relative ranking of poker hands change? They don't all scale the same way. Do most of the work by cancellation, so you don't have to perform a lot of the (tedious, error-prone) multiplication.
- Simple bits of probability, especially conditional probability from games -- card games (poker), dice games (craps), whatever. For example, understanding why it is harder to make your point the hard way (with a pair) when it is 8 rather than 4. As a grad student, I spent a lot of time teaching basic concepts to undergrads (at MIT!) that I mastered in middle school because i thought about the games i spent my time playing.
- Consider the mathematics of triage versus parity policies as described by Garrett Hardin in Chapter 4 of "Promethean Ethics," University of Washington Press, 1980.
- If you try something unlikely a few times, you might fail every time -- but it's commonly said "Even if the odds are a thousand to one against you, you try it a thousand times, you're sure to get it." Right? Wrong. If the odds are 50-50, and you can have two tries, you've got a 75% chance of a win. But it's downhill from there. One in a thou chance with up to a thousand tries? Only a 63.23% chance of a win. One in a million over a million tries? You're down to 63.21%, and it keeps dropping from there. How low can it go?
- Xeno's paradox came about because the Ancient Greeks did not know how to sum an infinite series. I've always used it to illustrate the concept of limits approaching infinity, because it puts the complex math on the side of common sense.
- For "Exponential Growth and Human Populations" set the end point as filling up the Americas by a colonizing group of 100 people, it's more interesting.
- Richard Dawkins has an interesting calculation on human ancestors in the book "River out of Eden" (1995). Figuring 20 years per generation, calculate the number of ancestors you have 2000 generations ago if none of your ancestors appear more than once in your family tree (no inbreeding).
- If the kids are into science fiction, have them work out dimensions of their favorite space ship based on extrapolation from the sizes of particular features (e.g. if the bridge of the Enterprise is so many feet across, how long is the whole ship?). Have them make upper and lower bound estimates to teach them error margins. It's not so great for web page presentation, though...
- Small business economics. Next time you and the kids are at the ice cream shop or other restaurant, have them work out the typical number of customers per hour (from typical customers-in-store and customer-visit-time). From this and the amount spent by a typical customer you get typical revenue. Guess at employee wages and commercial space leasing costs. Ask them why the place closes at 9 instead of staying open all night.
- Bridges of Konigsberg. Requires an illustration. The fundamental problem of graph theory.
- Predator/Prey? balancing over time.
- The different coin problem. N coins or objects of the same weight, one object of a different weight (in the simpler form lighter or heavier is known, in the slightly more difficult, just that it is "different"), a scale, and a limited number of weighings. Teaches binary group comparions. (Similarly, the switchback problem - You are at a fork in the road. You know your destination lies an unknown distance from the fork down one fork. What is the fastest way to surely find your destination?)
- Gabriel's Horn. A mathematical object with finite volume, but infinite surface area. Thus you can conclude that if you wish to paint Gabriel's horn, it's much wiser (and less costly) to fill the horn with paint than to try to coat the outside. Full appreciation will require calculus experience. [Link] [Link]
Gabriel's Horn, alas! is too sophisticated for my purposes--but it is wonderful:
Gabriel's Horn - Wikipedia, the free encyclopedia: Gabriel's Horn (also called Torricelli's trumpet) is a figure invented by Evangelista Torricelli which has infinite surface area, but finite volume. Gabriel's horn is formed by taking the graph of y = 1/x, with the range x ≥ 1 (thus avoiding the asymptote at x = 0), and rotating it in three dimensions about the x axis...
Posted by DeLong at January 29, 2005 03:18 PM
I love the wiki page and use it relatively often in my first year undergraduate classes but, I would love to find a page compiling problems that relate to single and multi-variable calculus (e.g. Xeno's, Gabriel's horn, plus lots more). Do you (or anyone else out there) know of a such a collection?
I know a bunch of them but there are surely many many more...
