Exploring and noticing

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?

Investigate the totals you get when adding numbers in threes on the diagonal of this pattern.

How many ways can you find of tiling the square patio, using square tiles of different sizes?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.