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 openStated as a riddle, with no notation and no explicit question. Working out what is being asked is the first half 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…
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 …
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 →