Curry's Paradox:
How Is It Possible?
According to Martin Gardner, a New York city amateur magician Paul Curry invented the following paradox in 1953.
Off a right triangle with legs 13 and 5 one can cut two smaller triangles with legs 8, 3 and 5, 2, respectively. The small triangles could be fitted into the angles of the given triangle in two different ways. In one case a 5×3 rectangle of area 15 is left over. In the other case, we the left-over rectangle has dimensions 8×2 and area 16. Which is a wonder in its own right.
Similar dissections have been described yet by William Hooper in 1794 (Rational Recreations, vol. 4, p. 286, see [Gardner, p. 131] and a dynamic illustration). The applet below displays a single parameter (call it n) such that the dissection applies to a right triangle with legs F_{n+1} and F_{n-1}, where F_{k} is the k^{th} Fibonacci number. The two rectangles then have dimensions F_{n-1}×F_{n-2} and F_{n}×F_{n-3}, with areas that always differ by 1:
(1) | F_{n}·F_{n-3} - F_{n-1}·F_{n-2} = (-1)^{n}, |
which, like Cassini's identity, is a variant of
(2) | F_{n+1}·F_{m} - F_{n}·F_{m+1} = (-1)^{n} F_{m-n}, |
known as d'Ocagne's identity (after Philbert Maurice d'Ocagne (1862-1938)). (1) is obtained from (2) by setting
Paul Curry has observed that (for
Curry's paradox is presented by the applet bellow. (Drag the shapes from their positions in the upper portion of the applet into the designated locations in its lower portion.) Hooper's appears elsewhere.
What if applet does not run? |
(In the applet you can either drag the pieces manually or make them move automatically by pressing the "Animate" button.)
The resolution of the apparent paradoxes is quite simple. The appearances notwithstanding, the three triangles involved have different angles so that their hypotenuses have different slopes. For n > 4, the discrepancy is imperceptible. But the cases
(Another explanation not surprisingly invokes the golden ratio.)
References
- M. Gardner, Mathematics Magic and Mystery, Dover, 1956, pp. 139-150
Dissection Paradoxes
- Curry's Paradox
- Dissection of a 10×13 Rectangle into Two Chessboards
- A Faulty Dissection
- Hooper's Paradox
- John Sharp's Paradox
- Langman's Paradox
- Popping A Square
- Sam Loyd's Son's Dissection
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