Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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 describe this route to infinity? Where will the arrows take you next?

Collect as many diamonds as you can by drawing three straight lines.

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

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?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Can you picture how to order the cards to reproduce Charlie's card trick for yourself?

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

A collection of short Stage 3 and 4 problems on Visualising.