pythagorean theorem proof

who was the 8 year old who found a proof for the pythagorean theorem? and where can i get a copy of it?

Is this something that happened recently, or does it refer to a famous person of the past? If the latter, these things may not be very well documented.

For an eight-year-old prodigy, I present Karl Gauss (the mathematician not the SDMB poster) who is said to have been about that age when he came up with the sum
1+2+3+4+[sup] . . . [/sup]+(n-1)+n = n(n+1)/2
For an original proof of the Pythagorean theorem by a youngster, I present Einstein, who is said to have done it at the age of about 12.

President James Garfield is also said to have come up with an original proof of the theorem, but at a more advanced age.

Ther’s apparenty a book containing 100 different proofs of the P. Theorem. Maybe that one has it. I’ve never heard of a proof by an eight-year old. The World of Mathematics chart made up by IBM (And part of the exhibit they have at he Boston Museum of Science, the Chicago Museum of Science and Industry, the L.A. Museum of Science, and on the poster they produced of it) cites the briefest as by the Arab mathematician Bhaskara, who drew a sigle picture (I’m not going to try to put it here) with the single word “Behold!”

This is killing me! (Assuming that I know what CalMeacham is looking for…I may be way off base here.)

There is some very old theorem that has been around for a long time and was actually proven ages ago. I don’t think it was the Pythagorean theorem however since I believe the proof for that is well known and somewhat simple. However, the proof of whatever I’m thinking of was a huge and unweildly thing but it is what mathematicians have lived with all of this time.

A few years ago some young students actually working on a computer programming assignment stumbled upon an insanely simple and elegant proof that took half a page instead of dozens of pages.

Mathematicians were duly shocked. While pleased at the elegant and simple solution being revealed it stunned them that it had been overlooked for so long and that some young students figured it out.

The students were given a trip to present their findings at some mathematical society meeting where all the braniac math types meet once a year.

I know of a real simple geometric proof that has just one picture to reference.

Click here to get the picture, then I’ll explain.

Take the large outer square with sides of length a+b. Inscribed in it is a square with sides of length c. (If asked later, I’ll prove that the green area is indeed a square.)

The area of the large square is:

Expanding this algebraically gives:
a[sup]2[/sup] + 2ab + b[sup]2[/sup].

The area of the large square can also be expressed as a sum of the green area and 4 yellow areas:
c[sup]2[/sup] + 4([sup]ab[/sup]/[sub]2[/sub]) or
c[sup]2[/sup] + 2ab

Since these both represent the same area, they can be set equal:
a[sup]2[/sup] + 2ab + b[sup]2[/sup] = c[sup]2[/sup] + 2ab

Subtracting 2ab from both sides:
a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup]
Gasp! Pythagoras’ Theorem!

CalMeacham should be imonacomputer.

The proof that the young students I’m thinking of may have been a spinoff of the Pythagorean theorem (i.e. another theorem suggested by the Pythagorean theorem). All I remember about it was drawing a box (or rectangle) to include the original triangle and then bisecting one of the angles of the triangle with a line that met halfway down the long edge of the rectangle. If that sounds warped I’m not surprised as I’m certain I’m mixing things up but it was something along those lines.

Just a guess, but might this be the 4-color-map theorem?


Could you give us a citation for the story in your OP? Did you read it in a newspaper or magazine? Did you see it mentioned on a website? Was it in a E-mail full of amazing facts? Did you hear it in a drunken conversation with friends one night? As it stands, the story you tell is too contradictory and too vague to match up with anything any of us has ever heard of.


I don’t think any of the stories told so far match the proof of the Four Color Theorem very closely. That was accomplished by a couple of mathematics professors at the University of Illinois in the '70’s who were able to show that the theorem would be proved if it were possible to check a certain huge but finite number of particular cases. They set up a computer program to doing this checking. It ran a long time (about a year, I think), but finally finished, allowing the professors to announce that the theorem was proved. I don’t think that much computer time would have been available to anyone in the '70’s except some professors with strong backing from the administration of the university. Now, of course, it would run much faster on present-day computers.

Jeff_42 writes:

> The students were given a trip to present their findings
> at some mathematical society meeting where all the
> braniac math types meet once a year.

Sigh. . . the strange things that nonmathematicians write about mathematicians. There are hundreds of mathematics conferences each year, so this tells us almost nothing. There is no single pre-eminent one. It’s so bizarre when nonmathematicians assume that mathematicians are a small group of people who all know each other and who could all meet in a single conference hall. There are several hundred thousand mathematicians in the world. Being good at mathematics is not this obscure, rare skill, but one that quite a large number of people have.

Forgive me for not being up on Mathematician’s meetings. I have no doubt that there are dozens if not hundreds or thousands of formal meetings of mathematicians around the world. I wasn’t trying to suggest that there is only one meeting of the minds of the only 10 people in the world who can do more than add 2 and 2.

While there may not be a single, pre-eminent meeting of mathematicians the article I read (which I unfortunately cannot dig up) led me to believe that this particular meeting was bigger and more well respected than most. Sort of like what Nature is to science journals. Not the only game in town but certainly one of the more impressive ones to get published in if you can manage it. I would be seriously surprised to learn that professional mathematicians don’t have a meeting or two that are especially sought after to be a speaker at than most of the others (i.e. something that looks really good on a resume).

