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.

A right triangle with legs 13 and 5 can be cut into two triangles (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 a 5×3 rectangle of area 15 is left over. In the other case, we get an 8×2 rectangle of 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:

which, like Cassini's identity, is a variant of

known as d'Ocagne's identity (after Philbert Maurice d'Ocagne (1862-1938)). (1) is obtained from (2) by setting m = n - 2.

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.

(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 n = 4 and n = 3 demonstrate this fact forcibly.

(Another explanation not surprisingly invokes the golden ratio.)

References

1. 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

Fibonacci Numbers

1. Ceva's Theorem: A Matter of Appreciation
2. When the Counting Gets Tough, the Tough Count on Mathematics
3. I. Sharygin's Problem of Criminal Ministers
4. Single Pile Games
5. Take-Away Games
6. Number 8 Is Interesting
7. Curry's Paradox
8. A Problem in Checker-Jumping
9. Fibonacci's Quickies

Golden Ratio

1. Golden Ratio in Geometry
2. Golden Ratio in an Irregular Pentagon
3. Golden Ratio in a Irregular Pentagon II
4. Inflection Points of Fourth Degree Polynomials
5. Wythoff's Nim
6. Inscribing a regular pentagon in a circle – and proving it
7. Cosine of 36 degrees
8. Continued Fractions

Copyright © 1996-2010 Alexander Bogomolny