Visualising and representing

Ever thought of playing three dimensional Noughts and Crosses? This problem might help you visualise what's involved.

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

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you picture it?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

A blue cube has blue cubes glued on all of its faces. Yellow cubes are then glued onto all the visible blue facces. How many yellow cubes are needed?

Jack and Jill run at different speeds in opposite directions around the maypole. How many times do they pass in the first minute?