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 openNeeds a real idea, but one idea is enough.
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.
Which primes are sums of two squares?
Show that an odd prime p can be written as p = x^2 + y^2 with x, y integers if and only if p equiv 1 pmod 4. Fermat announced it and Euler first proved it. Since then it has been proved by d…
Pick's theorem
A lattice polygon is a simple polygon whose corners are all points with integer coordinates. Suppose it has I lattice points strictly inside and B lattice points on its boundary. Show that i…
Every Pythagorean triple
Call positive integers (a, b, c) a primitive Pythagorean triple if a^2 + b^2 = c^2 and gcd(a, b, c) = 1. Such a triple has exactly one even leg; call it b. Show that every primitive triple h…
Wilson's theorem
Show that an integer p > 1 is prime if and only if … The easy half and the hard half are quite different in flavour. One half is a statement about factors; the other is a statement about inv…
The number e is irrational
Let … Show that e is irrational. The standard proof assumes e = p/q and multiplies by q!. Is there a proof that does not argue by contradiction?
Six people, three of one kind
Any two people either know each other or do not, and knowing is mutual. Show that among any six people there are three who all know each other, or three who are all strangers to each other. …
Cayley's formula
Take n labelled points, named 1, 2, dots, n. A tree on them is a way of joining them with n-1 lines so that every point can be reached from every other. Two trees are different if they join …
Euler's formula for polyhedra
A convex polyhedron has V vertices, E edges and F faces. Show that … Many quite different proofs are known. One flattens the polyhedron into a map on the plane; another adds up all the angle…
Fermat's little theorem
Let p be a prime and a any integer. Show that … It can be proved by induction with the binomial theorem, by counting coloured necklaces, and by noticing that multiplying by a shuffles the no…
The Cauchy–Schwarz inequality
For real numbers a_1, dots, a_n and b_1, dots, b_n, show that … and say exactly when equality holds. Proofs worth comparing: the discriminant of a quadratic, Lagrange's identity, and a pictu…
Prove that the harmonic series diverges
Show that … does not converge. The terms go to zero, and the partial sums are in no hurry. They first pass 10 at around n = 12,367, and first pass 20 somewhere near n = 2.7 times 10^8. Every…
Prove that there are infinitely many primes
Show that the set of prime numbers is infinite. Euclid settled this in Book IX, in about five lines, and as close to perfectly as an argument gets. That has stopped no one. There are proofs …
Prove that √2 is irrational
Show that there is no pair of integers p, q with q neq 0 such that … Equivalently: the diagonal of a square is incommensurable with its side. That older phrasing is worth keeping in view, be…
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 →