Challenge Level

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Challenge Level

Find your way through the grid starting at 2 and following these operations. What number do you end on?

Challenge Level

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Challenge Level

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Challenge Level

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

Challenge Level

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

Challenge Level

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Challenge Level

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Challenge Level

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Challenge Level

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Challenge Level

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Challenge Level

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Challenge Level

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Challenge Level

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Challenge Level

What is the best way to shunt these carriages so that each train can continue its journey?

Challenge Level

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Challenge Level

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Challenge Level

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

Challenge Level

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Challenge Level

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Challenge Level

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Challenge Level

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Challenge Level

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?

Challenge Level

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Challenge Level

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

Challenge Level

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Challenge Level

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

Challenge Level

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

Challenge Level

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

Challenge Level

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Challenge Level

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Challenge Level

Watch this animation. What do you see? Can you explain why this happens?

Challenge Level

Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

Challenge Level

How many different triangles can you make on a circular pegboard that has nine pegs?

Challenge Level

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Challenge Level

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

Challenge Level

Can you find ways of joining cubes together so that 28 faces are visible?

Challenge Level

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

Challenge Level

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Challenge Level

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

Challenge Level

An activity centred around observations of dots and how we visualise number arrangement patterns.

Challenge Level

Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

Challenge Level

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

This article for teachers describes a project which explores the power of storytelling to convey concepts and ideas to children.

Challenge Level

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?