Mathematics for the curious · thethinkingboard.net
One theorem, hundreds of proofs.
Which one makes you see it?
Here is a curious fact. The Pythagorean theorem has been proved in hundreds of ways, and most of us were shown only one. Some of the others are plain, some are sly, and a few are so lovely they stop you in the middle of a sentence. This is a place to collect them, and to try your own. Pull up a chair.
A puzzle for the road: why is the sum of the first n odd numbers always a perfect square? Draw it, and you will not need algebra.
On the board
33 openResisted serious people for a long time, or needs machinery that has to be built before it can be used.
Waiting for a first proof
30 problemsThese have no proof yet, which makes themthe best seats in the house: whatever you write becomes the argument the board begins with.
π is irrational
Show that pi is irrational. It was first proved in the 1760s, and a short proof using only calculus is now known. A proof that short is suspicious. Show where the work went.
A fixed point for the disk
Let D be the closed unit disk in the plane and let f : D to D be continuous. Show that there is a point x in D with f(x) = x. Proofs go through the fact that a disk cannot be continuously sq…
Every number is a sum of four squares
Show that every non-negative integer can be written as the sum of four squares of integers. For example, 7 = 4 + 1 + 1 + 1, and no three squares will do. Why is four the right number?
Quadratic reciprocity
Let p and q be distinct odd primes. For an integer a and an odd prime p, the Legendre symbol left(fracapright) is 1 if a is a non-zero square modulo p, -1 if it is not a square, and 0 if p d…
The fundamental theorem of algebra
Show that every non-constant polynomial with complex coefficients has a complex root. It has been proved with complex analysis, with topology, and with algebra. Every one of those proofs use…
Prove that the sum of the reciprocals of the squares is π²/6
Prove that … Posed by Pietro Mengoli in 1650. Jakob Bernoulli could prove the series converges and is less than 2, and then wrote, in print, that he could not find its value and would be gra…
How it works
One
Bring a problem
Anyone may propose one. An editor reads it before it goes up, because a board is nicer as a small, well-kept garden than as an endless field. Editors are readers who asked for the job, and you can ask too.
Two
Show your proof
Write as you would explain it to a clever friend: plain words, and LaTeX where symbols help —$a^2+b^2=c^2$ inline, $$…$$ for a display line. Beside every problem is a room for the half-formed things: hints, objections, and the dead ends that taught you something.
Three
The room responds
Readers leave one of three marks: this convinced me, this is beautiful, I will remember this. Notice that none of them says “this is correct.” Correctness is the price of admission, not the prize.
Four
One stands out
The proof with the most convincing marks wears the brass rule. It is not a throne. It moves the moment the marks do, and a proof written ten years from now may take it.
The three marks
As many as you like
The argument that actually persuaded you. Give it to every proof that earns it — a problem can have several. The proof holding the most carries the board’s recognition. If one did not persuade you, the downward triangle cancels a convincing mark and counts against it instead.
One per problem
Not the clearest proof, the one you would show someone. You get a single beautiful mark per problem, so giving it to a second proof takes it off the first.
One a year
For the proof you are still thinking about a week later. One per reader per calendar year, anywhere on the board. You can take it back — but you only ever have the one.
Marks are attached to an account, so reading is open to everyone but marking asks you to sign in. About the board →