Sources page biographical material

6.AS. DISSECTION OF SQUARES INTO A SQUARE
Lorraine Mottershead. Investigations in Mathematics. Blackwell, Oxford, 1985. P. 102 asserts that dissections of squares to various hexagons and heptagons were known c1800 while square to rectangle dissections were known to Montucla -- though she illustrates the latter with examples like 6.Y, she must mean 6.AS.5.
6.AS.1. TWENTY 1, 2, 5 TRIANGLES MAKE A SQUARE

OR FIVE EQUAL SQUARES TO A SQUARE
The basic puzzle has been varied in many ways by joining up the 20 triangles into various shapes, but I haven't attempted to consider all the modern variants. A common form is a square with a skew # in it, with each line joining a corner to the midpoint of an opposite side, giving the 9 piece version. This has four of the squares having a triangle cut off. For symmetry, it is common to cut off a triangle from the fifth square, giving 10 pieces, though the assembly into one square doesn't need this. See Les Amusemens for details.

Cf Mason in 6.S.2 for a similar puzzle with twenty pieces.

If the dividing lines are moved a bit toward the middle and the central square is bisected, we get a 10 piece puzzle, having two groups of four equal pieces and a group of two equal pieces, called the Japan square puzzle. I have recently noted the connection of this puzzle with this section, so there may be other examples which I have not previously paid attention to -- see: Magician's Own Book, Book of 500 Puzzles, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury, Landells, Hanky Panky, Wehman.
Les Amusemens. 1749. P. xxxii. Consider five 2 x 2 squares. Make a cut from a corner to the midpoint of an opposite side on each square. This yields five 1, 2, 5 triangles and five pieces comprising three such triangles. The problem says to make a square from five equal squares. So this is the 10 piece version.

Minguet. 1755. Pp. not noted -- ??check (1822: 145-146; 1864: 127-128). Not in 1733 ed. 10 piece version. Also a 15 piece version where triangles are cut off diagonally opposite corners of each small square leaving parallelogram pieces as in Guyot.

Vyse. Tutor's Guide. 1771? Prob. 6, 1793: p. 304, 1799: p. 317 & Key p. 357. 2 x 10 board to be cut into five pieces to make into a square. Cut into a 2 x 2 square and four 2, 4, 25 triangles.

Ozanam Montucla. 1778. Avec cinq quarrés égaux, en former un seul. Prob. 18 & fig. 123, plate 15, 1778: 297; 1803: 292-293; 1814: 249-250; 1840: 127. 9 piece version. Remarks that any number of squares can be made into a square -- see 6.AS.5.

Catel. Kunst-Cabinet. 1790.

Das mathematische Viereck, pp. 10-11 & fig. 15 on plate I. 10 piece version with solution shown. Notes these make five squares.

Das grosse mathematische Viereck, p. 11 & fig. 14 on plate I. Cut the larger pieces to give five more 1, 2, 5 triangles and five 5, 5, 2 triangles. Again notes these make five squares.

Guyot. Op. cit. in 6.P.2. 1799. Vol. 2: première récréation: Cinq quarrés éqaux étant sonnés, en former un seul quarré, pp. 40 41 & plate 6, opp. p. 37. 10 piece version. Suggests cutting another triangle off each square to give 10 triangles and 5 parallelograms.

Bestelmeier. 1801. Item 629: Die 5 geometrisch zerschnittenen Quadrate, um aus 5 ein einziges Quadrat zu machen. As in Les Amusemens. S&B say this is the first appearance of the puzzle. Only shown in a box with one small square visible.

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 8, pp. 25 & 84 & plate I, fig. 5, no. 1. = Vyse.

No. 10, pp. 25 & 84-85 & plate I, fig. 7, no. 1. Five squares to one. Nine piece version.

Rational Recreations. 1824. Feat 35, pp. 164-166. Usual 20 piece form.

Manuel des Sorciers. 1825. Pp. 201-202, art. 18. ??NX Five squares to one -- usual 10 piece form and 15 piece form as in Guyot.

Endless Amusement II. 1826?

[1837 only] Prob. 35, p. 212. 20 triangles to form a square. = New Sphinx, c1840, p. 141, with problem title: Dissected square.

Prob. 37, p. 215. 10 piece version. = New Sphinx, c1840, p. 141.

Boy's Own Book. The square of triangles. 1828: 426; 1828-2: 430; 1829 (US): 222; 1855: 576; 1868: 676. Uses 20 triangles cut from a square of wood. Cf 1843 (Paris) edition, below. c= de Savigny, 1846, p. 272: Division d'un carré en vingt triangles.

Nuts to Crack IV (1835), no. 195. 20 triangles -- part of a long section: Tricks upon Travellers. The problem is used as a wager and the smart-alec gets it wrong.

The Riddler. 1835. The square of triangles, p. 8. Identical to Boy's Own Book, but without illustration, some consequent changing of the text, and omitting the last comment.

Crambrook. 1843. P. 4.

No. 7: Egyptian Puzzle. Probably the 10 piece version as in Les Amusemens. See S&B below, late 19C. Check??

No. 23: Twenty Triangles to form a Square. Check??

