What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Can you guess the colours of the 10 marbles in the bag? Can you develop an effective strategy for reaching 1000 points in the least number of rounds?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Work out how to light up the single light. What's the rule?

This game challenges you to locate hidden triangles in The White Box by firing rays and observing where the rays exit the Box.

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

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

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

Can you decode the mysterious markings on this ancient bone tool?

A introduction to how patterns can be deceiving, and what is and is not a proof.

Can you make a hypothesis to explain these ancient numbers?

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

95% of people in Britain should live within 10 miles of the route of the Olympic Torch tour. Is this true?

How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions.

This problem offers you two ways to test reactions - use them to investigate your ideas about speeds of reaction.

Two video clips of classes organised into groups to work on Counting Cogs.

This article explores the process of making and testing hypotheses.