## 350-Year-Old Newton's Puzzle Solved By 16-Year-Old 414 414

First time accepted submitter johnsnails writes

*"A German 16-year-old, Shouryya Ray, solved two fundamental particle dynamic theories posed by Sir Isaac Newton, which until recently required the use of powerful computers. He worked out how to calculate exactly the path of a projectile under gravity and subject to air resistance. Shouryya solved the problem while working on a school project. From the article: 'Mr Ray won a research award for his efforts and has been labeled a genius by the German media, but he put it down to "curiosity and schoolboy naivety." "When it was explained to us that the problems had no solutions, I thought to myself, 'well, there's no harm in trying,'" he said.'"*
## That Moment (Score:5, Interesting)

We all had that moment in school when a teacher would pose an "impossible" problem, thought to ourselves "Well, they've never faced ME before!", spent a few minutes toying with it and finally giving up. This kid...did not.

Kudos all around! The rest of your life will, unfortunately, now no longer live up to something you accomplished when you were 16.

## Re:That Moment (Score:5, Interesting)

There are two things impressive about this. One is the fact that you mention, that the kid did not give up until he had the solution and was smart enough to solve a problem that stumped every mathemetician for 350 years. The second is that

people still try to solve difficult analytic problems at allinstead of just turning it into a computing problem.I don't know which surprises me more.

## Fermat & Poincaré (Score:5, Interesting)

## Re: (Score:3)

Of course, that's coming from a guy who continually messes up a number or sequence here or there.

## Re: (Score:3, Insightful)

If you're doing a half page proof that square root of 2 is irrational, then a computer would be pointless, but clearly you don't know that math is more complicated than that.

And to head off potential flames, I completely respect people who want to and are able to work through tho

## Re:Fermat & Poincaré (Score:4, Insightful)

Half a page? If (x/y)^2 = 2, then x^2 = 2y^2, so x is even. Let z = x/2, now we have 2z^2 = y^2, so y is also even. Thus, any fraction that's equal to the square root of 2 cannot be expressed in lowest terms, so cannot exist. That's, what, three lines at most?

I agree with the main point, though; quite a few of the proofs I do are just boring churning through tens of possible cases. Up to 100 or so it's plausible to do it by hand, although tedious and it's easy to make mistakes; significantly beyond that, though, you're going to want to automate it.

## Re: (Score:3)

It seems pointless to you because you are totally ignorant of math. A lot of these "hundreds of pages of mind-numbingly dense mathematics" proofs are long but tedious derivations which a computer can grind through in seconds.

If you're doing a half page proof that square root of 2 is irrational, then a computer would be pointless, but clearly you don't know that math is more complicated than that.

And to head off potential flames, I completely respect people who want to and are able to work through those derivations by hand, but to think doing it with a computer is pointless just shows your ignorance.

Most importantly, if there are hundreds of pages of dense computation to prove X, if I'm writing a function and I have some invariant, I can just write a comment,

"And invariant Y remains satisfied because of X, see the fun proof at..."

I don't really give a damn about the details, It Just Works.

## Re:Fermat & Poincaré (Score:5, Interesting)

Another famous example is Grigori Perelman who solved the Poincaré conjecture - with hundreds and hundreds of pages of mind-numbingly dense mathematics vs computer search.

Perelman's three primary papers ("The entropy formula for the Ricci flow and its geometric applications" http://arxiv.org/abs/math.DG/0211159 [arxiv.org], "Ricci flow with surgery on three-manifolds" http://arxiv.org/abs/math.DG/0303109 [arxiv.org], and "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" http://arxiv.org/abs/math.DG/0307245 [arxiv.org]) on modifying Hamilton's Ricci flow program to deal with singularities and proving Thurston's geometrization conjecture only span 68 pages, with the actual proofs/meaningful remarks comprising about 45 pages of that.

## Re:Fermat & Poincaré (Score:4, Informative)

Well, that is true but then after those papers appeared there was a several year effort by 3 groups to fill in the details and make it more digestible. Each of the resulting books/documents are several hundred pages long.

Some problems just require longer proofs.

## Re:Fermat & Poincaré (Score:4, Insightful)

This is why I read /.

## Re: (Score:3)

## Re: (Score:3, Insightful)

Computing tends to be a brute force analysis of all the possible inputs. That doesn't work well for NP hard problems and is often impossible with problems dealing with infinity... Not all problems are solvable by computers yet and instead need the analytical approach. Also computers may not find the most elegant solutions, for example there are problems which have been solved but required the invention of a new type of math to do so.

## Re:That Moment (Score:5, Insightful)

Computing tends to be a brute force analysis of all the possible inputs.

Hello? We've had symbolic computing ever since 1960's. There are many software tools today to assist mathematicians with creating and verifying proofs (e.g, Coq is probably the best known one). What's wrong with using them? Not to do that would be like using a pencil and paper instead of typing when you're preparing a publication – I'd think that brain power and time should be used constructively.

## Re: (Score:3)

Business wise I'd type, fiction wise I prefer to use pen and paper.

## Re:That Moment (Score:5, Funny)

I just wanted to say that I LOVE Coq.

## Re:That Moment (Score:4, Insightful)

What's wrong with using them? Not to do that would be like using a pencil and paper instead of typing when you're preparing a publication – I'd think that brain power and time should be used constructively.

It's not a matter of the tool is wrong. It's a matter that assuming one tool is always best is wrong.

