curry's paradox

Curry's Paradox:
How Is It Possible?

The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, because


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 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.