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Here is a 4 by 4 dotty grid:
How many different triangles can you draw by joining dots on the grid?
We count two triangles as the same if you can cut one out and fit it exactly on top of the other.
Can you come up with a convincing argument that you have found all the possible triangles?
Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?