Maths puzzles

Here is a list of mathematical puzzles that I like.

1. Let the operator S(n) sum the digits in a number (e.g., S(10) = 1; S(92) = 11; S(101) = 2). What is S(S(S(333^333)))?

2. I live in a house on a road where the houses are numbered 1, 2, ..., n. If I sum the numbers on the houses to one side of me they equal the sum of the numbers of the houses on the other side. Which number do I live at and how many houses are on the street? Is it possible to determine uniquely?

3. The corners of a regular tetrahedron are labelled A, B, C, D. An ant starts at A and moves randomly to either B, C, D, and then keeps moving randomly in the same way. What is the probability that the ant returns to A after 5 moves?

4. At noon the hour hand and the minute hand of a clock are exactly aligned. What is the exact next time at which this occurs?

5. Given n people in a room. How large does n have to be before it is likely that two people share a birthday?

6. What is the greatest common factor of all numbers of the form p^2-1, where p is prime and p > 5?

7. An ant lives on the outside surface of a cube and wants to crawl between the opposite corners. What is the shortest path for the ant to take?

8. Show that if n is an integer, n^3-n is divisible by 6.

9. A thin hoop of diameter d is thrown randomly on to a large chequerboard with squares of side L. What is the chance of the hoop enclosing a portion of the boundary between squares? What if the chequerboard is replaced with a board of tessellated regular hexagons with side-length L?

10. How many unique ways are there to colour the 6 faces of a cube using 6 different colours, where each way is not related to any other via rotation?

11. What is the ratio of the volume of a cube to a sphere, where the cube is the largest that can fit entirely within the sphere?

12. If an n sided regular polygon is inscribed within a circle, what is the ratio of the area of the polygon to that of the circle?

13. N ants are dropped on a log of length L. Each ant travels either left or right with constant speed c. When two ants collide, they bounce off each other, reversing their direction. When an ant reaches the end of the log, it falls off. The initial positions of each ant are independent and uniform across the length of the log. What is the longest possible time that it could take for all N ants to fall off the log?