So Where Does the Extra Square Come From?

[Yazata]

[Image: 220px-Missing_Square_Animation.gif]

The complete "triangle" isn't really a legit right-triangle, either before or after. Look closely at where the diagonal line (the hypoteneuse) crosses the grid squares before and after the rearrangement. It moves, because the red and blue triangles don't have the same slope. The blue triangle is steeper at 5/2, than the red at 8/3. So the apparent hypotenuse is bent inwards before the mystery square appears and bent out afterwards.