Richard Duncan's area puzzle has the following explanation.
The upper "triangle" in Richard's diagram is actually the
quadrilateral figure ACDG (Please refer to the attached image).
The area of ACDG is
triangle ACH
12
triangle
CDE 5
rectangle CEGH 15
total
32
The lower "triangle" is actually the slightly larger
quadrilateral figures ABDG.
The area of ABDG is
triangle ABI 5
triangle BDF 12
rectangle BFGI 16
total 33
triangle ABI 5
triangle BDF 12
rectangle BFGI 16
total 33
The difference between the two figures is the quadrilateral ABDC, which is
exactly one unit in area.
This puzzle is based on the fact that 5 x 13 - 8 x 8 =1. The three
numbers 5,8,13 are consecutive terms from the Fibonnaci series. This particlular
example and its generalization have apparently engaged the attention of a number
of mathematicians, Lewis Carrol among them. See "Riddles in
Mathematics" , Eugene P Northrop, Penguin Books 1961, page 54.
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