A bamboozlement is a dissection of a plane shape and subsequent rearrangement of the pieces that appears to result in a change of the total area. Use the activity below to explore some bamboozlements of a square and learn about the amazing connection to the Fibonacci numbers.


Fibonacci Bamboozlement
Let's do something impossible!

Drag the coloured pieces from the square to the rectangle and see what happens to the total area…

What is going on?

(Press and hold the background to bring up the magnifying glass.)



About the activity

The dimensions of the square and the associated rectangle can be recognised as three consecutive terms of the Fibonacci sequence:

        1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

leading to a discussion of Fibonacci numbers in nature and the golden ratio in art and architecture.

The paradox can be explained by considering the ratios of successive terms:

3⁄2 = 1.5
5⁄3 ≈ 1.6667
8⁄5 = 1.6
13⁄8 = 1.625
21⁄13 ≈ 1.6153
34⁄21 ≈ 1.6190
55⁄34 ≈ 1.6176

These values are simply related to the mis-matched slopes in the dissection, and rapidly approach the Golden Ratio ≈ 1.6180339887499….

Wouldn't it be nice if this could be used to create an infinite block of chocolate!

Pick's Theorem is relevant to calculating polygonal areas on maps, and can be used to demonstrate that the missing or extra area is exactly 1 square. To use Pick's theorem on a polygon with all b vertices at grid points (as we have here), count the grid points inside the polygon (call this i), then the area is equal to i + b⁄2 − 1. For the dissection, there are always 4 vertices and no internal grid points, giving an area of 0 + 4⁄2 − 1 = 1 in each case.