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 Fn+1 and Fn-1, where Fk is the kth Fibonacci number. The two rectangles then have dimensions Fn-1×Fn-2 and Fn×Fn-3, with areas that always differ by 1:

(1) Fn·Fn-3 - Fn-1·Fn-2 = (-1)n,

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

(2) Fn+1·Fm - Fn·Fm+1 = (-1)n Fm-n,

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

Paul Curry has observed that (for n = 5) a 5×3 rectangle can be cut into two shapes that after a rearrangement fill an 8×2 rectangle with one square left out. Curry himself has been interested in rearrangements that create holes entirely inside the resulting figure. But the variant with a square hole on the perimeter of the figure seems to me more popular nowadays.

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.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


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 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

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
  10. Fibonacci Numbers in Equilateral Triangle
  11. Binet's Formula by Inducion
  12. Binet's Formula via Generating Functions
  13. Generating Functions from Recurrences
  14. Cassini's Identity
  15. Fibonacci Idendtities with Matrices
  16. GCD of Fibonacci Numbers
  17. Binet's Formula with Cosines
  18. Lame's Theorem - First Application of Fibonacci Numbers

|Activities| |Contact| |Front page| |Contents| |Geometry| |Fallacies| |Store|

Copyright © 1996-2017 Alexander Bogomolny

 62338904

Search by google: