"Here's a cool problem": a collection of short questions with clever solutions

Question

This game will be familiar to many mathematicians, and it is always good fun to play.

I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one could solve pretty much anywhere: over dinner, while taking a walk - essentially, without pen & paper, and where the solution lies in spotting a clever trick or fact (rather than via some monstrously power theorem).

To get an idea of what I mean, here is an example:

Q. A real number is called repetitive if its decimal expansion contains arbitrarily long blocks which are the same. Prove that the square of a repetitive number is repetitive.

(please do not post a solution, since I am sure anyone can figure it out given enough time)

Does anyone have similar chestnuts to offer?

Answer 1

Evaluate:

$$\int_0^1\!\!\int_0^1\! \dfrac{1}{1-xy} \, \mathrm{d}x\mathrm{d}y$$

Answer 2

$1$, $z$, $z^2$, and $z^3$ (all distinct) lie on the same circle on the complex plane. What is the center of that circle?

Answer 3

Consider a 8x8 chessboard and 2x1 dominos. Clearly one can easily cover the whole chessboard with 32 dominos.

Now omit the squares on the bottom-left and top-right. There are 62 squares left on the chessboard. Can you cover them using 31 dominos?

Answer 4

Someone plays chess at least once a day and no more than 10 times in 7 consecutive days. Prove that there is a period of n consecutive days where the man or woman played 23 times.

Answer 5

A rectangular room in the museum has a floor tiled with tiles of two shapes: $1 \times 4$ and $2 \times 2$. The tiles completely cover the floor of the room, and no tile has been damaged, cut in half, or otherwise had its integrity interfered with. One day, a heavy object is dropped on the floor and one of the tiles is cracked. The museum handyman removes the damaged tile and goes to the storage closet to get a replacement. But he finds that there is only one spare tile, and it is of the other shape. Can he rearrange the remaining tiles in the room in such a way that the spare tile can be used to fill the hole?

Answer 6

This is an old putnam problem, that I think is an excellent example of this.

Let $a_n = 1111...1 $($n$ ones) for instance $a_2 = 11$. Find all polynomials $f$ such that for all $a_n$, there is some $a_k$ where $f(a_n) = a_k$.

Hint: Reduce the problem to finding polynomials $g$ where $g(10^n)=10^k$.

Answer 7

If the probability of rain on Saturday is 50% and the probability of rain on Sunday is 50%, which is the probability of rain on the weekend? It can be very fun during dinner with non-mathematicians friends...