Your premise is based on: using a computer is easier and better for 100% of humans. That's not true. Allow me to introduce you to my parents. Allow me to introduce you to senior engineers who can craft new formulas on a whiteboard faster than juniors can wake their laptops.

Different areas of the brain are involved with the act of handwriting than with touch typing or pecking. Make LCARS speech recognition a reality and we have a winner. Solving problems that stump otherwise intelligent humans for *hundreds* of years, *clearly* requires some creatively alternate use of the brain, and not Microsoft Clippy. ("I see you're trying to solve an unprovable theorem, would you like to Quit without Saving?") I don't even need to cite sources that say poor UIs slow people down. That's how it is. Computers add cruft, otherwise there wouldn't be a market for applications that remove distractions when writing.

...like using a pencil and paper instead of typing when you're preparing a publication...

Poor analogy. Publication implies mass reproduction and distribution. An *author* can write however they want to form their ideas, the result is the same. How the idea gets distributed is irrelevant to the core point. (Also there are such things as shorthand.)

## Re: (Score:3)

They are not helpful. Automatic proof, or automatic proof-verification is a research field, and has been so for decades, and has still YET to come up with something helpful to anyone doing real mathematical proofs. They have only barely reached the ability to help with play-thing problems handed to high school students, and even them the computer generated result (or input), is obtuse and stupid - not helpful in any way.

## Re: (Score:3)

is your software proven to be fault-free?

is your hardware proven to be fault-free? (have you ever read the list of errata for a modern CPU?)

if your software puts out a human readable (and verifiable) form of the proof everything is fine, if not you can only talk about probabilities not about proof.

First, you have to ask the same thing about mathematicians' brains and abilites (organic HW and SW). Second, of course that a proof assistant has to be able to print out a trace of the reasoning process. That's the whole point (or one of them).

## Re:That Moment (Score:5, Insightful)

Ha! Inventing a new mathmatical system in order to solve a problem is cheating! But it works.

Not only is it cheating, it's tradition. We have many great branches of mathematics because of it.

## Re: (Score:3)

In fact, Newton did this himself.

I recall a story of some mathematical puzzle or hypothesis which had been unsolved by a number of mathematicians for many years. It was brought to Newton's attention, whereupon over the course of a few days (maybe a weekend?) he invented a new branch of mathematics and solved the puzzle. He published his results anonymously, but no-one was fooled and immediately (if somewhat resignedly) congratulated Newton on his genius (again).

Can't remember the hypothesis or the resulting

## Re:That Moment (Score:4, Interesting)

I suspect you're thinking of the brachistochrone problem, posed by Johann Bernoulli in 1696, and solved the next day by Newton (also by several other mathematical giants of the time, very quickly).

## Re:That Moment (Score:5, Insightful)

Analytic solutions are far superior to computed approximations. They are far easier to calculate--computers have made computed approximations far easier, but most of the time that doesn't mean that they're *easy*--only that they're now possible. Being able to obtain the answer in a small fraction of the time is still a big advantage. They are more precise and do not require initial parameters. And they provide much greater understanding and insight into the underlying phenomenon. There is no surprise at all that people are still looking for analytic solutions.

## Re: (Score:3)

It depends on what you are trying to do. I'm a mechanical engineer and engineering is all about good enough. You have to economize resources to get a job done. While solving a dynamics problem analytically may give you more understanding into the solution it does take take to work out real world multidimensional problems. Numerical solutions to differential equations are very useful. I would have preferred to spend more time in Diff Eq setting up problems than solving problems analytically.

## Re: (Score:3)

Analytic solutions are far superior to computed approximations. They are far easier to calculate--computers have made computed approximations far easier, but most of the time that doesn't mean that they're *easy*--only that they're now possible. Being able to obtain the answer in a small fraction of the time is still a big advantage.

Have fun with solving the Navier-Stokes equations then ;)

## Re:That Moment (Score:5, Insightful)

Analytic solutions are

notnecessarily easier to calculate.Analytic solutions tend to involve special functions for which the computer can only compute an approximation anyway. Have you ever tried to write code to evaluate the error function over the entire domain of floating point numbers? (Yes, I know, it's now in the standard library; ten years ago, it wasn't.) That's one of the easier ones.

Even if there are no special functions, analytic solutions are still often harder to calculate if the problem is big enough. Think of solving systems of linear equations, one of the standard workhorses of numeric programming. We're talking really big ones; hundreds of thousands of equations in hundreds of thousands of unknowns or bigger. In the real world, this problem would almost certainly be solved using successive approximations,

even though high school students know how to solve them analytically.Finally, and most importantly, the

problem statementis usually an approximation. Take the OP as an example. What this kid almost certainly solved was an analytic solution to the problem of a particle in a gravitational field with linear air resistance. Well, air resistance is not linear. At low velocities, and for projectiles with a sufficiently small cross-section, it's close enough. But it's still an approximation.The advantages of analytic solutions are almost always not computational. What they buy you is

understanding. The methods of obtaining the solution, and the form of the final equations, often reveal some deep insights about the problem. For many situations, that's far more valuable. And it's certainly something that no computer can give you.## Re: (Score:3)

There are two things impressive about this. One is the fact that you mention, that the kid did not give up until he had the solution and was smart enough to solve a problem that stumped every mathemetician for 350 years. The second is that

people still try to solve difficult analytic problems at allinstead of just turning it into a computing problem.I don't know which surprises me more.