Boy's Own Book. 1843 (Paris): 436 & 441, no. 5: "Cut twenty triangles out of ten square pieces of wood; mix them together, and request a person to make an exact square with them." As stated, this is impossible; it should be as in Boy's Own Book, 1828 etc., qv. = Boy's Treasury, 1844, pp. 425 & 429. = de Savigny, 1846, pp. 353 & 357, no. 4. Also copied, with the error, in: Magician's Own Book, 1857, prob. 29: The triangle puzzle; Book of 500 Puzzles, 1859, prob. 29: The triangle puzzle; Boy's Own Conjuring Book, 1860, prob. 28: The triangle puzzle. c= Hanky Panky, 1872, p. 122.

Magician's Own Book. 1857.

How to make five squares into a large one without any waste of stuff, p. 258. 9 piece version.

Prob. 29: The triangle puzzle, pp. 276 & 298. Identical to Boy's Own Book, 1843 (Paris).

Prob. 35: The Japan square puzzle, pp. 277 & 300. Make two parallel cuts and then two perpendicular to the first two so that a square is formed in the centre. This gives a 9 piece puzzle, but here the central square is cut by a vertical through its centre to give a 10 piece puzzle. = Landells, Boy's Own Toy-Maker, 1858, pp. 145-146.

Charles Bailey (manufacturer in Manchester, Massachusetts). 1858. An Ingenious Puzzle for the Amusement of Children .... The 10 pieces of Les Amusemens, with 19 shapes to make, a la tangrams. Sent by Jerry Slocum -- it is not clear if there were actual pieces with the printed material.

The Sociable. 1858.

Prob. 10: The protean puzzle, pp. 289 & 305-306. Cut a 5 x 1 into 11 pieces to form eight shapes, e.g. a Greek cross. It is easier to describe the pieces if we start with a 10 x 2. Then three squares are cut off. One is halved into two 1 x 2 rectangles. Two squares have two 1, 2, 5 triangles cut off leaving triangles of sides 2, 5, 5. The remaining double square is almost divided into halves each with a 1, 2, 5 triangle cut off, but these two triangles remain connected along their sides of size 1, thus giving a 4, 5, 5 triangle and two trapeziums of sides 2, 2, 1, 5. = Book of 500 Puzzles, 1859, prob. 10, pp. 7 & 23-24.

Prob. 42: The mechanic's puzzle, pp. 298 & 317. Cut a 10 x 2 in five pieces to make a square, as in Vyse. = Book of 500 Puzzles, 1859, prob. 16, pp. 16 & 35.

Book of 500 Puzzles. 1859.

Prob. 10: The protean puzzle, pp. 7 & 23-24. As in The Sociable.

Prob. 42: The mechanic's puzzle, pp. 16 & 35. As in The Sociable.

How to make five squares into a large one without any waste of stuff, p. 72. Identical to Magician's Own Book.

Prob. 29: The triangle puzzle, pp. 90 & 113. Identical to Boy's Own Book, 1843 (Paris).

Prob. 35: The Japan square puzzle, pp. 91 & 114.

Indoor & Outdoor. c1859. Part II, prob. 11: The mechanic's puzzle, pp. 130-131. Identical to The Sociable.

Boy's Own Conjuring Book. 1860.

Prob. 28: The triangle puzzle, pp. 238 & 262. Identical to Boy's Own Book, 1843 (Paris) and Magician's Own Book.

Prob. 34: The Japan square puzzle, pp. 240 & 264. Identical to Magician's Own Book.

Illustrated Boy's Own Treasury. 1860.

Prob. 9, pp. 396 & 437. [The Japan square puzzle.] Almost identical to Magician's Own Book.

Optics: How to make five squares into a large one without any waste of stuff, p. 445. Identical to Book of 500 Puzzles, p. 72.

Vinot. 1860. Art. LXXV: Avec cinq carrés égaux, en faire un seul, p. 90. Nine piece version.

Leske. Illustriertes Spielbuch für Mädchen. 1864?

Prob. 174, pp. 87-88. Nine piece version.

Prob. 584-6, pp. 287 & 405. Ten piece version of five squares to one.

Hanky Panky. 1872.

The puzzle of five pieces, p. 118. 9 piece version.

Another [square] of four triangles and a square, p. 120. 10 x 2 into five pieces to make a square.

[Another square] of ten pieces, pp. 121-122. Same as the Japan square puzzles in Magician's Own Book.

[Another square] of twenty triangles, p. 122. Similar to Boy's Own Book, 1843 (Paris), but with no diagram and less text, making it quite cryptic.

Mittenzwey. 1880. Prob. 175, pp. 33-34 & 85; 1895?: 200, pp. 38 & 87; 1917: 200, pp. 35 & 84. 10 pieces as in Les Amusemens. See in 6.AS.2 and 6.S.2 for the use of these pieces to make other shapes.

See Mason, 1880, in 6.S.2 for a similar, but different, 20 piece puzzle.

S&B, pp. 11 & 19, show a 10 piece version called 'Egyptian Puzzle', late 19C?

Lucas. RM2. 1883. Les vingt triangles, pp. 128 129. Notes that they also make five squares in the form of a cross.

