As in Arthur C. Clarke’s “The Nine Billion Names of God” (1953), mathematics can provide a shortcut to the Day of Judgement. But... is it worthwhile?

The idea came to my mind during a congress, when I gave a talk just before **Gianni Sarcone**, who showed this vanishing area puzzle:

I needed to design a puzzle for my book about the end of the world in 2012 and I asked Gianni to reshape the picture of 7/8 eggs to support the narrative of 12/13 crystal skulls. The result was published in *2012: The end-of-the-world game* (1) :

The first example of vanishing area puzzles was reported in the book *Libro d’Architettura Primo* by **Sebastiano Serlio** (1475-1554), an Italian architect of the Renaissance. (2) :

Sometimes strange accidents happen to the architect: in situations like the one here described, the methods of mathematicians will help. He has a single panel 10 feet × 3 feet and he needs to convert it into a 7 feet × 4 feet panel. Cutting it vertically, its height will double and will reach 6 feet but he needs 7. At the same time, cutting a lateral slice of 3 feet, it will not satisfy the need because the remaining part will have a size of 7 feet × 3 feet. In order to get the result, he has to follow these instructions. The panel, 10 feet × 3 feet, has four vertices A, B, C, D: cut it on its diagonal CB and slide the upper part towards B until the distance between the point A and its projection F will be 4 feet. The same will happen with the distance ED, and the distance AE will be 7 feet, so that the new panel AEFD will be 7 feet × 4 feet. Two additional triangles allow to create a small door: CFC and EBG. (3)

Sebastiano Serlio, *Libro d’Architettura Primo*, 1566, p. 16 (read it here)

He cites the case of a 3×10 rectangle which can be converted into a rectangle 4×7 and two triangles 1×3. Curiously, Serlio didn’t noticed that the sum of the three resulting areas (31 square feet) was greater than the original one (30 square feet):

In 1769 **Edme Gilles Guyot** (1706-1786) described the vanish area paradox in the second volume of his *Nouvelles récréations physiques et mathématiques* (read it here): a 10 × 3 square can be divided in four pieces which, rearranged, define a 5 × 4 and a 6 × 2 rectangle. (4)

Edme Gilles Guyot, *Nouvelles récréations physiques et mathématiques*,

Vol.2, planche 6, 1799 edition (see here the *planche* 6)

Guyot suggested to draw a coin on each square and to use this paradox to tease alchemists and their claim of producing gold in mystical ways. Curious detail: the diagram given in the first edition of the book was wrong, and it was corrected in the second edition.

In the 1774 edition of *Rational Recreations* (5) , a math puzzle book written by William Hooper (1742-1790) which partly plagiarized Guyot’s work (read it here), the error was repeated:

William Hooper, *Rational Recreations*, Volume 4, 1774, pp. 286-287. (read it here)

It was corrected in the 1782 edition of Hooper’s book (read it here).

The second rectangle was 3×6 instead of 2×6.

In 1979 **Mitsunobu Matsuyama** created “Paradox”, a simple vanishing area puzzle involving a playing card: the back of a playing card, when reversed, shows a King of Diamonds, augmented of a small additional square coming from nowhere.

It was published by Karl Fulves in the issue 18 of *Chronicles* (6)

**Winston Freer** (1910-1981) created a more intricate version, used in this video by John Rogers to explain relativity theory: it is still more astonishing because it involves three different steps.

The structure of Freer’s puzzle is described in detail by Peter Tappan in “FuTILE Subtraction”. (7)

In the article, **Peter Tappan** describes an improved version of the vanishing area puzzle, which in 1994 won the I.B.M. Originality Contest.

In 1980 **Tofique Fatehi** created “The eleven holes puzzle”, an extended version of Winston Freer’s puzzle involving a progression of 11 vanishing squares!

It starts from here:

A detailed description of its structure can be downloaded from his website.

Robert Page marketed a “Business Card Paradox”, a simplified version of Matsuyama’s Paradox in which the rectangular vanishing area was replaced with a personal business card.

A tutorial is available to create a similar puzzle (see it on YouTube).

In 1998 Gianni Sarcone created *TangraMagic*, an extended Tangram involving a vanishing square, cut in foam. The 10 pieces (7 original and 3 additional) can be rearranged so that only 9 pieces fit the same area in a frame. (8)

His website offers a tutorial to create it.

*Many thanks to Max Maven for his precious contributions to this article. (9) Gianni Sarcone also provided useful information about the topic.*

1. Mariano Tomatis, *2012 è in gioco la fine del mondo*, Iacobelli, Roma 2010.

2. Greg N. Frederickson, *Dissections: Plane and Fancy*, Cambridge University Press, 1997, pp.271-273.

3. Sebastiano Serlio, *Libro d’Architettura Primo*, 1566, p. 16.

4. Edme Gilles Guyot, *Nouvelles récréations physiques et mathématiques*, Vol.2, planche 6, 1799 edition, pp. 38-39. This reference was found by Douglas Rogers.

5. William Hooper, *Rational Recreations : In which the Principles of Numbers and Natural Philosophy are Clearly and Copiously Elucidated, by a Series of Easy, Entertaining, Interesting Experiments. Among which are All Those Commonly Performed with the Cards*, L. Davis; J. Robson; B. Law; and G. Robinson, 1774, pp.286-287.

6. Mitsunobu Matsuyama, “Paradox” in *Chronicles* #18, 1979, pp.1235-1238.

7. Peter Tappan, “FuTILE Subtraction”, *Linking Ring*, october 2000, pages 112-119.

8. Gianni Sarcone, “Paradoxical Tangram and Vanishing Puzzles”, *Journal of Recreational Mathematics*, #29:2, 1998, pp. 132-133; problem 2424.

9. Post on Genii Forum “Amazing Geometric Vanish – MAGIC magazine”, 11/07/2010 and personal communications.