Patterns in chaos
Suppose you have an infinite sequence of random letters, and you are searching for a certain word in it. What is the expected (mean) value of the index of first occurence of the word? The interesting answer is that it is an integer (if the letters are drawn uniformly), and can be calculated using the Markov chain theory and string finite automata. If you are interested, find out more.
The seven cylinders
An interesting problem has been proposed by Littlewood: How many identical long rods (infinite cylinders) can be arranged in space so that they touch each other? The problem is still unsolved; the answer is believed to be seven. You can find an attempt for numerical solution here. (An earlier work in Mathematica. I no longer have access to Mathematica, so I can't correct any mistakes).
