The four figures (the yellow, red, blue and green shapes) total 32 units of area.
According to Martin Gardner,[3] this particular puzzle was invented by a New York City amateur magician, Paul Curry, in 1953.
The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers, which leads to the exact unit area in the thin parallelogram.
When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged.
The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one.
Animation of the missing square puzzle, showing the two arrangements of the pieces and the "missing" square
Both "total triangles" are in a perfect 13×5 grid; and both the "component triangles", the blue in a 5×2 grid and the red in an 8×3 grid.
What the "magician presentation" does not show. The angles of the hypotenuses aren't the same: they are not
similar triangles
. It is fairly trivial to prove that the triangles must be dissimilar for this form of the puzzle to work in the plane.
Splitting the thin parallelogram area (yellow) into little parts, and building a single unit square with them
There are two distinct and "false hypotenuses" for the total triangle.
