If you’ve seen these before it may be old news, but I only just stumbled across the seeming impossibility of two identical sets of shapes having a different combined area.
Click here to view a particularly amazing Missing Square Puzzle (aka “Curry’s Paradox”). Why does #5 fit on the top configuration but not the bottom?
a=w*h. This guides the bounding square which contains all the pieces. Yet one seems to contain an extra 4 units of area than the other? I first thought it must be some sort of trickery wherein pieces were only slightly smaller in some way on the second square. However, cutting all the pieces out shows that they are in fact identical.
Stumped as to how it works? Then read here for an interesting analysis of missing square puzzles at Wikipedia