### Fun math problems

These problems have solutions that are fairly basic (one requires calculus, another set theory). Some of the answers are counterintuitive.

It these are too easy for you, check out Ponder This.

Consider the grid of points (x,y) with integer x,y >= 0. You start with stones on (0,0), (1,0) and (0,1). A "legal move" replaces a stone at (x,y) with stones at (x+1,y) and (x, y+1) if these points are empty.

Question: is there a sequence of moves that leaves the original 3 points empty?

(Suggestion: get a checkerboard or Scrabble set and experiment).

First an easy problem:

A is looking at B and B is looking at C. A is married. C is unmarried. Is a married person looking at an unmarried person?

Now a related harder problem:

Are there numbers A, B and C such at A^B = C, A and B are irrational, and C is rational?

I give you a deck of cards and ask you to pick any five. I examine the cards. I select one and put it face down, then put the other four face up in a row. Then my assistant, who has not seen the cards, enters the room, examines the face-up cards, and correctly announces which card is face down.

Is this possible? How?

Note: only the order of the cards matters, not their orientation or precise position.

Take one cup of coffee, one cup of tea. Transfer one teaspoon of coffee to the tea cup. Mix well. Transfer one teaspoon from the tea cup back to the coffee cup. Which is greater, the amount of coffee in the tea cup or the amount of tea in the coffee cup?
Take an 8x8 grid and remove two diagonally opposite corner squares. Is it possible to cover the remaining 62 squares with (non-overlapping) 1x2 tiles?
N teams compete in a double-elimination tournament (i.e., a team is eliminated when it is beaten twice). At the end of the tournament, how many total games have been played?
There are three identical doors. Behind one is a prize, behind the others nothing. You pick a door. One of the other two doors, with nothing behind it, is then opened. You are given the option of either sticking with your original choice, or switching to the third door. What should you do to maximize your probability of getting the prize? What is the probability?
I flip two coins. If at least one is a head, I show it to you. What is the probability that the other one is also a head?
A river 1 mile wide runs at a speed of X miles per hour. A man swims at a speed of Y miles per hour, Y > X. Does it take longer for the man to swim directly across the river and back, or to swim upstream 1 mile and back? (Requires calculus).
You meet three men. You know that one always lies, another always tells the truth, the third lies randomly, but you don't know which is which. The three men do. You get three yes/no questions, each directed to a single man, to decide which is which.
There are N red points and N blue points in the plane, all distinct. Is it always possible to pair up the points, one red and one blue in each pair, so that if we draw a line segment between the points in each pair, there are no intersections between these segment? (This is a Putnam exam problem).
A pin one inch long is placed randomly on a plane ruled with parallel lines one inch apart. What is the probability that the pin crosses one of the lines? (Requires calculus).
(This one requires knowing set theory). P(Z) is the power set of the integers. We all know that P(Z) is uncountable. Is there an uncountable subset X of P(Z) that is totally ordered? (i.e. for all a and b in X, either a is a subset of b or b is a subset of a)
The plane is tiled by a set of polygons. Each polygon has area 1.5 and contains exactly one lattice point in its interior (and none on its boundary). Show that there is no upper bound on the diameter of the polygons.
Five pirates (P1...P5) have stolen 100 doubloons. P1 must propose a division of the booty, and all the pirates then vote on this proposal. If it fails to get a strict majority, P1 is executed, P2 proposes a division, and so on. Each pirate follows a strategy whose descending priorities are: 1) remaining alive 2) getting maximum gold 3) killing others. What division does P1 propose?
Show that any triangulation of a pentagon has an odd number of triangles.
An NxN grid is covered with layers L1...Lm of red and green cubes. We say that the configuration is "complete" if each red cube has an even number of green neighbors, and each green cube has an odd number of red neighbors. (Neighbors in adjacent layers are counted; diagonal neighbors are not counted). Is there a layer L1 such that there are are no layers L2...Lm, for any m, such that L1...Lm is complete?
Consider a rectangle X from which a subrectangle Y (not necessarily horizontal/vertical) has been removed. Clearly there are lines that bisect the resulting area X-Y. Give a simple procedure for determining such a line.
[The Induction Tribe]. A certain tribe has the following customs: a) If a married woman commits adultery, everyone except her husband learns about it the next day; b) If a man learns that his wife has committed adultery, he kills her that night.

There hasn't been any adultery for a long time, but one day the chief announces "adultery occurred in our tribe last night". (He doesn't say how many instances). What happens next?

There are 12 weights that look identical, but one of them is either lighter or heavier than the others. You have a balance. In no more than three weighings on the balance you must figure out which of the 12 weights is different, and whether it's lighter or heavier.
Finally, here's an entertaining puzzle that connects words and numbers.