What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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?

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

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

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?

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

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

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?

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

Find out what a "fault-free" rectangle is and try to make some of your own.

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?

An investigation that gives you the opportunity to make and justify predictions.

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

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

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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.

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred 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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

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

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

What happens when you round these three-digit numbers to the nearest 100?

What happens when you round these numbers to the nearest whole number?

Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?