Can you make a 3x3 cube with these shapes made from small cubes?

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

What shape is made when you fold using this crease pattern? Can you make a ring design?

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?

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

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?

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

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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?

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,. . . .

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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?

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

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. . . .

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?

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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

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?

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Exchange the positions of the two sets of counters in the least possible number of moves

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

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

Can you fit the tangram pieces into the outline of the rocket?

Make a cube out of straws and have a go at this practical challenge.

Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

On which of these shapes can you trace a path along all of its edges, without going over any edge twice?

Can you fit the tangram pieces into the outlines of the candle and sundial?

In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .

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

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

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

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?

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?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

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.

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

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

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

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

Can you fit the tangram pieces into the outline of this telephone?

See if you can anticipate successive 'generations' of the two animals shown here.

Make a flower design using the same shape made out of different sizes of paper.

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

Can you fit the tangram pieces into the outline of Little Ming?