Explaining, convincing and proving

problem

Integral inequality

Age

16 to 18

Challenge level

An inequality involving integrals of squares of functions.
problem

Proof of Pick's theorem

Age

16 to 18

Challenge level

Follow the hints and prove Pick's Theorem.
problem

Rarity

Age

16 to 18

Challenge level

Show that it is rare for a ratio of ratios to be rational.
problem

The great weights puzzle

Age

14 to 16

Challenge level

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?
problem

Distinct in a line

Age

11 to 14

Challenge level

This grid can be filled so that each of the numbers 1, 2, 3, 4, 5 appears just once in each row, column and diagonal. Which number goes in the centre square?
problem

Square LCM

Age

14 to 16

Challenge level

Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?
problem

Knights and knaves

Age

11 to 14

Challenge level

Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?
problem

To run or not to run?

Age

11 to 14

Challenge level

If an athlete takes 10 minutes longer to walk, run and cycle three miles than he does to cycle all three miles, how long does it take him?
problem

Digital counter

Age

14 to 16

Challenge level

When the numbers from 1 to 1000 are written on a blackboard, which digit appears the most number of times?
problem
Favourite

Always, sometimes or never? KS1

Age

5 to 7

Challenge level

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?