Easy Solution to Dr Ron Knott's Fibonacci Puzzle

A Fibonacci Jigsaw puzzle or How to Prove 64=65 using Trignometry

First visit this page for the puzzle, after which see my solution below.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles2.html#jigsaw1

In the second figure, red coloured one,(http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles2.html#jigsaw1) where the area is coming to be 65, look at the upper right triangle.

Side AC = 8
CD = 3
AB = 5
BC = 3
BE = 2

Now since the triangle ABE ~ ACD we have

AB        AC
----- = ------
BE         CD

5      8
-- = ---
2      3

ie 15 = 16, which is not true.

Now lets find the actual length of BE,
using the above deduction,

AB        AC
----- = ------
BE         CD

5           8
---- = -----
BE         3

BE = 15/8 = 1.875

Now if BE is 1.875 and not 2 then there must be some gap between the pieces, making the total area 65 instead of 64.

Same way the explanation for the second jigsaw puzzle (green one, How to Prove 64=63!! ) can be given.

This puzzle I had solved in October 1999, and had received complements from Mr Knott at that time. He had asked me to put it on my web page, but I didnt had one at that time, and now since I have one I am putting the solution right here.

Akhilesh Singh
09 Feb 2006