Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

This challenge is a game for two players. Choose two numbers from the grid and multiply or divide, then mark your answer on the number line. Can you get four in a row before your partner?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?

Follow this recipe for sieving numbers and see what interesting patterns emerge.

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

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

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Which numbers can we write as a sum of square numbers?

How would you judge a competition to draw a freehand square?

A weekly challenge concerning prime numbers.

A function pyramid is a structure where each entry in the pyramid is determined by the two entries below it. Can you figure out how the pyramid is generated?

A weekly challenge concerning the interpretation of an algorithm to determine the day on which you were born.

A selection of intriguing questions to consider on mechanics, particularly surrounding the ideas concerning impulse and momentum.

Can you massage the parameters of these curves to make them match as closely as possible?

Can you find the differential equations giving rise to these famous solutions?

Several of you from different schools clearly explained how you used the charts to sort out the data.

Josh clearly explained how to calculate the radius of a running track and the position of the staggered starts for the 200m and 400m races.

Here we look back at the year with NRICH and suggest mathematical summer holiday activities for students, parents and teachers.