Sources page biographical material

6.P.2. OTHER TYPES
In several early examples, the authors appear unaware that area has vanished!
Pacioli. De Viribus. c1500. Ff. 189v - 191r. Part 2. LXXIX. Do(cumento). un tetragono saper lo longare con restregnerlo elargarlo con scortarlo (a tetragon knows lengthening and contraction, enlarging with shortening ??) = Peirani 250-252. Convert a 4 x 24 rectangle to a 3 x 32 using one cut into two pieces. Pacioli's

description is cryptic but seems to have two cuts, making d c

three pieces. There is a diagram at the bottom of f. 190v, badly k f e

redrawn on Peirani 458. Below this is a inserted note which Peirani

252 simply mentions as difficult to read, but can make sense. The g  

points are as laid out at the right. abcd is the original 4 x 24 h a o b

rectangle. g is one unit up from a and e is one unit down from c.

Cut from c to g and from e parallel to the base, meeting cg at f. Then move cdg to fkh and move fec to hag. Careful rereading of Pacioli seems to show he is using a trick! He cuts from e to f to g. then turns over the upper piece and slides it along so that he can continue his cut from g to h, which is where f to c is now. This gives three pieces from a single cut! Pacioli clearly notes that the area is conserved.

Although not really in this topic, I have put it here as it seems to be a predecessor of this topic and of 6.AY.

Sebastiano Serlio. Libro Primo d'Architettura. 1545. This is the first part of his Architettura, 5 books, 1537-1547, first published together in 1584. I have seen the following editions.

With French translation by Jehan Martin, no publisher shown, Paris, 1545, f. 22.r. ??NX

1559. F. 15.v.

Francesco Senese & Zuane(?) Krugher, Venice, 1566, f. 16.r. ??NX

Jacomo de'Franceschi, Venice, 1619, f. 16.r.

Translated into Dutch by Pieter Coecke van Aelst as: Den eerstē vijfsten boeck van architecturē; Amsterdam, 1606. This was translated into English as: The Five Books of Architecture; Simon Stafford, London, 1611 = Dover, 1982. The first Booke, f. 12v.

3 x 10 board is cut on a diagonal and slid to form a 4 x 7 table with 3 x 1 left over, but he doesn't actually put the two leftover pieces together nor notice the area change!

Pietro Cataneo. L'Architettura di Pietro Cataneo Senese. Aldus, Venice, 1567. ??NX. Libro Settimo.

P. 164, prop. XXVIIII: Come si possa accresciere una stravagante larghezza. Gives a correct version of Serlio's process.

P. 165, prop. XXX: Falsa solutione del Serlio. Cites p. xxii of Serlio. Carefully explains the error in Serlio and says his method is "insolubile, & mal pensata".

Schwenter. 1636. Part 15, ex. 14, p. 541: Mit einem länglichten schmahlen Brett / für ein bräites Fenster einen Laden zu machen. Cites Gualtherus Rivius, Architectur. Discusses Serlio's dissection as a way of making a 4 x 7 from a 3 x 10 but doesn't notice the area change.

Gaspar Schott. Magia Universalis. Joh. Martin Schönwetter, Bamberg, Vol. 3, 1677. Pp. 704-708 describes Serlio's error in detail, citing Serlio. ??NX of plates.

I have a vague reference to the 1723 ed. of Ozanam, but I have not seen it in the 1725 ed. -- this may be an error for the 1778 ed. below.

Minguet. 1755. Pp. not noted -- ??check (1822: 145-146; 1864: 127-128). Same as Hooper. Not in 1733 ed.

Vyse. Tutor's Guide. 1771? Prob. 8, 1793: p. 304, 1799: p. 317 & Key p. 358. Lady has a table 27 square and a board 12 x 48. She cuts the board into two 12 x 24 rectangles and cuts each rectangle along a diagonal. By placing the diagonals of these pieces on the sides of her table, she makes a table 36 square. Note that 36² = 1296 and 27² + 12 x 48 = 1305. Vyse is clearly unaware that area has been created. By dividing all lengths by 3, one gets a version where one unit of area is lost. Note that 4, 8, 9 is almost a Pythagorean triple.

