Exploring and noticing

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

How many 10-digit numbers containing only 1s, 2s and 3s can you write?

Have a go at this 3D extension to the Pebbles problem.

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?