How many moves does it take to swap over some red and blue frogs? Do you have a method?

Can you find the connections between linear and quadratic patterns?

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?

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

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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

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

Can you figure out how sequences of beach huts are generated?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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?

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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?

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

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

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

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

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

A collection of short problems on patterns and sequences.

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