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?

This activity investigates how you might make squares and pentominoes from Polydron.

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

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

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

Can you draw a square in which the perimeter is numerically equal to the area?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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

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

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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

What happens when you try and fit the triomino pieces into these two grids?

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

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?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

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

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

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

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?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

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 fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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

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

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

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?

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. . . .

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

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.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

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.

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

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

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

My coat has three buttons. How many ways can you find to do up all the buttons?

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