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

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

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 can you put five cereal packets together to make different shapes if you must put them face-to-face?

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?

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?

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

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

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

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

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?

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

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

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

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

How many trapeziums, of various sizes, are hidden in this picture?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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?

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?

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

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?

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.

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

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?

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

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

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

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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.

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. . . .

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

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

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

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 models can you find which obey these rules?

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

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

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?

If you had 36 cubes, what different cuboids could you make?

A challenging activity focusing on finding all possible ways of stacking rods.

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

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'.

In how many ways can you stack these rods, following the rules?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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?