Posted by: Scott Pauls at January 29, 2005 04:23 PM
This is all fun stuff -- a great Saturday afternoon diversion (man, that marks me as a geek).
I wish I had a kid to teach this stuff to. Time to invite the niece over and torture her... (well, I hope she'd enjoy it, but who knows).
Posted by: Timothy Klein at January 29, 2005 05:06 PM
I read with interest the question of the survival odds of a WWII bomber pilot. You might be interested to know that both the eminent British military historian, John Keegan, and the Allied Military got the answer wrong.
Apparently, during one part of WWII the odds of getting shot down were 1 in 20. The military would not raise the flight limits during that time because they believed (or, perhaps, felt that the pilots would erroneously believe, that anything above 20 flights was tantamount to a suicide mission. Interestingly, the pilots often resisted flying beyond the limits even when ordered to by dropping their bomb loads short of the assigned targets and simply returning to base.
Posted by: Stuart Levine at January 29, 2005 05:37 PM
Interesting addition to the problem. Hmmm.
Posted by: anne at January 29, 2005 05:54 PM
This is a GREAT page! Now I can put off work on my dissertation for another few days. As for wiki--NEVER go there; You'll get sucked in and won't return till...
Posted by: ac at January 29, 2005 06:28 PM
Can math be fun? Two-and-a-half words:
Texas Hold 'em.
Posted by: Matt Davis at January 29, 2005 07:12 PM
Posted by: at January 29, 2005 09:23 PM
It strikes me that Gabriel's horn can be represented by human lungs.
Posted by: Dave of Maryland at January 30, 2005 06:10 AM
In the Caesar example, this statement should not have the negative: "Since there are 2x10^-22 different molecules..."
Otherwise, great stuff.
Posted by: Will Kiblinger at January 30, 2005 06:15 AM
Brad DeLong's collaborative website writes:
> Small business economics.
...and then goes on to describe Little's Law, which I really do think needs to be cited by name here.
Anyway, these ideas really are great, but most of them seem to be only awesomely great if you use them early enough. By the time you reach is in college (or at least a large public university) the "math can be cool" gene, if there is one, has been permanently switched off in most people.
Anyway, here' one more.
Materials: could be hand drawn by kids, or you
might provide game sheets with 21 Turkeys on them,
arranged in 5 rows of 4 turkeys (pictures or just the letter "T") and one turkey in the last row.
Rules: Players alternate crossing off 1, 2, or 3
turkeys from the game sheet. The person who crosses off the LAST turkey LOSES.
Purpose: An interesting game because you can show that with optimal play it is a forced loss for the
first player. Once the insight is gained also the
9 Turkeys, 25 Turkeys, 41 turkeys, and, heck, even 401 Turkeys are also forced losses, I would venture that everybody in the class would have learned something about m*****r a****m***c. (If you haven't already, now you have. :-))
Posted by: Jonathan W. King at January 30, 2005 06:25 AM
Can you do predator/prey without differential equations?
Posted by: jam at January 30, 2005 06:48 AM
There's a definite lack of algebra/number theory type puzzles here, which is strange considering how much more approachable problems in discrete mathematics tend to be, and the odd bit of topology or geometry-related stuff would also be nice. I realize that calculus is useful for economists and engineers, but the above list gives the misleading impression that it constitutes the entirety of mathematics.
Posted by: Abiola Lapite at January 30, 2005 07:14 AM
Mendelbrot's "Fractal Geometry of Nature" and related fractal-art books could motivate someone of a more artistic or literary disposition. One of the things that come from that book is that many that we think of as intuitive and "irrational" can actually be mathematically represented -- odd textures, irregular patterns, and erratic, unpredictable processes.
Kids will not start off their math study with fractals, but the Lorenz equation is just three coupled differential equations. Mandelbrot's pictures could be a motivator.
When I was young math was taught to me from a pure engineering point of view, within a deterministic control metaphysic, and I resisted it. If math had then been taught in a more interesting way I'd be much smarter now than I am. (Mandelbrot and Prigogine both said that mathematicians simply ignored equations that didn't work netaly and deterministically, even though they proved to be very useful once looked at closely.)
Posted by: John Emerson at January 30, 2005 07:55 AM
I was good at math in school but never had much exposure to the wonders of "e" which is prominent in the questions I've looked at here.
Posted by: sm at January 30, 2005 08:14 AM
"How strong is this force? If we apply such a force to a one-kilogram object for one second, it will give the object a speed of 6.67 x 10-11 m/s. Recall that there are 86,400 seconds in a day, and 365 days in a year. Multiplying, we see that a speed of 6.67 x 10-11 m/s is a speed of 0.002 meters/year--or two millimeters a year."
This is very misleading. If the two 1kg masses were isolated in space, as this seems to assume, then an initial separation of 1 meter would result in a collision in less than 2 days. (They are accelerating towards each other with increaing acceleration -- the differential equation describing the motion is d^2u/dt^2 = -2 G / u^2)
Posted by: ErikR at January 30, 2005 08:23 AM
Someone needs to get inside the heads of formerly-math-averse post-menopausal women. There really is a change to a passion for accounting, math puzzles, and doing complicated sums in your head while walking the dogs. When it happened to me, I found myself collecting teach-yourself-algebra books and dumping the novels.
It's de riguer for us elderly arithmeticals to invest in a mega-lottery ticket now and then just for the pleasure of fantasizing/calculating: a) the total win minus penalty for refusing 20 year installments, b) minus the IRS' bite, and c) what a good portfolio would look like with x percent in bonds, another x in blue chips, and finally some adventurous growth stocks. Move on to income on investments, the tax bite minus charitable donations, etc. etc. All in one's fuzzy head. Sometimes while driving. (Now you know what's going on in the car ahead, driven by a seniorita at varying speeds while jumping lanes.)
I bet this is fodder for Larry Summers. Don't care. Fact of the matter is, it happens to us old trouts.
PS: Geometry? Fuggedabahtit! (Wimmins' brains can't imagine a solid shape from all sides, etc.)
Posted by: PW at January 30, 2005 08:37 AM
Gee, Stuart, do you think there might have been another reason than a math error? Like, I don't know, maybe the fact that 20 missions with 5% chance of death each, results in less than 36% chance of survival? More than 64% chance of dying doesn't sound so good to me.
Posted by: ErikR at January 30, 2005 08:41 AM
All these are interesting problems, but I wonder whether for some people that may not be the best approach. Mathematics after all has an esthetic appeal as well as practical uses.
I discovered this when I first took a course in geometry. I don't know how it's taught now, but back then it was axioms, theorems, proofs, with only a smattering of applications thrown in. For me the power of this sort of abstract logic was striking and fascinating. More than forty years later I still remember the impression it made.
Do we emphasize this sufficiently today, or are those of us who like this sort of thing not the problem?
Posted by: Bernard Yomtov at January 30, 2005 08:44 AM
Can someone please explain Gabriel's horn so that I can visualize the solution. I can do the calculus, but I can not visualize or conceptualize the problem. I have tried.
Posted by: Ari at January 30, 2005 09:10 AM
Convincing kids that math is important should be easy, and your examples should be enough to do it. Convincing them that it's interesting is another story. Most people just aren't interested in something as dry and abstract as mathematics.
Almost anybody would be turned off of math after taking K-12 math classes, and maybe some of them would actually like it if they had some exposure to what mathematicians actually do, instead of spending years imitating textbook examples in preparation for standardized tests. Then again, some wouldn't, and there's nothing wrong with that.
Posted by: rps at January 30, 2005 09:25 AM
B. Yomtov wrote what I was thinking. Math should provide an intrinsic pleasure. When my kids were little, something like 5 and 7, they were fascinated by converting numbers into different bases and, especially, by fractions represented by digits to the right of the "decimal" point. They especially loved a story their teacher told of the binary units having to go through a small door where they repeatedly got chopped in half. And they enjoyed competing to see which one could count on their fingers faster in binary. (Each finger represents a different place value with up and down being one and zero.)
