ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

What is the greatest number of squares you can make by overlapping three squares?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Square It game for an adult and child. Can you come up with a way of always winning this game?

If you move the tiles around, can you make squares with different coloured edges?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

What can you see? What do you notice? What questions can you ask?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

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?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

How many centimetres of rope will I need to make another mat just like the one I have here?

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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

A man paved a square courtyard and then decided that it was too small. He took up the tiles, bought 100 more and used them to pave another square courtyard. How many tiles did he use altogether?

The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .

A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?

Can you work out the area of the inner square and give an explanation of how you did it?

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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?

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

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?

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

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?

This interactivity allows you to sort logic blocks by dragging their images.

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

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

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

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

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

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

Look at how the pattern is built up - in that way you will know how to break the final pattern down into more manageable pieces.

Creating designs with squares - using the REPEAT command in LOGO. This requires some careful thought on angles

This LOGO Challenge emphasises the idea of breaking down a problem into smaller manageable parts. Working on squares and angles.

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

A Short introduction to using Logo. This is the first in a twelve part series.