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 openEasy through Difficult rank the proof, not the statement. Cryptic problems are posted as they were found — working out what is being asked is part of the problem.
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.
The chessboard with two corners missing
Two opposite corners of a chessboard have been cut away. Thirty-one dominoes lie on the table, each exactly the size of two adjacent squares. Can they cover what is left, with none overlappi…
Two who match
Friendship is mutual: if I count you as a friend, you count me. At any gathering of two or more people, two of them have exactly the same number of friends in the room. Why?
Two pancakes
Two pancakes of any shapes whatever lie flat on a plate. They may overlap, or lie far apart. One straight cut of a knife is wanted that leaves each pancake in two halves of equal area. Is th…
The monk and the mountain
A monk sets out from the foot of a mountain at dawn, follows the one path to the summit, and arrives at dusk. He sleeps there. At dawn the next day he sets out down the same path and reaches…
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…
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…
The bridges of Königsberg
A town has four districts, A, B, C and D, joined by seven bridges. Five bridges meet at A, and three bridges meet at each of B, C and D. Is there a walk that crosses every bridge exactly onc…
The arithmetic mean beats the geometric mean
Show that for all non-negative real numbers a and b, … with equality exactly when a = b. There is a one-line algebraic proof, and there is a picture with a semicircle in it. Notice that they…
The binomial coefficients add up to 2ⁿ
Show that for every integer n ge 0, … Try it three ways: from the binomial theorem, by counting the subsets of an n-element set, and by induction using Pascal's rule. One of them will feel l…
The angles of a triangle add up to a straight line
Show that the three interior angles of any triangle in the Euclidean plane add up to 180^circ. The usual proof draws a line through one vertex, parallel to the opposite side. There are other…
Two integers that differ by a multiple of n
Show that among any n+1 integers there are two whose difference is divisible by n. The whole proof is one idea. It is the same idea behind a thousand other arguments, which is why it is wort…
The handshake lemma
At a gathering, some pairs of people shake hands. No pair shakes more than once, and nobody shakes their own hand. Show that the number of people who shook an odd number of hands is even. Sa…
Gauss's sum: 1 + 2 + … + n
Show that for every positive integer n, … There is a proof by induction, a proof that pairs the terms from the two ends, and a proof that stacks the numbers into a staircase of dots and coun…
What the visitor said
An island holds two hundred people. Each of them has blue eyes or brown eyes. Each can see everyone else's eyes and never their own, and there is nothing on the island that reflects. No one …
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…
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…
The sum of the first n odd numbers is a square
Show that for every positive integer n, … This is the friendliest thing on the board, and it is here deliberately. It has a proof by induction, a proof by pairing, and a proof that is a pict…
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…
Prove the Pythagorean theorem
Let triangle ABC have its right angle at C. Write a = BC, b = CA, and c = AB. Prove that … Something over four hundred proofs are on record, and that abundance is itself the interesting fact…
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 →