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.

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

What's the greatest number of sides a polygon on a dotty grid could have?

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at. . . .

What shape and size of drinks mat is best for flipping and catching?

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?