A Paradoxical Dissection

by David Radcliffe

Is it possible to dissect an 8×8 square and rearrange the pieces to form a 13×5 rectangle? Common sense indicates that this cannot be done, because the area of the square is 64, while the area of the rectangle is 65. However, this animation seems to show that it is possible. What is going on?



Sam Loyd's Paradoxical Dissection





Since the rectangle is larger than the square, the pieces should not fit together precisely, but there should be a gap. In the previous image, the gap was covered up by making very small adjustments to the shapes. When the shapes are drawn accurately, the gap is revealed.

You might be wondering how we can trust that this drawing is accurate, because the previous drawing also looked pretty convincing. This is where math comes to the rescue! Notice that the slope of the hypotenuse of the green triangle is 3/8, but the slope of the longest side of the blue trapezoid is 2/5. Since 3/8 is less than 2/5, a slight gap exists between the green triangle and the blue trapezoid.

Incidentally, the fact that 3/8 is very close to 2/5 is not a coincidence, but is based on the properties of Fibonacci numbers. Note that 2, 3, 5, and 8 are consecutive Fibonacci numbers. One can create a similar paradoxical dissection by using any four consecutive Fibonacci numbers.

But a lingering doubt still remains. The gap looks pretty small — how do we know that it has area 1? There are many ways to verify this, but one of the most interesting ways is to use Pick’s Theorem. Pick’s theorem states that if a polygon is drawn on a grid such that all vertices are at grid points (points with integer coordinates), then the area is equal to i + b/2 − 1, where i is the number of interior grid points and b is the number of grid points on the boundary. In this case, there are four grid points on the boundary: (0,0), (8,3), (13,5), and (5,2). There are no grid points in the interior, so the area is 0 + 4/2 − 1 = 1.

This dissection is apparently due to Sam Loyd. The missing square puzzle is a famous variant that is based on dissecting a right triangle instead of a square.