Fractions

problem
Doughnut

Age

7 to 11

Challenge level

How can you cut a doughnut into 8 equal pieces with only three cuts of a knife?
problem
Racing odds

Age

11 to 14

Challenge level

In a race the odds are: 2 to 1 against the rhinoceros winning and 3 to 2 against the hippopotamus winning. What are the odds against the elephant winning if the race is fair?
problem
Archimedes and numerical roots

Age

14 to 16

Challenge level

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
$Unit fractions$

problem
Unit fractions

Age

11 to 14

Challenge level

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.
problem
Bull's eye

Age

11 to 14

Challenge level

What fractions of the largest circle are the two shaded regions?
problem
3388

Age

11 to 14

Challenge level

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.
problem
Look before you leap

Age

16 to 18

Challenge level

Relate these algebraic expressions to geometrical diagrams.
problem
Plutarch's boxes

Age

11 to 14

Challenge level

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?
problem
Comparing continued fractions

Age

16 to 18

Challenge level

Which of these continued fractions is bigger and why?
problem
Do unto Caesar

Age

11 to 14

Challenge level

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?