

Bogus geometry problem
Name: murphy
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
It is possible to divide up a square into four pieces and rearrange them to
give the square a greater area. Of course, you laugh, but it works, like it
or not. A square that 8" on each side has an area of 64 square inches. Now
cut down three inches from the top straight across. Cut this top rectangle
(8x3") diagonally, so two triangles a are made whose two straight bases are
3 and 8 inches each. They share a common diagonal. Still having the
overall square form, cut 3 inches from the bottom left hand corner, to
exactly three inches in from the left right hand side, at three inches down
from the right hand corner at the top. (Note: The cut from the bottom left
is on the bottom of the square, not the side of it.) If you do this right
you get two triangles and two quadrilaterals. By arranging these so the
three inch face of the quads, matches the three inch base of the triangles,
you get a rectangle with an area of 65 square inches. Yes, it is true! So
much for law of conservation of energy and matter! An extra cubic inch from
nowhere! Enjoy this one, fellows. It can make you rich at the local
hangout!
Replies:
BOGUS! This is from a recent issue of the International Journal of Theoret
ical Physics and has since been shown to be bogus. The trick is in the
rearrangement there are very small gaps.
hawley
Actually though, the axioms often used in mathematics allow something
essentially equivalent to this to be done, if you allow people to make cuts
of somewhat infinite complexity. There was a cute article describing this
in the "recreational mathematics" section of Scientific American some years
back. Sorry I do not have the exact reference. Of course, it is not a very
practical approach (particularly since materials are not infinitely divisi
ble  there are atoms down there after all) so this is really just something
for mathematicians to have fun with and make them worry about the consisten
cy of some of their axioms.
asmith
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Update: June 2012

