September 23, 2006

How can this be true?

Winners: Simin Habibian (Sad Afarin!) & William Angmar (Sad Afarin!)
Sent by Amirhosein Hazrati

Simin Habibian (and "higher experts") wrote: The "triangles" are not the same size. The difference is very small >>> See diagram

See calculation below:

1-The area of big triangle is (13 X 5) : 2 = 32.5
2- The area of the smaller green triangle is (5x2) : 2 = 5
3- The area of the larger green shape is 8
4-The area of the red triangle is (8x3) :2 = 12
5- The area of the yellow shape is 7

The sum of 2, 3, 4, and 5 = 32

Also the 4 components do not fill the big triangle entirely. There is a very small space left between the hypotenuse of the big triangle and the hypotenuse of the red triangle. The area of that spece is 0.5 unit.

The sum of 0.5 and the (32.5 -32) = 1 and that is the one unit that is seen as vacant in the lower picture >>> See diagram

***

William Angmar wrote: The key to the puzzle is the fact that neither of the 13x5 "triangles" has the same area as its component parts.

The four figures (the yellow, red, blue and green shapes) total 32 units of area, but the triangles are 13 wide and 5 tall, which equals 32.5 units. The blue triangle has a ratio of 5:2, while the red triangle has the ratio 8:3, and these are not the same ratio. So the apparent combined hypotenuse in each figure is actually bent.

The amount of bending is around 1/28th of a unit, which is very difficult to see on the diagram of this puzzle, though just about possible.

According to Martin Gardner, the puzzle was invented by New York city amateur magician Paul Curry in 1953. Ever since it has been known as Curry's Paradox. The principle of a dissection paradox has however been known since the 1860s.

The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers. Many other geometric dissection puzzles are based on a few simple properties of the famous Fibonacci sequence.

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