Posted by: SusanJ at January 30, 2005 10:30 AM
Bridges of Konigsberg isn't the 'fundamental' problem of graph theory, just the first one.
Even though I'm a former graph theorist, I'd be hard-pressed to say whether graph theory even HAS a 'fundamental problem'. Maybe the P/NP problem is closest - can Hamilton cycles, etc. be found in polynomial time, or not? Since that problem's just one problem in a very large equivalence class of similar problems, that's what it would make it more fundamental than other problems in the field.
Posted by: RT at January 30, 2005 10:33 AM
I'm not sure I agree that the best way of making math fun is to turn computational problems into word problems. I think the most joyous, exulting times in mathematics are when you realize some powerful concept that makes you feel like you've known it all your life. (Why are there infinitely many primes? How d'ya find the eigenvalue or eigenvector of a matrix? Why do the rational numbers, integers, and natural numbers all have the same cardinality?)
Posted by: Julian Elson at January 30, 2005 11:53 AM
thanks brad, math has bored me since the 6th grade but those were some awesome problems. and i asked the same question while reading "Pride and Prejudice" (much less boring than science!) Thanks for the weekend reading.
Posted by: matt at January 30, 2005 11:59 AM
You should put this in a wikibook! http://en.wikibooks.org
Posted by: Macneil at January 30, 2005 12:14 PM
Concerning One Hundred Interesting Mathematical Calculations, Number 3: The Strength of Gravity, here's where I am stuck.
If 6.02 X 10^24 kilograms is the mass of the earth then 6.02 X 10^21 is the mass of the earth in metric tons. If we round the number of metric tons to 6.0 X 10^21 and divide by the number of people on the planet, 6.0 X 10^9, then we end up with the ratio of 1.0 X 10^12 metric tons of earth-stuff per person. That's a trillion metric tons of earth-stuff per person. Brad seems to come up with 1.0 X 10^15 (one quadrillion) metric tons of earth-stuff per person. By my calculation that would be the ratio for kilograms of earth-stuff per person.
Upon further review of the old post one of the commenters did address errors and it appears Brad corrected the the one about total metric tons but failed to correct the one about tons of earth-stuff per person. Anyway, if I got this wrong would someone show me the errors of my ways and if I got it right would someone (Brad?) edit the old post and stop frustrating the odd dummy or two of us who are devotees of this site.
Posted by: CMike at January 30, 2005 12:14 PM
Thanks Brad, I'll be referencing this post for about 2 weeks to figure it out (already bookmarked).
Posted by: peBird at January 30, 2005 12:18 PM
Should math be fun ?
Posted by: Andrew Boucher at January 30, 2005 02:02 PM
The graph of a single bump, zero outside a finite radius, in the plane has infinite surface area and finite volume, no? Or am I missing something?
Posted by: Ralph at January 30, 2005 02:44 PM
Here's another one, which I made up myself. It is perhaps not mathematically interesting enough for your idea here, but it is certainly timely.
About how many days would it take to fill one empty, hollow, (ex-) World Trade Center tower with crude oil at the current rate of U.S. crude oil consumption?
Data: Each WTC tower was unvarying in horizontal (eg., plan) cross section all the way to the top. The base area of each tower was one acre. The roof height of each building was 415 meters.
The whole world uses approximately 80 million barrels of oil per day.
An official barrel (abbreviation: bbl) of oil is defined as 42 U.S. gallons.
The U.S., with 5% of world population, uses approximately 25% of that crude oil.
(That's right. The rest of the world -- not part of this math problem, but just so you'll know -- 95% of the global population -- gets to divvy up the other 75% of the crude oil.)
That is all.
Hint: do use Google's units calculator -- but it does not know the "bbl" unit, so substitute 42 gallons.
