in Dissections: Plane & Fancy, by Greg N. Frederickson

- David Singmaster's article on vanishing area puzzles
- Erroneous spelling!
- Abu'l-Wafa exposed cheats
- A double-cheat by Hooper?
- More discussion of the chessboard paradox
- A related paradox
- Oskar Schlömilch
- Torsten Sillke's article on bamboozlement
- Peter Tappan's tile puzzle

David Singmaster (London South Bank University) has just recently published a wonderful article:"Vanishing area puzzles",The scholarship and the historical illustrations are just superb!Mathematics Magazine, No. 1, (March 2014), pp. 10-21.

Sherman Stein asked if "tournay" should not in fact be "tourney". This may be the use of an archaic spelling in Fiske's book, and unfortunately I do not have photocopies of the relevant pages to check. The Oxford English dictionary lists "tournay" as an obsolete form of "tourney", and gives examples of the use of "tournay" in 1820 and 1878. I also ran the chapter from the manuscript through a spelling checker, which I must have forgotten to do back in 1996. The checker picked up misspellings "erronious" and "irresistable", which (of course) should be "erroneous" and "irresistible". Of the two, a copy editor appears to have caught the latter. It seems symbolic, and almost irresistible, that a chapter on dissection fakes should contain erroneous spelling.

(This is corrected in the paperback edition.)

A fascinating article has appeared that discusses Abu'l-Wafa's manuscript "On the Geometric Constructions Necessary for the Artisan". It is "Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World", by Alpay Özdural, has appeared inHistoria Mathematica, vol. 27 (2000), pp. 171-201. Follow this link to view the abstract. In the manuscript, Abu'l-Wafa discussed two incorrect methods that Islamic artisans had been using to dissect three squares to one.

One of these methods used just six pieces: One square was left uncut, and one square was cut on its diagonal. The third square was cut into three pieces, of which one is an isosceles right triangle of area one quarter of the square, and the other two pieces are mirror images of each other. With a little bit of thought, you will see that these pieces assemble to give something close to, but not exactly, a square. For an artisan, this would be good enough, because he could hide the approximation with the grout lines in an actual tile floor.

Douglas Rogers has found that Hooper's "Geometric Money" paradox was apparently taken from the 1769-1770 collectionNouvelles récréations physiques et mathétiquesby the French author Edmé Gilles Guyot. Actually, Guyot's work contained a mistake (a 3x6 rectangle rather than a 2x6 rectangle), that Guyot corrected before Hooper's first edition in 1774. Hooper's first edition still contained that original mistake, which was corrected in later editions. So it looks like eighteenth-century dissection enthusiasts got a double dose from William Hooper!

About a decade after Oskar Schlömilch published the chessboard paradox, V. Schlegel, in Waren, Germany, wrote an article that generalized the paradox, analyzed it in terms of recurrence relations, and elucidated the link to the Fibonacci numbers and (indirectly) the golden ratio:Schlegel, V. (1879). Verallgemeinerung eines geometrischen Paradoxons.Zeitschrift für Mathematik und Physik24, 123-128.

Professor Noam Elkies displayed a related paradox on one of his webpages: Noam's Mathematical Miscellany, in which he warned us "to beware of dissection proofs!" His example "proof" purports to prove that 88=89=90=91, giving (supposedly) four 9 x 20 rectangles formed, respectively from two right triangles with legs of length 4 and 9, two right triangles with legs of length 5 and 11, and two rectangles. The two rectangles are of four different combinations: {two 4 x 11 rectangles}, {one 4 x 11 rectangle and one 5 x 9 rectangle}, {two 5 x 9 rectangles}, {one 4 x 9 rectangle and one 5 x 11 rectangle}.

Professor Elkies supplied an illustration that seems to back up the claim that 88=89=90=91. But how has he bamboozled us? His drawing is correct only if a right triangle with legs of length 4 and 9 is similar to right triangle with legs of length 5 and 11. And therein lies the hornswoggle! And yes, Noam. We promise to be careful with dissection proofs!

Oskar Schlömilch was born in 1823 in Weimar, Germany. After study in Jena, Berlin, and Vienna, he earned a Ph.D. in Jena in 1842, where he became a privatdozent and then an ausserordentlicher professor. In 1849 he was appointed professor of higher mathematics and analytic mechanics at the polytechnic in Dresden. He was the founder (in 1856) and longtime publisher ofZeitschrift für Mathematik und Physik,and published a textbook with Osmar Fort on analytic geometry. Oskar Schlömilch died in 1901.

Torsten Sillke has prepared an interesting article with a substantial bibliography on bamboozlement: Jigsaw Paradox.

Harold Cataquet alerted me to some fascinating work on bamboozlement by the magician Peter Tappan. The puzzle consists of 11 pieces that fit together to form a 7x11 rectangle. Three of the pieces are 1x1 squares, and for three consecutive times Tappan can dispose of one of these squares and yet paradoxically still be able to form a 7x11 rectangle! Tappan formed several of the pieces with slanted cuts, and he exploits these slanted cuts as he rearranges the pieces. He also made the slanted cuts at slightly different angles, and he exploits that by rotating first one, and then another, of the pieces by 180 degrees. The paradox is certainly niftier than the ones that I mention in my book.

Tappan discussed his puzzle in his article, "FuTILE Subtraction",The Linking Ring, vol. 80, no. 10 (October, 2000), pp. 112-119. I cannot present any of the illustrations here because the article is under copyright. In the article, Tappan also described an earlier version of the same sort of puzzle, but not as impressive, that was created by the late magician Winston Freer.The Linking Ringmagazine is published monthly by the International Brotherhood of Magicians.

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