How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

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

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

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

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?

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

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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

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

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?

This challenge extends the Plants investigation so now four or more children are involved.

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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

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?

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

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

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

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

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?

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

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

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

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

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

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

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?

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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

Can you find all the different ways of lining up these Cuisenaire rods?

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

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?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

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?

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

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

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

Number problems at primary level that require careful consideration.

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

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?

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?

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

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?

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

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