^^^^ This.

I think the most impressive part is that even though we hear all the time about "X-teen year old invents BLAH" we're like "Great!" but secretly think "BIG Deal! Who can't invent something? How is that challenging, really? Oh look their dad's an electrical engineer that works at XYZ... hmmmm....." but this 16-year-old actually solved something that the best mathematicians on Earth haven't been able to solve for 350 years.

Major kudos kid! Only way that can be topped is if a teen cures canc

## Re:That Moment (Score:5, Interesting)

## Re: (Score:3)

It's people like this kid who pop-up very rarely in the world that will eventually improve our mathematical understanding and all the technology based on advanced mathematics. That's a positive thing.

## Re: (Score:3)

A common mistake that people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools. (Douglas Adams)

## Re:That Moment (Score:5, Funny)

The rest of your life will, unfortunately, now no longer live up to something you accomplished when you were 16.

Imagine the freedom of no longer having to live up to anybody's expectations. ;)

## Re: (Score:3, Interesting)

## Gave Up (Score:3)

And the submitter gave up right while copying the name of the kid from the article to slashdot.

"Shouryya Ray" became "Shouryya Ra" and samzenpus also let it through without any corrections.

## Re:That Moment (Score:5, Informative)

Germany still produces some rays of light.

To be accurate... he was born in India and moved to Germany with his family at age 12. He did not speak a word of German when he arrived.

While credit must be given to the German school system, I think most of his accomplishment comes from him and possibly his family.

## Re: (Score:3, Insightful)

While credit must be given to the German school system

Must it? The school system could be garbage and still have the occasional intelligent person go through it. Perhaps it's not the school system that must be given credit, but something else (like the child himself, for instance).

## Re:That Moment (Score:5, Interesting)

Germany still produces some rays of light.

To be accurate... he was born in India and moved to Germany with his family at age 12. He did not speak a word of German when he arrived.

While credit must be given to the German school system, I think most of his accomplishment comes from him and possibly his family.

And maybe from not being in Europe or the western world the first twelve years of his life, adopting beliefs or creating a mental attitude that stuff like this cannot be done. And I'm not criticizing the Germans.

## Re: (Score:3)

Possibly relevant here (in some minor way) is that thinking in a foreign language allows people to be more rational [wired.com].

## Re:That Moment (Score:5, Informative)

## Re:That Moment (Score:5, Informative)

I'd reserve your hosannas until this kid's magic formula gets published, along with a formal statement of the problem.

the formula has already been published, here: https://www.jugend-forscht.de/images/1MAT_67_download.jpg [jugend-forscht.de]

(photo of the formula taken on May 18th)

article source:

https://www.jugend-forscht.de/index.php/projectsearch/detail/6038.4568 [jugend-forscht.de]

and

http://www.jufo-dresden.de/projekt/teilnehmer/matheinfo/m1 [jufo-dresden.de]

i can't find the full paper yet though, but on reddit some users claim that the formula works in Maple

e.g.

http://www.reddit.com/r/worldnews/comments/u7551/teen_solves_newtons_300yearold_riddle_an/c4szejb [reddit.com]

where f is constant on the path the particle makes in the space of velocities: /(2*u^2 ) + a*(g/2)*(v*sqrt(u^2 +v^2 )/(u^2 ) + arcsinh(v/u)));

f:=(g^2

## terrible article (Score:5, Insightful)

The article itself is mathless. It doesn't tell you what the solution was, or even present the exact problem that was solved.

## Re:terrible article (Score:5, Interesting)

The article itself is mathless. It doesn't tell you what the solution was, or even present the exact problem that was solved.

And running a search for the kid's name turns up the same article fifty fucking times over. Google did some work on link farms... they need to do some work deduping / despamming press releases.

## Re: (Score:2, Insightful)

With all due respect to this brilliant student, I wouldn't worry too much about that - the problem isn't actually solved until its been peer- reviewed and thd other mathematicians agree that his approach is correct.

## Re: (Score:2, Funny)

The answer was 42 . . . now what was the question?

## Re:terrible article (Score:5, Informative)

You are right. This article is awful, conveying no sense of the nature of the problem or its complexity, and giving no idea of the solution at all.

The only equations I'm aware of for a falling particle subject to air resistance take the form

m v' = -mg -a*v-b*v^2which is a constant coefficient Riccati differential equation for the velocity v. I'm reasonably sure this would have an analytic solution.

Maybe complications arise in the 2D motion case, or perhaps the problem includes a particle which is also spinning. Maybe the drag terms take more complicated forms. I don't know. The article is pretty dreadful to be honest.

## Re:terrible article (Score:5, Informative)

## Re:terrible article (Score:5, Informative)

That's "Analytische lösung von zwei ungelösten fundamentalen Partikeldynamikproblemen" or, in English, "Analytical solution of two fundamental unsolved problems of particle dynamics".

But that doesn't seem to be a paper published in a peer-review journal, but rather the title slide of a presentation he gave on March 1, presumably when when he received the Jugend Forscht ("Young Researchers") award.

And the kid is Indian, not German (as long as we can tell from the article).

And this is a problem in Physics, not in Mathematics. It shocks me that people get that mixed up.

And the kid looks 30 years old, but I would never hold that against him.

