The two figures shown below are divided into four parts. The upper figure has two triangles, A and B, and two L-shaped areas, C and D. The lower figure has the same four parts arranged in a different order. Both figures have a width of 13 and a height of 8. However there is also a 1-by-1 hole in the lower figure. Where did the extra space come from?
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The slanting upper edge of the upper figure is not a straight line. It is made up of two lines that meet where they touch area C. This edge bends downward, so the area of the figure is 32. For the lower figure, the slanting upper edge is also made up of two lines that meet where they touch area C. This edge bends upward so the area of the figure is 33. The difference accounts for the hole.
The hypotenuses of triangles A and B have different slopes, therefore, both upper and lower 5x13 figures are quadrilaterals, not a triangles. The triangles A and B have areas of 12 and 5 and the other two pieces have areas of 7 and 8. The total area of the four pieces is 32 for each of two the figures. If the upper figure is rotated 180° and moved over the lower figure as shown below, the resulting rectangle has an area of 65. This is twice the area of 32 for the four pieces plus the area of the hole.
This puzzle can be analyzed by using Fibonacci numbers (also here), the series 1, 1, 2, 3, 5, 8, 13, 21,... The two figures have a height of 5 and base of 13, both Fibonacci numbers. The tangents of the two triangles in the figures are 2/5 and 3/8, also Fibonacci numbers. A similar puzzle can be made with the next sequence of Fibonacci numbers. The figures would have a height of 8 and base of 21. The tangents of the two triangles would be 3/8 and 5/13. Then the slopes of the two triangles would be closer together and the sag in the slanting upper edge would be even less noticeable.
Also see: -- ¤ Scott's Amazing Card Trick
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