Google's calculator works like this:
Q: 1 cup in ml
A: 236.588237 ml
Q: 1 cup / 236.6 ml
(note that this last result is a "pure" or unitless number, since both dividend and divisor are in units of volume)
Posted by: Ralph at January 30, 2005 03:24 PM
How about: You have $20 and you want to drive to the mall which is 50 miles away. The car only has a quarter tank of gas. Do you have enough money to buy gas and stop for a McD's value meal on the way home?
Posted by: pragmatic_realist at January 30, 2005 05:13 PM
This proposal (how to illustrate "Crucial Math") is a very good idea.
But, if we are *really* serious about this - we need more of an emphasis on sex and/or money. Math can be applied to both easily.
Like it or not (and I don't particularly like it) there is a reason why TV focuses almost exclusively on these two topics (namely, evolution has ensured that these issues will immediately get our attention).
Money is easy. Any example of compound interest will illustrate how easy it is to become a millionaire if you start saving/investing at 15. Most kids/teens (like adults) are obsessed with money. Math mission accomplished.
Sex may be a little bit trickier.
How about calculating the odds of pregnancy (sexually transmitted disease?) per incident of unprotected sex?
Or for the Match.com crowd - how about examining some of the research looking into "the mathematics of beauty" - that is, how calculable facial symmetry plays a *big* role in commonly held definitions of beauty...
Just some thoughts...
Posted by: Chris Sgarlata at January 30, 2005 06:45 PM
Some comments about Caeser's last breath...(I cannot leave a comment on that page - I get a server error)
If you make the student calculate the number of molecules in Caeser's lungs and / or the number in the earth's atmosphere (a bit harder!) you have a very nice freshman chemistry problem.
The mixing time for the troposphere, which contains ~90% of the molecules of the atmosphere, is only a couple of years. So one can make it a bit more current when one selects the dying person....
I guess I should read innumeracy...but it is interesting to think about the teaching goals here. Typically one does not learn such approximation techniques without learning where they are from (I think the innumeracy approach disagrees with this?). My instinct would be to either attack the large exponential with logs, or to actually explain the taylor series.
Thanks to Jack (see the comment on that page) for a good mnemnotic.
Lastly, can I vote for right-justified comments and non-invisible links?
Posted by: BoulderDuck at January 30, 2005 08:07 PM
In a nutshell: the volume is finite because the series 1/n^2 (i.e., 1/1^2 + 1/2^2 + 1/3^2 + ...) is finite. The surface area is infinite because the series 1/n is infinite.
Posted by: Yoram Gat at January 30, 2005 08:57 PM
<Can someone please explain Gabriel's horn so that
<I can visualize the solution. I can do the
<calculus, but I can not visualize or
<conceptualize the <problem. I have tried.
Think in terms of a sphere. The volume varies as the cube of the raduis, but the surface area varies as the square of the raduis. You might think this means the volume is much larger than the surface area, and this is true when the raduis is big, but when the raduis is small, the surface area is much larger than the volume.
(raising a small number to a high power is a form of euthanasia)
In Gabriel's horn, if you look at a cross section of the horn (perpendicular to the x-axis) you see a circle. The further out you go, the smaller the radius is. The volume is obtained by "adding" the areas of all the circles, which goes to zero fast since the radius is going to zero and the area varies with the radius squared. However the surface area is obtained by
"adding" all the circumferences (which goes to zero slower).
The point is that the surface area diverges like
the series 1+ 1/2 +1/3 +1/4 +......
(this is called the harmonic series, and this may be what is causing your problems. Just becase the sequence (1/n) is going to zero doesn't mean the harmonic series converges)
while the volume coverges like the seres
1 + 1/4 +1/9 +1/16 +....
Posted by: elspi at January 30, 2005 09:04 PM
Best math problem:
How can you tell that the sun is not as dense as the moon. Hint: How big do they look, and which one causes the tides.
