You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon?
On which of these shapes can you trace a path along all of its edges, without going over any edge twice?
Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?
Find all the different shapes that can be made by joining five equilateral triangles edge to edge.
Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.
A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?
Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.
Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.