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

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Can you find the values at the vertices when you know the values on the edges?

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

Can you describe this route to infinity? Where will the arrows take you next?

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

Can all unit fractions be written as the sum of two unit fractions?

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

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

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

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

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

A collection of short Stage 3 and 4 problems on Exploring and Noticing Structure.