Posted by: elspi at January 30, 2005 09:26 PM
From the science department, physics (math) problems that are light-hearted and not dated:
Posted by: Link at January 30, 2005 09:33 PM
Elspi, in response to your "best" math problem:
If I hold a nickel at arm's length, it looks as big as the moon. My nickel does not cause tides. Therefore, the moon is more dense than my nickel.
Or, if I were lying on the ground, looking up, your head might appear to be the same size as the moon. Therefore, you are less dense, so are not responsible for the tides. However, being less dense, you might float on a pond, like small rocks (or even a duck), and therefore should probably be burnt as a witch.
For some real info on tides, the sun and the moon:
Posted by: Huh at January 30, 2005 09:54 PM
You played craps in middle school? You gangster.
Posted by: Carleton Wu at January 30, 2005 09:54 PM
A great list, but I have a problem with your explanation of the Grass is Greener paradox. A good analysis shouldn't require any assumptions about finite amounts of money or intuitions about amounts being hig or low (unless you want your kids to turn out to be Bayesians). The real problem lies in confusion among x, 2x, and x/2. To do it right, just describe that in one envelope is x dollars and in the other is 2x dollars. Switching envelopes gets you 2x-x dollars if you got the big one first (with a probability = 1/2) and gets you x-2x dollars if you got the small one first (with probability = 1/2). Thus a net of nothing. As it should be. People get confused by always taking what they have to be x and the other being either 1/2 or double that. What isn't accounted for is that the value of x is different for those two cases.
Posted by: AP at January 30, 2005 10:00 PM
"Switching envelopes gets you 2x-x dollars if you got the big one first (with a probability = 1/2) and gets you x-2x dollars if you got the small one first (with probability = 1/2)."
Of course, I just got that backwards.
Or you can think about the generalization (for a more serious brain chug). The woman just tells you that there are two envelopes with money in them without telling you anything about the amounts at all. You could claim that one has x dollars and the other has alpha*x dollars, and you'd be off on the same goose-chase as before since 1/2*((alpha*x)+x/alpha) is larger than x for any alpha greater than one. Alternatively, if you formalized it by saying that one has x dollars and the other has x+alpha dollars, you avoid the whole problem since 1/2*((x-alpha)+(x+alpha)=x for any alpha. And that the two methods suggest different answers you know something's gotta be whack. And it is. The first method is bogus for the reason described above.
Posted by: AP at January 30, 2005 10:10 PM
Sure math can be fun (and just looking at the listed problems brings back memories of skills I used to use much more often - very head-clearing!). OTOH, there's nearly zero incentive for American kids to play with math. I used to work on an NSF-funded project at the University of Chicago that sought out Eastern European (esp. Soviet) mathematical texts - many of them recreational and aimed at kids - for translation and subsequent publication here. There was nothing comparable to that literature in the US 25 years ago, and I daresay there isn't now.
Unless you're trying to calculate how many angels can dance on the head of a pin, I don't see this as a values issue in education, either. One of the attracttions of math and the hard sciences in the Old Evil Empire was how their study could skirt the brain-shriveling orthodoxy. Something to keep in mind now, for us.
Posted by: grishaxxx at January 30, 2005 10:29 PM
Some late night, possibly idiotic, brainstorming: isn't the real challenge how to convince teenagers that things other than math etc. are not so interesting?
My wise wife, the other night, was remarking that there is not a single toy in house that isn't educative. My 3 year old daughter is getting a blast from learning how to read and count. But, then again, I know that is to go away when she will interact with other teenagers... or does it need to be that way?
Tonight she saw me and a friend playing music and my wife told me she looked at us like she thought we were gods. I think we were genuinely terrible, but maybe there is something to be learned from this experience... i.e. she needs to see me enjoy and perform the things I'd her to...
Posted by: Jean-Philippe Stijns at January 30, 2005 10:51 PM
I always like volume calculations: if Evans Hall was a (water) tank filled with orange juice, would there be enough to give everyone in the US a glass? And also (slightly off color), when the train is in the station, you cannot use the toilet, but you can when it is moving. At 30mph, what is the waste flux on the track (the toilets drain right to the track).
