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 openA complete proof is within reach of anyone who has met the definitions. A good place to post first.
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 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…
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 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 →