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?

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

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?

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'.

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?

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

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. . . .

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?

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

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

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?

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?

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?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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?

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

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?

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.

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

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?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

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?

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?

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?

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?

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?

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

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

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?

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,. . . .

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

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

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

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

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

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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

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?

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?

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?

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

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 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.

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

Reasoning about the number of matches needed to build squares that share their sides.

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?