As to 6: Xeno's paradox NO NO NO. It is not that the greeks could not sum infinite series. I do not know if they could be that is not relevant (as to that point, google eudoxus -- but he was quite later than Xeno). Xeno had four paradoxes to cover the four possibitilies: time and distance are each either continuous or have a minimum quantum (as we would say), making 4 possibilities.
The paradox we are all used to (tortoise and achilles) is time continuous, distance quantum -- since the tortoise was moving and has to move at least a quantum each time period, when you get time periods small enough, that means the tortoise is always at least distance quantum ahead.
Cannot get anywhere is distance continuous time quanta: it takes at least time quanta to go 1/2 the distance, and time quanta to go 1/4, and time quanta to go 1/8, etc. Because time quanta are a specific finite but non-zero and NON-DECREASING number, Xeno was able to add the infinite series and get infinity correctly, so we cannot go anywhere.
Time and distance quanta: the chariots passing each other in opposite directions.
Each continuous: the arrow always actually stopped in flight (newton would say Xeno did not undertstand force and momentum, which is true).
Xeno had four paradoxes to show motion was impossible. One presumes he still got out of bed and went to the market and the agora, so his conclusions were a comment on the inadequacy of science to describe phenomena, but they had NOTHING to do with being able to sum infinite series.
Posted by: David Margolies at January 31, 2005 09:48 AM
Here is a problem that I gave to my girlfriend's little brother and sister.
In two dimensional space, one line cannot make an intersection point, two lines can make at most one intersection point, three lines can make at most three. What is the maximum number of lines that can be made by an arbitrary number of points?
This problem is of more abstract concern, it deals with graphing, and it is a good example of using recursion to figure out a problem.
Posted by: Andy at January 31, 2005 01:44 PM
Correction. That last line should be:
What is the maximum number of intersection points that can be produced by an arbitrary number of lines?
Posted by: Andy at January 31, 2005 01:48 PM
Most glaring omission: Monkeys on typewriters. How many monkeys (or randomly typing monkey robots designed by Fafblog) would we need to type "small cheer and great welcome makes a merry feast " with their first 50 keystrokes on a simplified 27-key Fafwriter if monkey robots randomly type at X keys per minute?
The "fundamental" graph theory problem is probably the four color map.
A simple automata theory problem: You have three playing cards in front of you, with either face (F) or back (B) up. You can turn either left and middle (LM) or right and middle (RM) card. You can't turn single cards, all cards or left and right card.
From the starting configuration FFF, create a decision tree with the actions LM and RM that reaches: 1. BFB, 2. FBF, and 3. BBB.
Now try to solve the same problem using an automaton.*
*An automaton is a decision tree where same states are represented by the same node.
Posted by: ogmb at January 31, 2005 04:21 PM
"What is the maximum number of intersection points that can be produced by an arbitrary number of lines?"
Looks more like an exercise in algebraic geometry than graph theory to me ...
Posted by: Abiola Lapite at January 31, 2005 08:24 PM
I forgot to mention the obvious answer to the challenge: n*(n-1)/2.
Posted by: Abiola Lapite at January 31, 2005 08:45 PM
"If the economy grows at 3% and the s&p 500 at 5% and P/E stays constant at 20, when will profits be all of GDP?"
Useful trivia: take growthrate and divide 72 by it to get number of periods to double. In this case, 2% extra growth will get a doubling in 36 years, so, taking earnings at 12.5% of GDP today, it has to double 3 times (25%, 50%, 100%), so in 108 years, profits will be all GDP.
Posted by: Dinsky at February 1, 2005 11:50 AM
I didn't say that it was graph theory, only that it is easily solved with a graph. And yeah, the answer isn't extremely hard if you know what you're doing, but I wouldn't expect a child to come up with the closed form solution like that. I would be most surprised if a child came up with the recursion...
Posted by: Andy at February 1, 2005 09:02 PM
Posted by: at February 17, 2005 06:04 AM