I only mentioned the whole bit in the hopes that someone familiar with these sorts of meetings might narrow it down to two or three candidates to allow a more targeted search for the identity of these kids. They would certainly figure prominently in the schedule of whatever meeting invited them in.


I think I know what you’re talking about. The two kids were in high school. They were asked to prove a certain theorem for extra credit in a high school geometry class. They came up with a novel one instead of the one the teacher expected.

I don’t recall their proof being any better than the existing one–in fact it looked a bit more complicated to me. The only real info I can give is that the story was written up in the WSJ, but you can’t search for old articles on-line. One more thing: the theorem was a very old geometric theorem, something from Greco-Roman times. That might narrow it down a bit.

Well, there are a lot of legends about mathematical proofs. The best one I read was about some grad student who came late to a lecture, and started frantically copying down the equations on the blackboard. Then the next day, he came to the professor with his daily work, and the professor was dumbfounded at the student. What the student had missed was the professor’s preamble, that he was demonstrating problems for which no known solution existed. And the grad student, thinking they were homework, sat down and solved them, not knowing they were impossible. This is supposedly true, I recall reading it in a biography of some Nobel prizewinning mathematician but I can’t remember who it was…maybe it was Hilbert…
Anyway, I hear that this tradition of presenting insoluble problems to students persists. When I visited MIT about 25 years ago (back when I was trying to get admitted) a student showed me the basic Calculus book. He showed me the problems in the back of the book, which were used as homework assignments. Since I had already taken basic calculus in High School (which got me up to about page 5 in the MIT textbook) he invited me to take a crack at the last 3 problems in the book. I threw up my hands in frustration, it was way beyond me. And he said that if I HAD solved the problems, I would be due a Nobel Prize nomination. These insoluble problems were deliberately put in the book on the hopes that some math prodigy would solve them by accident.

I should be imonacomputer?

How do I do this? Why would I want to?

Chas. E.,

The student who solved the up-to-then unsolved problem that the professor had put on the blackboard was George Dantzig. Here’s a URL that gives a fuller description of the incident:

You further write:

> And he said that if I HAD solved the problems, I would be
> due a Nobel Prize nomination.

There is no Nobel Prize in math. The closest thing is the Fields Medal (although it’s not quite the same thing).


My best guess based on what’s been discussed so far is that some high school geometry students found a new proof for a long-known geometry theorem, and they presented it at some mathematics conference.

What you said was:

> The students were given a trip to present their findings
> at some mathematical society meeting where all the
> braniac math types meet once a year.

No, it’s not true that there’s any single most important mathematics conference (and in any case the students’ new proof of an old geometry theorem isn’t a significant breakthrough). In any case, the use of the term “brainiac” is offensive. It’s like referring to athletes as “a bunch of overdeveloped, muscle-bound, physical freaks.”

It is true the students I am thinking of did not advance the field of mathematics beyond where it was at the time. They did however find a stunningly simple and elegant solution to what had been a huge and ugly proof. Something no one else had managed in hundreds of years and is therefore noteworthy…especially since the discovery came from an unlikely source (some bright but not especially gifted teenagers).

That doesn’t seem like a particularly offensive term to me but if being told you have unusual brainpower is offensive to you then I apologize. Perhaps some people automatically equate unusual intelligence with freakishness but I’d say that’s their problem and not something to be ashamed of.

I do think it is safe to assume that professional mathematicians are probably (generally speaking) a cut above the average Joe citizen in the intelligence department. That may be unfair as well but I don’t see how a person can be a successful mathematician and not possess above average intelligence.


Check out the following URL:

Clearly “brainiac” is being used as an insult. Furthermore, the term comes from an alien villain in the
Superman comics. The implication was that he was too smart to be human. I remember a lot of this sort of veiled insults when I was young. People would say things that showed that while they were overtly impressed, they covertly hated people who were smarter than them. If you were better at athletics than others, you were their hero, but if you were better at academics, you weren’t quite human. Indeed, you were a freak.

Whatever you mean, it’s better not to use unnecessary slang terms in your posts.

Another pictoralproof


The book that CalMeacham mentions is called The Pythagorean Proposition, by Elisha Scott Loomis. IIRC, it contains well over 300 proofs. For those that are interested, it is definitely worth the time to look through it and find a favorite dozen or so. I believe it’s out of print though; a library is where I found it. And for those of you wondering, no, I did not read all the proofs :).

Do you know if there is a site that has that book on it or that has more than just 30 proofs because if u or anyone else does it would help a lot.

Like Pies said, the book’s out of print, but Amazon has this “Out of print book searcher” that might help you. Here’s the link.

Ackkk!!! It’s killing me, too. I believe I remember what you’re talking about (although it’s uncertain if that’s what imonacomputer is talking about). Seems like this happened about five years ago? I heard a radio account of the story:

If I remember correctly (and that’s a big IF), some high school students were given the extra credit geometry assignment of trisecting a line (using the classical compass-and-a-straightedge-only approach). They started fooling around with a computer (graphing calculator?) trying to solve it, and wound up discovering a novel, elegant proof that involved constructing a rectangle and then mumblemumble intersecting lines mumblemumble dropping a perpendicular mumblemumble and voila! a trisected line. The neatest part about the proof was that the same basic technique could be used to cut the line into any number of equal pieces- five, eight, eleven, whatever.

Now, does anyone else remember this? I ask because 1) I’d like to know if I’ve embarassingly mangled any of the particulars mentioned above, and 2) I’d like to re-discover just what the proof actually was.