## Re: (Score:3)

How does Slashdot accept such a crappy post?!I believe they welcome this stuff with open arms, and add an obscure summary with sensational headline to boot.

And slashdotters tear it aprt even while complaining. Win-win for everyone.

## I thought these were pretty much known already (Score:2)

Should make for even better gaming physics...

## Re: (Score:2, Informative)

There is no problem solving the equations numerically. This kid found analytical solutions to the equation of motion (or at least, that's how I read TFA). Punching in the exact solution is faster and more accurate than taking a zillion small but discrete steps, which is what you're stuck doing right now. Well, that depends on the complexity of the solution, but as a general rule...

## Re: (Score:2)

## Re:I thought these were pretty much known already (Score:5, Informative)

You forgot a lot of things:

-gravity is not a constant vector force downward. It is a radial force inward toward the center of the Earth, and its intensity varies with altitude.

-air resistance is not constant either. It depends on air pressure which varies with altitude as well.

-air resistance is not perfectly proportional to v^2, especially at transonic and supersonic speeds.

-if the projectile is spinning, it may cause a net aerodymamic force in a direction other than -v. Like a curveball.

-the earth is a spinning frame of reference, which results in various annoying effects.

-the air is not necessarily stationary. Wind exists.

and so on.

But we don't know whether this dude accounted for any of this stuff or not, because the goddamn article doesn't tell us.

## artillery can go 26 miles (Score:3)

Unassisted shells have pretty decent range (26 miles or so), and specialty weapons can go even further.

## Specifics? (Score:5, Insightful)

Can anyone actually find the problems in question somewhere? I've been scouring Google and the whole thing is very vague -- no story really goes into depth about the actual problem he solved and how.

## Re: (Score:2)

I've been scouring Google and the whole thing is very vague - no story really goes into depth about the actual problem he solved and how.Looks like he's just invented the recursive puzzle!

## Re:Specifics? (Score:5, Interesting)

## Re:Specifics? (Score:5, Informative)

This is an article from 1983. I believe it explains the problem.

http://www.annualreviews.org/doi/pdf/10.1146/annurev.fl.15.010183.000245

## Re:Specifics? (Score:4, Interesting)

Here's a post where someone determined what the original equations were and verified Ray's answer (in the picture of him holding a solution) in Maple:

http://www.reddit.com/r/worldnews/comments/u7551/teen_solves_newtons_300yearold_riddle_an/c4szejb [reddit.com]

## Difference between Germany and the US (Score:4, Insightful)

German media praise math geniuses, while american media praise hollywood actors/actresses (read: human rubbish) and reality show weirdos. In the US a "genius" is someone who makes millions, especially with lower education and without being able to do anything. That's "free market economy", and "supply and demand", right?

"The land of the free and of the brave" (with some fat on the belly).

## Re: (Score:3)

"In fact, neuroscience and psychology points the opposite direction: happiness leads to success. If we could grasp that one fact we'd all be better off."

That's philosophy. Let's stick to the facts: what can a math or physics genius become in the US? Maybe a university professor, making 100-150 K$ a year. Or maybe the R&D leader of a major company, but the salary would be nearly the same, the only way to get "rich" would be with stock options, which depend on factors that have nothing to do with R&D (marketing makes a company more profitable than R&D). An hollywood weirdo makes 10 millions per movie instead.

That's the obvious consequence of the mighty law of "supply and demand" that nobody wants to oppose: people are retarded and spend lots of money to go to the movies rather than financing scientific research. That's the "demand", so the "supply" will act accordingly. And who doesn't agree with this system is considered a "communist".

Now, who's more useful to mankind, a physicist or an actress? If answering "a phycicist" makes me a communist, well I'm proud to be one.

No, that's not philosophy. That's science. The facts are on my side, not yours. Read "The Happiness Advantage" for details. I'm not denying supply and demand, arguing that a physicist makes more than Tom Cruise (although in general physicists make more than actors), or anything else you might think I'm saying. I'm saying as a matter of fact, based on good science, that the human brain is generally more productive and powerful when it's happy, which leads to increased success, but having success does no

## Re: (Score:3)

The problem isn't with the proof, the problem is with the AXIOMS. Very good and convincing proofs of the existence of God are there, if you take a particular set of axioms as the basis for your outlook. That's the faith part.

So does that, ultimately, amount to "you will be convinced of the existence of God if you make assumptions about the world that require the existence of God"? Unless there's a non-faith-based reason to make those assumptions, the proof isn't going to be convincing to people who don't make those assumptions, making it just an entertaining exercise for those who happen to make those assumptions, not something to take seriously as a reason to believe.

It's a real problem in our world today that people take math and science as gospel. Everyone seems to forget that all of math and science are based on axioms, things that we assume must be correct because there's no way to prove it. We have to make those assumptions, though, to do anything at all. You might say "well, so far nothing has shown those assumptions to be wrong, so we must be right!", but that's only good to a point.

Yes, math is a subject where you start with a set of axioms

## Re: (Score:3)

"So I'm sorry, but I decline to offer my proofs."

Then you have no evidence you dare produce for debate.

