Sources page biographical material

6.AY. DISSECT 3A x 2B TO MAKE 2A x 3B, ETC.
This is done by a 'staircase' cut. See 6.AS.
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 now 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.P.2.

Cardan. De Rerum Varietate. 1557, ??NYS. = Opera Omnia, vol. III, p. 248 (misprinted 348 and with running head Lib. XII in the 1663 ed.). Liber XIII. Shows 2A x 3B to 3A x 2B and half of 3A x 4B to 4A x 3B and discusses the general process.

Kanchusen. Wakoku Chiekurabe. 1727. Pp. 11-12 & 26 27. 4A x 3B to 3A x 4B, with the latter being square. Solution asserts that any size of paper can be made into a square: 'fold lengthwise into an even number and fold the width into an odd number' -- cf Loyd (1914) & Dudeney (1926) below.

Minguet. 1733. Pp. 117-119 (1755: 81-82; 1822: 136-137; 1864: 114-115). 3 x 4 to 4 x 3. Shows a straight tetromino along one side moved to a perpendicular side so both shapes are 4 x 4.

Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. See 4.B.1 for more details. F. 4r is "Analysis of the Essay of Games". F. 4v has an entry "8½ c Prob of figure" followed by a staircase piece. F. 145-146 show two pieces formed into both rectangles. There are other dissection problems adjacent on F. 4v -- see 6.F.3, 6.F.4, 6.AQ, 6.AW.1.

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 7, pp. 24 & 83-84 & plate I, fig. 4. 9 x 16 to 12 x 12.

No. 12, pp. 25 & 85 & plate I, fig. 9. 4 x 9 to 6 x 6.

No. 14, pp. 26 & 86 & plate I, fig. 10. 10 x 20 to 13 1/3 x 15.

Endless Amusement II. 1826?

Prob. 1, p. 188. 5A x 6B to 6A x 5B and to 4A x 7B with two A x B projections. The 6A x 5B looks to be square. = New Sphinx, c1840, p. 136.

Prob. 22, pp. 200-201. 16 x 9 to fill a 12 x 12 hole. Does it by cutting in four pieces -- one 12 x 9 and three 4 x 3. = New Sphinx, c1840, pp. 136-137.

Prob. 33, pp. 210-211. Take a rectangle of proportion 2 : 3 and cut it into two pieces to make a square. Uses cut from 4A x 5B to 5A x 4B, but if we make the rectangle 4 x 6, this makes A = 1, B = 6/5 and the 'square' is 5 x 24/5. = New Sphinx, c1840, p. 140.

Nuts to Crack II (1833), no. 124. 9 x 16 to fill a 12 x 12 hole using four pieces. = Endless Amusement II, prob. 22.

Young Man's Book. 1839. Pp. 241-242. Identical to Endless Amusement II, prob. 22.

Boy's Own Book. 1843 (Paris): 436 & 441, no. 6. 5A x 6B to 6A x 5B and to 4A x 7B with two A x B projections. The 6A x 5B looks to be square. = Boy's Treasury, 1844, pp. 425 & 429. = de Savigny, 1846, pp. 353 & 357, no. 5. Cf de Savigny, below.

de Savigny. Livre des Écoliers. 1846. P. 283: Faire d'une carte un carré. View a playing card as a 5A x 4B rectangle and make a staircase cut and shift to 4A x 5B, which will be nearly square. [When applied to a bridge card, 3.5 x 2.25 in, the result is 2.8 x 2.8125 in.]

Magician's Own Book. 1857. Prob. 2: The parallelogram, pp. 267 & 291. Identical to Boy's Own Book, 1843 (Paris).

The Sociable. 1858. Prob. 19: The perplexed carpenter, pp. 292 & 308. 2 x 12 to 3 x 8. = Book of 500 Puzzles, 1859, prob. 19, pp. 10 & 26. = The Secret Out, 1859, p. 392.

Book of 500 Puzzles. 1859.

Prob. 19: The perplexed carpenter, pp. 10 & 26. As in The Sociable.

Prob. 2: The parallelogram, pp. 81 & 105. Identical to Boy's Own Book, 1843 (Paris).

Charades, Enigmas, and Riddles. 1860: prob. 29, pp. 60 & 64; 1862: prob. 30, pp. 136 & 142; 1865: prob. 574, pp. 107 & 155. 16 x 9 to 12 x 12. All the solutions have an extraneous line in one figure.

Boy's Own Conjuring Book. 1860. Prob. 2: The parallelogram, pp. 229 & 254. Identical to Boy's Own Book, 1843 (Paris).

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 584-4, pp. 286 & 404. Looks like 3 x 4 to 2 x 6. The rectangles are formed by trimming a quarter off a playing card. The diagrams are not very precise, but it seems that the card is supposed to be twice as long as wide. If we take the card as 4 x 8, then the problem is 3 x 8 to 4 x 6.

Hanky Panky. 1872. The parallelogram, p. 107. "A parallelogram, ..., may be cut into two pieces, by which two other figures can be formed." Shows 5A x 4B cut, but no other figures.

Mittenzwey. 1880. Prob. 253 & 255, pp. 45-46 & 96-97; 1895?: 282 & 284, pp. 49 & 98-99; 1917: 282 & 284, pp. 45 & 93-94. 4 x 9 to 6 x 6. 16 x 9 to 12 x 12.

Cassell's. 1881. The carpenter's puzzle, p. 89. = Manson, 1911, p. 133. 3 x 8 board to cover 2 x 12 area.

