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

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?

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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

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?

How many different symmetrical shapes can you make by shading triangles or squares?

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

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

How many models can you find which obey these rules?

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

Use the clues about the symmetrical properties of these letters to place them on the grid.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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. . . .

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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?

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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?

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?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

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?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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'.

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?