Exploring and noticing

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

Why does the tower look a different size in each of these pictures?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?