## When in Doubt... (Score:5, Informative)

...go to the source! The German articles I've scoured seem to have a little more information about the problem itself and what he actually accomplished. The oldest one only records that he "claims" to have solved them (earlier this month), but so far no actual data. Close.

http://www.enso-blog.de/jugend-forscht-drei-arbeiten-aus-ostsachsen-beim-bundeswettbewerb [enso-blog.de]

http://www.morgenpost.de/vermischtes/article106358144/16-jaehriger-Schueler-loest-uraltes-Mathe-Problem.html [morgenpost.de]

## Gotcha! (Score:5, Informative)

http://jugend-forscht-sachsen.de/2012/teilnehmer/fachgebiet/id/5 [jugend-for...sachsen.de]

Text is in German. It all stems from a Youth Research competition he entered into back in March of this year. This is, so far, the best summary I've found -- there is a paper, apparently, but no link just yet.

'Two problems in classical mechanics have withstood several centuries of mathematical endeavor. The first problem is therefore to calculate the trajectory of a body thrown at an angle in the Earth's gravitational field and Newtonian flow resistance. The underlying power law was discovered by Newton (17th century). The second problem is the objective description of a particle-wall collision under Hertzian collision force and linear damping. The collision energy was derived in 1858 by Hertz, a linear damping force has Stokes (1850) is known. This paper has so far only the analytical solution of this approximate or numerical targets for the problems solved. First, the two problems are solved fully analytically. For the first problem will be investigated further using the analytical solution, the physical behavior of the system and set up outline solutions for generalized models. For the second problem is carried out in order to increase efficiency and convergence control a semi-analytical optimization. Finally, the analytical results are compared with numerical solutions so as to validate accuracy and convergence to numerically."

## Re:Gotcha! (Score:5, Interesting)

On a sad note, he only placed 2nd in the overall competition :(

## Re:Gotcha! (Score:5, Interesting)

Number one cured cancer AND solved the world's energy problem. That's hard to top. :)

## Re:Gotcha! (Score:5, Informative)

That helps a little, but still doesn't really clarify completely what he did. I'll explain a little about what I know about the projectile problem and what I can figure out about what he might have accomplished here.

In the Principia [wikisource.org], Newton poses three closely related problems. One is projectile motion under the influence of a frictional force that's proportional to velocity (book II, section I). Next he considers the case where the friction is proportional to the square of the velocity (book II, section II), and finally the case where it's of the form av+bv^2, where a and b are constants (book II, section III). Let's call these cases 1, 2, and 3.

Case 1 is pretty straightforward. The x and y motions are decoupled, and each of the motions is governed by a first-order, linear, inhomogeneous equation.

Case 2 is actually of more physical interest than case 1 for most real-world projectiles. For example, when you toss a baseball in air, its Reynolds number is about 10^4 or 10^5, and in that regime, a force proportional to v^2 is a pretty decent approximation. There is a well known closed-form solution for the one-dimensional subcase (I actually had a student a few years back who figured it out for herself, which was impressive), which is y=A ln[cosh(t sqrt(g/A))].

A hint is that this page [jugend-forscht.de] has a photo of him holding up a large sheet of paper with his closed-form solution on it. The equation is clearly visible, and reads g^2/(2u^2)+(alpha g/2)[v sqrt(u^2+v^2) / u^2 + arsinh |v/u|] = const. The notation isn't explained, but clearly u and v are the components of some vector, probably the velocity vector. If so, then the constant alpha has to have units of inverse meters.

This makes me think that what he's solved is the full two-dimensional version of case 2. It can't be case 3, because besides g there is only the one constant alpha appearing in his equation. If you write down the equation of motion, a=F/m=(mg-bv^2)/m=g-(b/m)v^2, the constant that naturally occurs is b/m, which has units of inverse meters. It also makes sense that his solution has a hyperbolic trig function in it, since the y(t) for the one-dimensional version of case 2 has a hyperbolic trig function in it.

If my interpretation is right, then you should get a correct one-dimensional result from his equation when u=0. Unfortunately his equation blows up to infinity in that case, so I'm not sure how to extract any sane interpretation from it. By setting alpha=0, you should also get the case with zero friction. That does sort of make sense, since it says u is a constant, which it should be in that case.

It would be interesting to see if my interpretation is right by doing a numerical simulation and seeing if his expression really does seem to be a constant of the motion.

One thing to point out is that he may not have actually solved the full problem as set by Newton. He hasn't found the equation of the trajectory in closed form (which I think was what Newton was most interested in), and he also hasn't found the position in closed form as a function of time. (This is all assuming my interpretation is right.)

## Re:Gotcha! (Score:4, Informative)

You're right, he's demonstrated a constant of motion (i.e. a first integral) in the 2D version of Newton's Case 2. The constant alpha in his equation is what you called b. Gravity points in the -v direction.

You can easily check this by differentiating his equation with respect to time, and then eliminating the derivatives of u and v using the expressions

du/dt = -b u sqrt(u^2 + v^2)

dv/dt = -b v sqrt(u^2 + v^2) - g

His solution can probably be extended to Case 3 quite easily, if anyone feels like a challenge :)

## Re:Gotcha! (Score:4, Interesting)

Doing a reply-to-self because I checked my interpretation using a numerical simulation. I wrote some python 3 code, which does a reasonably realistic simulation of a baseball being hit for a home run. Slashdot's lameness filter wouldn't let me post it, so I put it here: http://ideone.com/yeP4y [ideone.com]

The results:

The notation is that u and v are the x and y components of the velocity vector, "Ray" is the expression that Ray seems to be claiming is a constant of the motion, and the final column is the total energy, which should be conserved.

