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

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.

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?

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?

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?