What Is Wrong?
(Fibonacci Bamboozlement)

Bamboozlement is a term suggested by Greg Frederickson to describe plane dissections, followed by rearrangement of pieces that result in a figure of supposedly different area. One such bamboozlement, besides Frederickson's book, can be found in a couple of M. Gardner books [Puzzles, Magic] and the classical one by W. W. Rouse Ball.

Drag the pieces from the square on the left to the rectangle on the right. Compute and compare areas of the square and the rectangle. Hmm. Something is wrong. What is it?

Copyright © 1996-2010 Alexander Bogomolny

When the pieces put together to form a rectangle, the inner edges of the polygons do not fit together. This is clearly noticeable for small values of the control parameter. For larger values the discrepancy is virtually imperceptible. The explanation involves the famous Fibonacci numbers.

There is a "dishonest" version of the applet that makes it more difficult to notice the discrepancies.

Copyright © 1996-2010 Alexander Bogomolny