I tested my code two ways: (1) Energy is very nearly conserved. (2) If I turn off air friction, the range is very nearly as calculated by theory.

Let R be the expression that Ray says is a constant, under my interpretation of his variables. Then dR/dt appears to be very nearly zero early on in the simulation. However, later on it starts to drift upward. So I suspect that one of the following is true: (1) Ray is wrong; (2) my interpretation of his notation is wrong; or (3) my simulation doesn't use good enough numerical techniques to demonstrate with good precision that Ray is right.

Anyone who's got Runge-Kutta, etc., on the tip of their tongue want to try a better simulation of this?

## Re: (Score:3)

## Flash journalism (Score:5, Insightful)

Think about that for a moment. A story supposedly lionizing science lacking the most basic facts that would permit substantial verification, or falsification, of that science. This is just flash journalism at work.

## Re:Flash journalism (Score:5, Informative)

Since we're linking to comments from Reddit: people also found out that this solution was known since at least 1860 [reddit.com], and was published in a modern journal in as recently as 1977 [reddit.com].

It's great that a 16 year old discovered this, and it could have been a cute (but not as flashy) story. But the reporter didn't even bother to talk to someone familiar with the field.

## This bright Dude comes across as down to earth (Score:3)

This longer piece (German) [www.welt.de] quotes him pointing out that he is very weak in Graph theory and Combinatorics. Nevertheless he skipped two classed in school and will be able to start university this fall.

Won't be the last time we heard form this guy.

## Is his last name Ray or Ra? (Score:3)

## overhyped; not new, not a solution (Score:3)

As often seems to be the case with these news articles about teenage prodigies, this has been overhyped. It turns out that what he did is not new and is not a complete solution to the problem.

Parker, Am J Phys 45 (1977) 606 [df.uba.ar] has a summary of the preexisting results. The expression immediately after equation 23 is the constant of the motion that Ray rediscovered.

A reddit user has a nice simple derivation: http://redd.it/u74no [redd.it] (Note that there is an error because he claims to have proved it in general, but it's only valid when v (the vertical velocity) is positive.)

For more on the history of the problem:

Synge and Griffith, Principles of Mechanics, p.~154 http://archive.org/details/principlesofmech031468mbp [archive.org]

Whittaker, A treatise on the analytical dynamics of particles and rigid bodies, p.~229 http://archive.org/details/treatisanalytdyn00whitrich [archive.org]

According to Whittaker this was first done by D'Alembert in 1744.

## Re:Explain the mind of a genius? (Score:4, Interesting)

Concepts of mathematics (calculus) are actually very simple.

Most confuse the trivia of solving problems (knowing many rules) and how to apply them with understanding of basic mathematical principles.

Teach your kid about 'x' and abstract thinking in relation to rates of change. The rest follows quite naturally. (IMO).

## Re:Explain the mind of a genius? (Score:5, Interesting)

I was not a prodigy, but a really smart kid who was in many environments with prodigies or near prodigies.

My experience has been that most pre-teen children with this history don't understand the material very well, and there tends to be a lot of exaggeration about it. Smart kids are good at mimicking things and that is all that is really need to "do" the first year or two of college math.

Occasionally, but very occasionally you get someone really young who later goes on to do decent, or even more rarely great things, like Norbert Wiener or Terry Tao. But I would like to hear those people give their opinions of the depth of their understanding at that age.

I knew Nadine Kowalsky, who in HS would essentially just remember everything she heard in class and got 100 on every exam. (She wasn't the only one though. I knew a number of other people like that though that didn't do as well as Nadine did.) She later went on to get a Ph.D. from Chicago and published her thesis in the Annals of Math. That is a journal most mathematicians can't get a paper in. Like publishing in Nature or Science. Nadine was the real deal, but sadly she died of cancer not long after finishing her Ph.D. But I don't believe that Nadine was doing calculus until she was 15. And that was certainly on purpose. She, and her parents apparently, knew what was a good idea to do, and not to do, with a super smart kid. (This last sentence is conjecture on my part.)

But I think most cases of pre-teens you hear about are really not what they are made out to be. Once you get to 12 or 13 those, I think things do change a lot.

## Re: (Score:3)

I think you're right. Mozart was not a genius, and arguably not a prodigy. He was however raised in a house of means by the greatest musical pedagogue of his age who started teaching him music before the kid could talk. I've been teaching long enough (and not very long at all) to realize that any kid could be taught to do these things if they were raised in a stable home and taught intensively for many years. This kid doing this is no more impressive to me than 13 year old Chinese kids on the pommel hor

## Re:Explain the mind of a genius? (Score:5, Informative)

...it does publish great papers, but does require something of a personal connection to get into... Same for The Proceedings of the National Academy of SciencesActually, this isn't so true of PNAS any more. One of the previous editors decided in the late 1990s to raise the quality prestige of the journal by accepting more papers through a traditional peer-review route, as opposed to NAS members "communicating" or "contributing" articles (which would often have minimal peer review). This was very successful, and now most articles in PNAS get in through the front door, and they're slowly eliminating the back doors. The overall quality is pretty good - not as high-impact as Science or Nature or some of the top specialty journals, but it's definitely a journal that researchers are excited about publishing in if they can't get into the top tier. The fact that they're not part of Elsevier or one of the other big commercial publishers, and their open-access fee is very reasonable, is an added bonus. (Disclaimer: I've published there, so I'm not entirely unbiased.)

