Katie and Will have some balloons. Will's balloon burst at exactly the same size as Katie's at the beginning of a puff. How many puffs had Will done before his balloon burst?

Can you complete this jigsaw of the multiplication square?

These alphabet bricks are painted in a special way. A is on one brick, B on two bricks, and so on. How many bricks will be painted by the time they have got to other letters of the alphabet?

Susie took cherries out of a bowl by following a certain pattern. How many cherries had there been in the bowl to start with if she was left with 14 single ones?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Can you find a way to identify times tables after they have been shifted up or down?

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

Reasoning about the number of matches needed to build squares that share their sides.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

These grids are filled according to some rules - can you complete them?

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

What's the greatest number of sides a polygon on a dotty grid could have?

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

Which of these pocket money systems would you rather have?