Cool problem I've been working on:
You have five shapes on a two dimensional plane. Can they all touch each other? (With our definition of touch being sharing more than one points as borders) No overlapping, no third dimension.

Just to clarify, they can be any size, any shape. They don't have to be congruent (they can be 5 completely different shapes)

I think I have a proof of this not being possible, though its not something I can express through words. Not even sure if the proof is adequate. Eager to hear your thoughts.