Tom Tit, vol. 2. 1892. Diviser un carré en cinq carrés égaux, pp. 147 148. = K, no. 2: To divide a square into five equal squares, pp. 12-14. = R&A, Five easy pieces, p. 105. Uses 9 pieces, but mentions use of 10 pieces.

Hoffmann. 1893.

Chap. III, no. 21: The five squares, pp. 100 & 132 133 = Hoffmann-Hordern, p. 94, with photo. 9 piece version, as in Magician's Own Book. Photo on p. 94 shows: an ivory version, 1850-1900;, and a wood version, named Egyptian Puzzle, by C. N. Mackie, 1860-1890; both with boxes.

Chap. III, no. 24: The twenty triangles, pp. 101 & 134 = Hoffmann-Hordern, pp. 96-97, with photo. As in Boy's Own Book. Photo on p. 97 shows The Twenty Triangle Puzzle, with box, by Jaques & Son, 1870-1895. Hordern Collection, p. 64, shows Apollonius, with box, by W. X., Paris, 1880-1900, in a solution very different to the usual one.

Chap. III, no. 30: The carpenter's puzzle -- no. 1, pp. 103 & 136 137 = Hoffmann Hordern, p. 101. Cut a 5 x 1 board into five pieces to make a square.

Chap. X, no. 25: The divided square, pp. 346 & 384 = Hoffmann-Hordern, p. 242. 9 piece puzzle as a dissection of a square which forms 5 equal squares. He places the five squares together as a 2 x 2 with an adjacent 1 x 1, but he doesn't see the connection with 6.AS.2.

Montgomery Ward & Co. Catalog N^o 57, Spring & Summer, 1895. Facsimile by Dover, 1969, ??NX. P. 237 shows the Mystic Square, as item 25463, which is the standard 10 piece version.

Benson. 1904.

The carpenter's puzzle (No. 2), p. 191. = Hoffmann, p. 103.

The five square puzzle, pp. 196 197. = Hoffmann, p. 100.

The triangle puzzle, p. 198. = Hoffmann, p. 101.

Wehman. New Book of 200 Puzzles. 1908.

P. 3: The triangle puzzle. 20 pieces. = Boy's Own Book, omitting the adjuration to use wood and smooth the edges

P. 12: The protean puzzle. c= The Sociable, prob. 10, with the instructions somewhat clarified.

P. 14: The Japan square puzzle. c= Magician's Own Book.

P. 19: To make five squares into a large one. 10 piece version.

P. 27: The mechanic's puzzle. = The Sociable, prob. 42.

J. K. Benson, ed. The Pearson Puzzle Book. C. Arthur Pearson, London, nd [c1910, not in BMC or NUC]. [This is almost identical with the puzzle section of Benson, but has 13 pages of different material.] Juggling geometry, pp. 97-98. Five triangles, which should be viewed as 2, 4, 25. Cut one from the midpoint of the hypotenuse to the midpoint of the long leg and assemble into a square, so this becomes a six-piece or five-piece version as in Vyse, etc.

I have seen a 10 piece French example, called Jeu du Carré, dated 1900 1920.

I have seen a 9(?) piece English example, dated early 20C, called The Euclid Puzzle.

Dudeney. Perplexities column, no. 109: A cutting-out puzzle. Strand Magazine 45 (No. 265) (Jan 1913) 113 & (No. 266) (Feb 1913) 238. c= AM, prob. 153 -- A cutting-out puzzle, pp. 37 & 172. Cut a 5 x 1 to make a square. He shows a solution in five pieces and asks for a solution in four pieces. AM states the generalized form given in 6.AS.5.

Rohrbough. Puzzle Craft. 1932. Square "T", p. 23 (= The "T" Puzzle, p. 23 of 1940s?). 1 x 1 square and two 1 x 2 rectangles cut diagonally can be formed into a square or into a T.

Gibson. Op. cit. in 4.A.1.a. 1963. Pp. 71 & 76: Square away. A five piece puzzle, approximately that formed by drawing parallel lines from two diagonally opposite corners to the midpoints of opposite sides and then cutting a square from the middle of the central strip. As drawn, the lines meet the opposite sides a bit further along than the midpoints.

Ripley's Puzzles and Games. 1966.

Pp. 58-59, item 6. Five right triangles to a square. Though not specified, the triangles have sides proportional to 1, 2, 5. Solution is as in Benson.

Pp. 60-61, item 5. Start with the large square, which is 25 on a side. Imagine the 9 piece puzzle where one line goes from the upper left corner to the midpoint of the right side. Number the outer pieces clockwise from the upper left, so that pieces 1, 3, 5, 7 are the small triangles and 2, 4, 6, 8 are the trapezia. The pieces of this puzzle are as follows: combine pieces 1, 8, 7 into a 5, 25, 5 triangle; combine pieces 2, 3, 4 into an irregular hexagon; take separate pieces 5 and 6 and the central square. These five pieces form: a square; a Greek cross; a 5 x 4 rectangle; a triangle of sides 25, 45, 10; etc. I think there are earlier versions of this, e.g. in Loyd, but I have just observed the connection with this section.