Now, as with any journal, knowing the right people always helps - sadly, this is true at any level.

## Re: (Score:3)

This isn't just integral calculus, though; it's differential equations. Finding an analytic solution to a nonlinear differential equation is often difficult, sometimes (provably) impossible.

## Re:Explain the mind of a genius? (Score:4, Interesting)

The principles of differential equations are also simple and there are many simple physical systems that can be used to demonstrate them in a way that is easy to grasp. Even by relatively young children.

The idea is not to confuse the understanding of principles with their applications, as those can be (and are) horribly complex.

Math is not hard. Math is very elegant and simple. Much like language, the same words that are in children's books also comprise the classics.

## Re:Explain the mind of a genius? (Score:5, Insightful)

I have to agree with your comment about learning DE, I failed differential equations the first time I took the class (a D-grade) I was taking engineering course work at the time that required them - and what they actaully "meant" clicked in an electrical networks class - when I took the class again (my university had a 1 time grade forgiveness policy) I got an A - it seemed trivial and simple the second time around in a different context. I general I have mathematics makes mroe sense to me personally when I can relate it to a real world problem - Mathematics taught as rote learning is a horrible thing - some of us can't do it that way....

## Re:Explain the mind of a genius? (Score:5, Interesting)

Exactly. As a kid, I had a dog that understood when I threw a ball up on the roof of our garage, which caused it to disappear from her sight, that it would roll along the slope of the roof and and reappear further down the roofline. She actually got fairly good at predicting where the ball would reappear, repositioning herself along its path over time so she would meet it at its eventual drop point. Does that mean my dog understood calculus, or solved Newton's problem? Well, she recognized a pattern and was able to apply a repeatable solution.

That tells me that the brain is capable of recognizing complex patterns around us, and is actually already very capable of deriving and applying practical solutions. ("So easy a dog could do it.") Applying abstract mathematical models to them, however, is not so easy.

What I'd be most interested in in this whole saga is "what methods did his father use to teach him math?" Obviously they were highly effective.

## Re: (Score:2)

With all these things the kid probably started at 6 and got it all wrong rather then claim to be able to understand it at 6. Otherwise I think he's full of shit.

## Re: (Score:2, Interesting)

I'll bet you that any 6 year old can solve the problem of where a ballistic projectile will be, even accounting for air resistance, in real time without a computer.

Don't believe me? Toss them a ball. The rest is just notation.

## Re:Explain the mind of a genius? (Score:5, Insightful)

No. The problem is to determine the trajectory from the initial position and velocity. A human tracks the ball as it moves, which is a completely different problem.

## Re: (Score:3)

Catching a ball is a feedback mechanism. See where the ball is, compare to where the ball was, move (hands or feet, depending on how far off you are). Repeat as necessary.

## Re: (Score:3)

Of course it's a different problem.

The first is a prediction from a known initial state, the second is an exercise in analytical approximation that just means you have to get your hands to reach the same position in space and time as the ball, based upon a continuous stream of information of ever-increasing accuracy about the relationship between said hands and the ball over time.

Wildly different exercises.

## Re: (Score:3)

Path of a projectile at the Earth's surface without air resistance in a uniform gravitational field is a parabola (or quadratic curve). Go into Earth orbut and you also get ellipse curves.

Air resistance would slow horizontal and vertical velocity by a fraction per unit of time, so I would guess that it is an integral of a power sequence.

## Re:Explain the mind of a genius? (Score:5, Insightful)

I was doing advanced Geometry and Algebra at age 8, yes I'm a slow fool compared to this kid. but it's mostly the quality of teachers (his dad) and the willingness to keep giving a kid what they want and challenging them.

The american school system is designed to DISCOURAGE this. Smart kids are told to be happy with the A they got without trying. If they challenge their teachers knowledge they are told they are wrong. Mostly because Grade-High-school education in the USA is simply following a lesson out of a book and not teaching it from an expert. the Gym teacher teaches computer class, The English teacher teaches Chemistry, and all of it creates a ho hum boring as hell experience for the children.

Here in the USA we do NOT want geniuses, we want good factory and office workers. Mediocre will not challenge authority.

yes I am jaded at the education system here. I was one of them that got bad grades because the teachers were idiots. I challenged my math teacher who could not believe that a kid can do multiplication and simple geometry in his head. I proved it on several occasions, but I was given failing grades for not doing the busywork of writing it all out. Plus I refused to learn his technique. It sucked and was harder than what I was using that came from college text books. So I ended up being a pissed off moody kid hating the education system because all I saw was idiots and morons trying to tell me they knew more than Me and I knew that they were wrong. I was reading at a 14th grade level when I was 12 years old. I read 1984 and understood the concepts and hidden meanings. I was devouring Vonnegut with a passion. I was told that the books were "too grown up for me" Everyone talked down to me and all it did was piss me off.

Sadly I did not have rich parents, so I had to suffer through the waste of time that the American Public School system is. College I slept through and aced it, at least they were not morons requiring me to turn in worthless busy work. It was in college where I ran into real education, educators that actually knew what they were talking about and would actually hold a discussion with me and help me learn more.

This is the problem here in the USA. If you are smart, you have a sack put over your head to slow you down to match the rest of the other students.

## Re:Explain the mind of a genius? (Score:5, Insightful)

Here in the USA we do NOT want geniuses, we want good factory and office workers. Mediocre will not challenge authority.

