Riddles with a mathematical twist

Question

I am looking for riddles that are understandable for everyone(so especially non-mathematicians) but require mathematical knowledge or deep abstract ideas to be solved.

The best answer will be the riddle that is most understandable( especially not contain any abstract math at all, so for example fermat's last theorem is NOT what I am looking for) but most mathematically demanding at the same time. (I hope it is clear how I will try to objectively assess the answers, so that no one has to vote for closing this thread)

Also it would be nice, if the riddle you recommend is not very famous.

Answer 1

Although not needing deep mathematical knowledge...

• A blindfolded man is handed a deck of 52 cards and told that exactly 10 of these cards are facing up. How can he divide the cards into two piles (possibly of different sizes) with each pile having the same number of cards facing up?

An old-fashioned implication one:

• Mrs Claus always sneezes just before it starts snowing. She just sneezed. “This means that it’s going to start snowing”, thinks Santa. Is he correct?

Robbed these from http://math.alamzy.com/wp-content/uploads/2012/10/Handbook.pdf

Answer 2

Hilbert's hotel questions could be what you are looking for: suppose that Hilbert has a hotel which has inifinitely many rooms. The only rule is only one person can accomadate in a room. A bus comes with infinitely many seats to the hotel. etc.

Answer 3

The four color theorem is easy to explain but hard to prove. However, it's quite famous.

As a variation, you could ask how the following picture can be repainted with only four colors.

(Source: Wikipedia)

Answer 4

A small company (say $N$ people) are going to have a team-building exercise. The exercise manager tells the group to stand in a line, and explains that he will hang a red or a green water balloon above the head of each one. Everyone will be able to see the balloons in front of him/her, but cannot see his/her own or the ones behind.

The exercise manager will ask each person in order, from the back, what colour the balloon above his/her head is. If the answer is wrong, the balloon is popped, otherwise it is not popped. The group are to discuss beforehand for a few minutes to come up with a strategy for what colour each one should say, but once they are in the line no more communication is allowed.

If they want to minimise the number who get wet, what strategy should they employ? What is the smallest number of popped balloons that they can guarantee?

(Source: I'm not sure, but I believe it was in the Swedish onboard train magazine Kupé.)

Answer 5

Puzzles based on Ramsay Theory could qualify. For example, there are 9 people in a room for a meeting. Amongst any three there is at least one pair who have never met before. Show that there is a group of four people amongst the nine who were mutual strangers before the meeting.