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

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

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

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.

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?

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

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

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

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

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?

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

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?

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

A Sudoku with clues given as sums of entries.

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?

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

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

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.

Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's there is one digit, between the two 2's there are two digits, and between the two 3's there are three digits.

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

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

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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.

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

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?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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

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?

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

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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

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?

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

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

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?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Can you fill in the empty boxes in the grid with the right shape and colour?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?