Here is an interesting problem.
There are cards numbered 1, 2, 3, … , 2000 face down on a table. If I choose one at random, what is the probability the number on it will contain a 5?
Do not be scared away by the word “probability”; in this question: it’s just the ratio of the count of numbers that contain digit “5” to the total count of given numbers (which is 2000). — For your warm-up: if you cannot solve the problem right away, can you work on a smaller-scaled version? Say, the cards are numbered 1,2,3, .. until 200.
The question below is a truly mind blower. It has shown you some of the most amazing facts through division of a shape which seems to create something from nothing!
[This link leads to an external video.]
The Banach-Tarski Paradox
This video is a rich discussion about infinity. It starts from countable infinity, to uncountable infinity, to Cantor’s diagonalization system and comes to Hilbert’s hotel, and the Banach-Tarski Paradox.