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.

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

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.

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

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

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?

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

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?

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

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?

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.

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

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

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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?

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

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?

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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?

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

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

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

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

What fractions can you divide the diagonal of a square into by simple folding?

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

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

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

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

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

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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?

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?

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

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

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

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?

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.

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

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

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

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