William Hooper. Rational Recreations. 1774. Op. cit. in 4.A.1. Vol. 4, pp. 286 287: Recreation CVI -- The geometric money. 3 x 10 cut into four pieces which make a 2 x 6 and a 4 x 5. (The diagram is shown in Gardner, MM&M, pp. 131 132.) (I recently saw that an edition erroneously has a 3 x 6 instead of a 2 x 6 rectangle. This must be the 1st ed. of 1774, as it is correct in my 2nd ed. of 1782.)

Ozanam-Montucla. 1778. Transposition de laquelle semble résulter que le tout peut être égal à la partie. Prob. 21 & fig. 127, plate 16, 1778: 302-303 & 363; 1803: 298-299 & 361; 1814: 256 & 306; 1840: omitted. 3 x 11 to 2 x 7 and 4 x 5. Remarks that M. Ligier probably made some such mistake in showing 17² = 2 x 12² and this is discussed further on the later page.

E. C. Guyot. Nouvelles Récréations Physiques et Mathématiques. Nouvelle éd. La Librairie, Rue S. André des Arcs[sic], Paris, Year 7 [1799]. Vol. 2, Deuxième récréation: Or géométrique -- construction, pp. 41 42 & plate 6, opp. p. 37. Same as Hooper.

Manuel des Sorciers. 1825. Pp. 202-203, art. 19. ??NX Same as Hooper.

The Boy's Own Book. The geometrical money. 1828: 413; 1828-2: 419; 1829 (US): 212; 1855: 566 567; 1868: 669. Same as Hooper.

Magician's Own Book. 1857. Deceptive vision, pp. 258-259. Same as Hooper. = Book of 500 Puzzles, 1859, pp. 72-73.

Illustrated Boy's Own Treasury. 1860. Optics: Deceptive vision, p. 445. Same as Hooper. Identical to Book of 500 Puzzles.

Wemple & Company (New York). The Magic Egg Puzzle. ©1880. S&B, p. 144. Advertising card, the size of a small postcard, but with ads for Rogers Peet on the back. Starts with 9 eggs. Cut into four rectangles and reassemble to make 6, 7, 8, 10, 11, 12 eggs.

R. March & Co. (St. James's Walk, Clerkenwell). 'The Magical Egg Puzzle', nd [c1890]. (I have a photocopy.) Four rectangles which produce 6, 7, ..., 12 eggs. Identical to the Wemple version, but with Wemple's name removed. I only have a photocopy of the front of this and I don't know what's on the back. I also have a photocopy of the instructions.

Loyd. US Patent 563,778 -- Transformation Picture. Applied: 11 Mar 1896; patented: 14 Jul 1896. 1p + 1p diagrams. Simple rotating version using 8 to 7 objects.

Loyd. Get Off the Earth. Puzzle notices in the Brooklyn Daily Eagle (26 Apr   3 May 1896), printing individual Chinamen. Presenting all of these at an office of the newspaper gets you an example of the puzzle. Loyd ran discussions on it in his Sunday columns until 3 Jan 1897 and he also sold many versions as advertising promotions. S&B, p. 144, shows several versions.

Loyd. Problem 17: Ye castle donjon. Tit Bits 31 (6 & 27 Feb & 6 & 20 Mar 1897) 343, 401, 419 & 455. = Cyclopedia, 1914, The architect's puzzle, pp. 241 & 372. 5 x 25 to area 124.

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. Discusses and shows Get Off the Earth.

Ball. MRE, 4th ed., 1905, pp. 50-51: Turton's seventy-seven puzzle. Additional section describing Captain Turton's 7 x 11 to 7 x 11 with one projecting square, using bevelled cuts. This is dropped from the 7th ed., 1917.

William F. White. 1907 & 1908. See entries in 6.P.1.

Dudeney. The world's best puzzles. Op. cit. in 2. 1908. Gives "Get Off the Earth" on p. 785.