Richard A. Proctor. Some puzzles; Knowledge 9 (Aug 1886) 305-306 & Three puzzles; Knowledge 9 (Sep 1886) 336-337. Cut 4 x 3 to 3 x 4. Discusses general method for nA x (n+1)B to (n+1)A x nB and notes that the shape can be oblique as well as rectangular.

Lemon. 1890. Card board puzzle, no. 58, pp. 11 12 & 99. c= The parallelogram puzzle, no. 620, pp. 77 & 120 (= Sphinx, no. 706, pp. 92 & 121). Same as Boy's Own Book, 1843 (Paris). In the pictures, A seems to be equal to B.

Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 136, no. 8. 9 x 15 to 12 x 12. No solution.

Tom Tit, vol. 2. 1892. Les figures superposables, pp. 149-150. 3 x 2 to 2 x 3.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. Chinese Geometrical Puzzles No. 1, pp. 108 & 111. Same as Boy's Own Book, 1843 (Paris).

Hoffmann. 1893.

Chap. III, no. 8: The extended square, pp. 91 & 124 125 = Hoffmann-Hordern, p. 80. As in Boy's Own Book, 1843 (Paris), but A is clearly not equal to B.

Chap. III, no. 31: The carpenter's puzzle -- no. 2, pp. 103 & 137 = Hoffmann-Hordern, p. 101. 12 x 36 to 18 x 24.

Clark. Mental Nuts. 1897, no. 27. The leaking ship. 12 x 12 to 9 x 16.

Benson. 1904.

The extended square, p. 190. 5A x 6B square, but the other two figures are 8A x 7B and 8A x 5B with two A x B projections.

The carpenter's puzzle (No. 1), pp. 190 191. = Hoffmann, p. 103.

Anon [possibly Dudeney??] Breakfast Table Problems No. 331: A carpenter's dilemma. Daily Mail (31 Jan & 1 Feb 1905) both p. 7. 16 x 9 to 12 x 12.

Pearson. 1907. Part II, no. 1: The carpenter's puzzle, pp. 1 2 & 185 186. 2 x 12 to 3 x 8.

Wehman. New Book of 200 Puzzles. 1908.

P. 6: The perplexed carpenter. 2 x 12 to 3 x 8. c= The Sociable.

P. 9: The carpenter's puzzle. "A plank was to be cut in two: the carpenter cut it half through on each side, and found he had two feet still to cut. How was it?" This is very vague and can only be recognised as a version of our present problem because the solution looks like cutting a 2 x 6 to make a 3 x 4.

P. 16: The parallelogram. Identical to Boy's Own Book, 1843 (Paris).

P. 17: Another parallelogram. Takes a parallelogram formed of a square and a half a square and intends to form a square. He cuts 5A x 4B and makes 4A x 5B. But for this to be a square, it must be 20 x 20 and then the original was 25 x 16, which is not quite in the given shape.

M. Adams. Indoor Games. 1912. A zigzag puzzle, p. 349, with figs. on 348. 5A x 6B square, but the other two figures are 5A x 4B and 7A x 4B with two A x B projections.

Loyd. Cyclopedia. 1914.

The smart Alec puzzle, pp. 27 & 342. (= MPSL1, prob. 93, pp. 90 91 & 153 154.) Cut a mitre into pieces which can form a square. He trims the corners and inserts them into the notch to produce a rectangle and then uses a staircase cut which he claims gives a square using only four pieces. Gardner points out the error, as carefully explained by Dudeney, below. Since Dudeney gives this correction 1n 1911, he must have seen it in an earlier Loyd publication, possibly OPM?

The carpenter's puzzle, pp. 51 & 345. Claims any rectangle can be staircase cut to make a square. Shows 9 x 4 to 6 x 6 and 25 x 16 to 20 x 20. Cf Kanchusen (1727) and Dudeney (1926).

Dudeney. Perplexities. Strand Magazine 41 (No. 246) (Jun 1911) 746 & 42 (No. 247) (Jul 1911) 108. No. 45: Dissecting a mitre. "I have seen an attempt, published in America, ..." Sketches Loyd's method and says it is wrong. "At present no solution has been found in four pieces, and one in five has not apparently been published."

Dudeney. AM. 1917. Prob. 150: Dissecting a mitre, pp. 35 36 & 170 171. He fully describes "an attempt, published in America", i.e. Loyd's method. If the original square has side 84, then Loyd's first step gives a 63 x 84 rectangle, but the staircase cut yields a 72 x 73½ rectangle, not a square. Dudeney gives a 5 piece solution and says "At present no solution has been found in four pieces, and I do not believe one possible."

Dudeney. MP. 1926. Prob. 115: The carpenter's puzzle, pp. 43 44 & 132 133. = 536, prob. 338, pp. 116 117 & 320 321. Shows 9 x 16 to 12 x 12. "But nobody has ever attempted to explain the general law of the thing. As a consequence, the notion seems to have got abroad that the method will apply to any rectangle where the proportion of length to breadth is within reasonable limits. This is not so, and I have had to expose some bad blunders in the case of published puzzles ...." He discusses the general principle and shows that an n step cut dissects n² x (n+1)² to a square of side n(n+1). Gardner adds a note referring to AM, prob. 150. Cf Kanchusen (1727) & Loyd (1914)

Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane, The Bodley Head, 1939, p. 299. Mentions 12 x 12 to 9 x 16.

Harry Lindgren. Geometric Dissections. Van Nostrand, 1964. P. 28 discusses Loyd's mitre dissection problem and variations. He also thinks a four piece solution is impossible.