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

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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?

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

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.

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

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?

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

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

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

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.

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

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

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?

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?

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

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.

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?

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

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?

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

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?

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

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

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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?

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

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

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

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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

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?

How many models can you find which obey these rules?

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

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?

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?

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

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?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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

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

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

This article for primary teachers suggests ways in which to help children become better at working systematically.

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

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?

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?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?