Loyd. Teddy and the Lions. Gardner, MM&M, p. 123, says he has seen only one example, made as a promotional item for the Eden Musee in Manhattan. This has a round disc, but two sets of figures -- 7 natives and 7 lions which become 6 natives and 8 lions.

Dudeney. A chessboard fallacy. The Paradox Party. Strand Mag. 38 (No. 228) (Dec 1909) 676 (= AM, prob. 413, pp. 141 & 247). (There is a solution in Strand Mag. 39 (No. 229) (Jan 1910) ??NYS.) 8 x 8 into 3 pieces which make a 9 x 7.

Fun's Great Baseball Puzzle. Will Shortz gave this out at IPP10, 1989, as a colour photocopy, 433 x 280 mm (approx. A3). ©1912 by the Press Publishing Co (The New York World). I don't know if Fun was their Sunday colour comic section or what. One has to cut it diagonally and slide one part along to change from 8 to 9 boys.

Loyd. The gold brick puzzle. Cyclopedia, 1914, pp. 32 & 342 (= MPSL1, prob. 24, pp. 22 & 129). 24 x 24 to 23 x 25.

Loyd. Cyclopedia. 1914. "Get off the earth", p. 323. Says over 10 million were sold. Offers prizes for best answers received in 1909.

Loyd Jr. SLAHP. 1928. "Get off the Earth" puzzle, pp. 5 6. Says 'My "Missing Chinaman Puzzle"' of 1896. Gives a simple and clear explanation.

John Barnard. The Handy Boy's Book. Ward, Lock & Co., London, nd [c1930?]. Some interesting optical illusions, pp. 310-311. Shows a card with 11 matches and a diagonal cut so that sliding it one place makes 10 matches.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 24: A chessboard fallacy, pp. 28-29. 8 x 8 cut with a diagonal of a 8 x 7 region, then pieces slid and a triangle cut off and moved to the other end to make a 9 x 7. Clear illustration.

Mel Stover. From 1951, he devised a number of variations of both Get off the Earth (perhaps the best is his Vanishing Leprechaun) and of Teddy and the Lions (6 men and 4 glasses of beer become 5 men and 5 glasses). I have examples of some of these from Stover and I have looked at his notebooks, which are now with Mark Setteducati. See Gardner, MM&M, pp. 125-128.

Gardner. SA (May 1961) c= NMD, chap. 11. Mentioned in Workout, chap. 27. Describes his adaptation of a principle of Paul Curry to produce The Disappearing Square puzzle, where 16 or 17 pieces seem to make the same square. The central part of the 17 piece version consists of five equal squares in the form of a Greek cross. The central part of the 16 piece version has four of the squares in the shape of a square. This has since been produced in several places.

Ripley's Puzzles and Games. 1966. P. 60. Asserts that when you cut a 2½ x 4½ board into six right triangles with legs 1½ and 2½, then they assemble into an equilateral triangle of edge 5. This has an area loss of about 4%.

John Fisher. John Fisher's Magic Book. Muller, London, 1968.

Financial Wizardry, pp. 18-19. 7 x 8 region with £ signs marking the area. A line cuts off a triangle of width 7 and height 2 at the top. The rest of the area is divided by a vertical into strips of widths 4 and 3, with a small rectangle 3 by 1 cut from the bottom of the width 3 strip. When the strips are exchanged, one unit of area is lost and one £ sign has vanished.

Try-Angle, pp. 126-127. This is one of Curry's triangles -- see Gardner, MM&M, p. 147.

Alco-Frolic!, pp. 148-149. This is a form of Stover's 6 & 4 to 5 & 5 version.

D. E. Knuth. Disappearances. In: The Mathematical Gardner; ed. by David Klarner; Prindle, Weber & Schmidt/Wadsworth, 1981. P. 264. An eight line poem which rearranges to a seven line poem.

Dean Clark. A centennial tribute to Sam Loyd. CMJ 23:5 (Nov 1992) 402 404. Gives an easy circular version with 11 & 12 astronauts around the earth and a 15 & 16 face version with three pieces, a bit like the Vanishing Leprechaun.