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

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

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

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

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

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

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

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

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

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

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?

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

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?

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?

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

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

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

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

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

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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

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. . . .

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?

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

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?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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

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.

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

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

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?

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?

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

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

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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

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.

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

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?

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

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

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

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?

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?

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.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?