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?

These practical challenges are all about making a 'tray' and covering it with paper.

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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

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.

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?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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?

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?

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

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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

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?

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?

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

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. . . .

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?

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

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?

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

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

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?

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

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?

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

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?

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?

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

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

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

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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

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?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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

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

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?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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 to have an ordered approach.

An activity making various patterns with 2 x 1 rectangular tiles.

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

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

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

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