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.
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:
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.