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

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?

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?

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?

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

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

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?

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. . . .

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

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?

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?

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.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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

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

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

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

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?

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'.

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

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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 remove them. The winner is the last one to remove a counter. How you can make sure you win?

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

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?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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

How many models can you find which obey these rules?

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

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

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

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

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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

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

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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

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 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.

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?

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?