Hooper's Paradox:
How Is It Possible?

This page is dedicated to the valiant effort by Douglas Rogers to establish the true date of the first appearance of what Martin Gardner called Hooper's paradox. In his small book, M. Gardner indicates that the paradox can be found in the 4th volume (p. 286) of the 1794 edition of William Hooper's Rational Recreations. The point of the inquiry is that the reference is to the 4th edition of the book, the first having appeared 20 years earlier, in 1774. Was the paradox included (or even known) at the time of the original writing, or was it conceived some time in the intervening twenty years before the 4th edition? I trust we'll know the truth soon.

The truth, according to Douglas Rogers, is that the puzzle has been included in the first edition of Hooper's book with a conspicuous error in the diagram. Instead of a 2×6 rectangle Hooper drew a 3×6 shape. The error has been corrected in later editions. But a question arose, How does one make such a pronounced mistake? Driven by curiosity, Professor Rogers embarked on a search that led him to a discovery that the puzzle in this form has appeared in a 1769-1770 collection Nouvelles récréations physiques et mathématiques by the French author Edmé Gilles Guyot. Guyot has corrected his mistake in the second edition of his work even before the appearance of Hooper's first edition. Comparison of the two works by expert librarians at the University of London and University of Aberdeen proved beyond a reasonable doubt that Hooper cribbed the puzzle from Guyot's text probably along with much else.

So what is the puzzle?

A 3×10 rectangle is cut into two equal triangles that form a 2×6 rectangle and two equal trapezoids that combine into a 4×5 rectangle.

Thus an area of 30 square units is being transformed into 32 by mere cutting and rearrangement of the pieces. Not for nothing Hooper called his paradox "geometric money." As M. Gardner observed, Hooper's paradox can be given an infinite number of forms by varying the proportions of the figures and the degree of slope of the diagonal. It can be constructed so that the loss and gain is 1 square unit, 2, 3, 4, 5, and so on up to infinity. Obviously, the smaller the difference the more difficult is to detect the deceiving nature of the construction.

[Frederickson, pp. 271-273] credits Sebastiano Serlio (1475-1554), an Italian architect and mathematician, with one of the variants that transformed the same 3×10 rectangle into a 4×7 and 1×3 rectangles. Curiously, as Prof. David Singmaster has observed, Serlio has been solving a practical problem of converting one rectangle into another and has not noticed that the two had different areas.

The applet below presents a slight modification of Hooper's construction. The four shapes comprising the original rectangle can be dragged to the right into their designated positions on the right.

What if applet does not run?

Three parameters are under the user control: the sides of the rectangle and the length of the horizontal leg of the triangles.

References

  1. G. N. Frederickson, Dissections: Plane & Fancy, Cambridge University Press, 1997
  2. M. Gardner, Mathematics Magic and Mystery, Dover, 1956, pp. 131-132

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