Mathematical Puzzles

1. What is

$$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots?$$

What is

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\ldots?$$

2. You are invited on to a game show. There are 3 closed doors, behind one of the doors is a car and behind the other two are goats. You choose a door, the host then opens one of the remaining doors to reveal a goat. The host asks if you would like to stay with your initial choice of switch. You will win whatever is behind your final choice of door. Should you switch?

3. Prime numbers of the form 4n+1 are always the sum of two squares. For example:

5=2²+1², 13=3²+2², 17=4²+1², 29=5²+2², 37=6²+1², 41=5²+4²

How would you prove this is always true? (This is by far the hardest problem on this page.)

4. In Königsberg there are 7 bridges connecting the town. Is it possible to plan a route going across each bridge exactly once and returning to the start?

5. Can you write $\sqrt2$ as a fraction $\frac{a}{b}$?

6. Your dominoes are just the right size to cover two squares of a chessboard. Clearly 32 dominoes will cover the whole board. Two opposite corners are now removed from the board. Is it possible to cover the remaining 62 squares by 31 dominoes?

Solutions to these may be obtained by sending a s.a.e. to the Mathematics & Statistics Department.

 

Back to the admissions page