Working systematically

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

There are lots of ideas to explore in these sequences of ordered fractions.

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

If you move the tiles around, can you make squares with different coloured edges?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Can you work out what step size to take to ensure you visit all the dots on the circle?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?