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

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,. . . .

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

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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. . . .

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?

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?

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?

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?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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?

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.

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

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?

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 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?

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?

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

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?

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

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

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.

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?

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

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?

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

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

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

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

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

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Can you visualise what shape this piece of paper will make when it is folded?

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outlines of these clocks?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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

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

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

Can you fit the tangram pieces into the outlines of these people?

Can you fit the tangram pieces into the outline of Granma T?

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

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

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

Can you fit the tangram pieces into the outlines of the chairs?

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

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

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

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

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?