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© Copyright 2000, Jim Loy
Here is a famous puzzle. I have divided up an 8x8 square into two
triangles and two trapezoids. To the right of that, I have rearranged the four
pieces to fit into a 5x13 grid. The area of the left figure is 64 (8x8), and
the area of the right figure is 65 (5x13). We seem to have gained a square. How
can that be?
Answer:
The answer is that the figure on the right (above) is not a rectangle. Instead it is a rectangle with a hole in the middle. If I had drawn the pictures larger, and more accurately, you would see that the hole is a very long, thin parallelogram (with an area of 1), like this (I've magnified it to 2x):
With the hole cut out of it, the rectangle has an area of 64.
Addendum:
Here is a similar problem, called the Curry triangle. We start with
an isosceles triangle with base 10 and height 12, and dissect it as shown. We
then rearrange the pieces to form the same triangle, as shown in the right side
of that diagram. But now there is a hole in the middle (the gray rectangle). We
seem to have gained some area. Any ideas?
Answer: The different right triangles in the first figure are all similar, so it may seem that the second figure (made from the same pieces) must be the same shape. But the base of the second figure is significantly longer (10.42), and there should be a tiny gap between the upper triangles of the second figure (nearly invisible as drawn). The two figures are not the same. Close, but no banana.
The difference between the two triangles depends upon how you draw
the two figures. To illustrate that, here is a very similar puzzle (slightly
different dimensions). In this one the pieces match up nicely, but there is
still a hole (in gray) in the second figure. The differences between the top
and bottom triangles will probably be a mystery to you, unless you measure
everything carefully. In the upper triangle, the vertical line segment labeled
"2" is much shorter than 2. If it were 2, and both large triangles are then
accurately drawn, then you might notice that the long hypotenuse of the upper
triangle will be bent downward, and the long hypotenuse of the lower triangle
will be bent upward. In fact, neither large triangle is an actual triangle. And
the area of the lower figure is one unit larger than the area of the upper
figure.
See Sam Loyd's Get Off the Earth.