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

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

Can you make five differently sized squares from the tangram pieces?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

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?

Complete the squares - but be warned some are trickier than they look!

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.

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?

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

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

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

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.

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

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?

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 pictures show squares split into halves. Can you find other ways?

This activity challenges you to make collections of shapes. Can you give your collection a name?

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

What do you think is the same about these two Logic Blocks? What others do you think go with them in the set?

Can you each work out what shape you have part of on your card? What will the rest of it look like?

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

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.

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.

Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?

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

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?

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

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?

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