Posted by Soroban on August 14, 2002 at 00:19:02:
In Reply to: REALLY HARD MATH posted by Spike Spiegel on August 13, 2002 at 15:19:55:
: Can anyone figure this out:
: http://challenger.users1.50megs.com/trimathpuzzle.jpg
BUS-ted! Subhotosh is absolutely correct!
Those two right triangles are not similar.
Their hypoenuses (?) do NOT form a straight line.
Consider the areas involved.
If the large right triangle is correct, then with legs of 5 and 13, the area would be 32.5
Add up the areas of the separate pieces. The yellow triangle has area 5, the blue triangle
has area 12. The red and green pieces have a total of 15. This is a total of only 32.
Hence, in the first arrangement there is a half-unit missing.
Where? Along the "hypotenuse" which actually dips imperceptibly inward.
In the second diagram, the areas of the pieces are unchanged. Yellow = 5, blue = 12, red &
green = 15. These pieces, plus the 1-unit hole, have a total of 33 square units.
There is a gain of a half-unit, spread along the "hypotenuse" which bulges outward.
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There is a large variety of this type of puzzle/paaradox. The most popular is the
8x8 chessboard which can be "assembled" into a 5x13 rectangle.
The dimensions are usually taken from the Fibonacci Sequence: 1, 1, 3, 5, 8, 13, 21,...