I've shared this and I'll share it again (and again...) but when I was in third grade I had an asshole, authoritarian teacher who I believe was only at my school for a couple of years. He was a lazy, arrogant, abusive asshole. When one was done with one's work one was to literally lay one's head down on one's desk and wait quietly for the other children to finish. I was in trouble on numerous occasions for "looking at the other children". I wrote so many lines I had wrist problems before I ever owned a computer or even discovered masturbation.

Sadly I did not have rich parents, so I had to suffer through the waste of time that the American Public School system is.

I went to a private school for a couple of years, before my parents broke up and there wasn't enough money because my dad was a deadbeat. I was about to be learning algebra, I was learning Spanish (I had great retention back then, and I never forgot some of the words I learned back then... though "ferrocarril" does have a fantastic ring to it, no?) and so on. Then I was placed literally into kindergarten due to my age and went from actually learning at a satisfying pace to being told lies about American colonization, making flags out of construction paper and placing Dead-President's-Head's stickers on them, and the like. After a year of that I spent two weeks in first grade before being bumped up to second, where I was

stilldoing work inferior to what I'd been doing in my previous school.This is the problem here in the USA. If you are smart, you have a sack put over your head to slow you down to match the rest of the other students.

Especially if you are smart, but your parents are dysfunctional and can't teach you how to blend in because they know fuck-all about how social situations work.

College I slept through and aced it, at least they were not morons requiring me to turn in worthless busy work.

Alas, I discovered life about the same time I went to college for the first time and besides, by that time I was prejudiced against education. What really shat upon my educational aspirations at that time, though, was a counselor who suggested I take a fully practical case load and save my electives for later. If I could remember who that was, I would send them a picture of my asshole right now. Hated it. Made school just a big bore of a chore. Most counselors don't give one tenth of one fuck about you as a person or even as a student, you're just a convenient unit that can be used to fill out slightly empty classes. What, am I bitter? Why do you ask?

Now I have a two-year degree from going back to school much later, but it wasn't convenient for me to matriculate to a four-year at the time and now what do I do with this extra piece of paper? It's too crisp to be good bumwad.

## Re:Explain the mind of a genius? (Score:5, Insightful)

"Here in the USA we do NOT want geniuses, we want good factory and office workers. Mediocre will not challenge authority."Exactly. And I tell you, is the same thing here in Brazil.

## Re: (Score:3)

I'm also from Brazil and I share the feeling... sadly, I think it's the same thing everywhere in the world.

I guess we're entering the next Dark Ages of knowledge...

## Re: (Score:3)

Yes, I'm much more interested in the story of how his dad taught him so well and effectively than I am in the solution itself.

And while I'm sorry you had such a crappy experience in public school, you might be heartened to know that not all public schools are equally horseshit as the ones you were unfortunate enough to attend. We have some absolutely stellar schools around us here, with teachers that actually care, and they try hard to challenge the kids to reach above their "expected potential". Not ever

## Re:Explain the mind of a genius? (Score:5, Insightful)

Teachers are doing this for your benefit, not theirs. If you can hand in your homework with just the answers and get them all correct, great, but if you hand in the homework and get some wrong, the teacher won't have any idea where you went wrong, whether you used the wrong method when solving it or if you just made a simple error with the arithmetic. 99.9% of kids, even the ones who think they don't need to show their working because they know to do it, will at some pointstruggle with something and need help.

The UK exam system drills this into you pretty early, only 1 mark out of 3 or 4 being awarded for the correct answer, the rest being awarded for the method used. By the time you get to A-level (High school) maths, you're even given the answer beforehand and asked to "show that x = 5".

Ultimately the working out is usually more important in maths than the answer. You won't win a Fields medal for "Fermat's late theorem : it was correct. The end"

## I don't think it's by design (Score:3)

I'm honestly not sure that the system is actually designed to discourage this (though it certainly feels like it). It's just an unintended consequence of the relatively low IQ levels of the teachers and administrators who design such systems, and the teachers who are actually doing the teaching. IQ, intelligence, call it what you will - is distributed in something approximated by a bell curve. If you had the brains to be doing advanced geometry and algebra at age 8, you are very, very likely to be smarter t

## Re:are those problems NP? (Score:5, Interesting)

-=Geoskd

## Re: (Score:2)

This is not a decision problem so the P-NP complexity classes do not apply.

## Re: (Score:3)

While P/NP is indeed pretty way offtopic here, P vs. NP doesn't necessarily apply solely to decision problems. Furthermore, many problems can be rephrased as decision problems; e.g. Does the cannonball need more than 10 second to complete its flight?

For a traditional P/NP example: the traveling salesman problem is about finding the shortest path, which is also not a decision problem.

## Re: (Score:3)

I've just realized that my example is wrong, because it seems to me that for the shortest-path version of TSP you can get away with a binary search over all the possible lengths (since the length of a path is upper bounded in a finite graph), which is just a (polynomial-time) iteration over the decision problem. I'm fairly certain that my comment on the differing complexities is true in general, but I'd rather someone else chime in with a correct example :)

## Re: (Score:2)

Good job there, providing zero evidence outside of hearsay and stereotyping. Because if there's one thing that will provide evidence for eugenics, it's the opinions of other people who want to provide evidence for eugenics.

## Re: (Score:3)

So the crazy old guy in that movie with the long title was right all along?