Sources page biographical material

Leske. Illustriertes Spielbuch für Mädchen. 1864?

Prob. 564-31, pp. 254 & 396. From a 6 x 6 array, remove 6 to leave an even number in each row. (The German 'Reihe' can be interpreted as row or column or both.) If we consider this in the first quadrant with coordinates going from 1 to 6, the removed points are: (1,2), (1,3), (2,1), (2,2), (6,1), (6,3). The use of the sixth column is peculiar and has the effect of making both diagonals odd, while the more usual use of the third column would make both diagonals even.

Prob. 583-5, pp. 285 & 403: Von folgenden 36 Punkten sechs zu streichen. As above, but each file ('Zeile') in 'all four directions' has four or six points. Deletes: (1,1), (1,2), (2,2), (2,3), (6,1), (6,3) which makes one diagonal even and one odd.

Mittenzwey. 1880. Prob. 154, pp. 31 & 83; 1895?: 177, pp. 36 & 85; 1917: 177, pp. 33 & 82. Given a 4 x 4 array, remove 6 to leave an even number in each row and column. Solution removes a 2 x 3 rectangle from a corner. [This fails -- it leaves two rows and a diagonal with an odd number. One can use the idea mentioned for Leske 564-31 to get a solution with both diagonals also being even.]

Hoffmann. 1893. Chap. VI, pp. 271-272 & 285 = Hoffmann-Hordern, pp. 186-187.

No. 22: The thirty six puzzle. Place 30 counters on a 6 x 6 board so each horizontal and each vertical line has an even number. Solution places the six blanks in a 3 x 3 corner in the obvious way. This also makes the diagonals have even numbers.

No. 23: The "Five to Four" puzzle. Place 20 counters on a 5 x 5 board subject to the above conditions. Solution puts blanks on the diagonal. This also makes the diagonals have even number.

Dudeney. The puzzle realm. Cassell's Magazine ?? (May 1908) 713-716. The crack shots. 10 pieces in a 4 x 4 array making the maximal number of even lines -- counting diagonals and short diagonals -- with an additional complication that pieces are hanging on vertical strings. The picture is used in AM, prob. 270.

Loyd. Cyclopedia. 1914. The jolly friar's puzzle, pp. 307 & 380. (= MPSL2, no. 155, pp. 109 & 172. = SLAHP: A shifty little problem, pp. 64 & 110.) 10 men on a 4 x 4 board -- make a maximal number of even rows, including diagonals and short diagonals. This is a simplification of Dudeney, 1908.

King. Best 100. 1927. No. 72, pp. 29 & 56. As in Hoffmann's No. 22, but specifically asks for even diagonals as well.

The Bile Beans Puzzle Book. 1933. No. 19: Thirty-six coins. As in Hoffmann's No. 22, but specifically asks for even diagonals as well.

Rudin. 1936. No. 151, pp. 53-54 & 111. Place 12 counters on a 6 x 6 board with two in each 'row, column and diagonal'. Reading the positions in each row, the solution is: 16, 34, 25, 25, 34, 16. Some of the short diagonals and some of the broken diagonals are empty, so he presumably isn't including these, or he meant to ask for each of these to have an even number of at most two.

M. Adams. Puzzle Book. 1939. Prob. C.179: Even stars, pp. 169 & 193. Same as Loyd.

Doubleday - 1. 1969. Prob. 61: Milky Way, pp. 76 & 167. = Doubleday - 5, pp. 85-86. 6 x 6 array with two opposite corners already filled. Add ten more counters so that no row, column or diagonal has more than two counters in it. Reading the positions in each row, the solution is: 13, 35, 12, 67, 24, 46. Some short diagonals are empty or have one counter and some broken diagonals have one or four counters, so he seems to be ignoring them. Hence this is the same problem as Rudin, but with a less satisfactory solution.

Obermair. Op. cit. in 5.Z.1. 1984. Prob. 37, pp. 38 & 68. 52 men on an 8 x 8 board with all rows, columns and diagonals (both long and short) having an even number.

6.AP . DISSECTIONS OF A TETRAHEDRON
6.AP.1. TWO PIECES
Richard A. Proctor. Our puzzles; Knowledge 10 (Feb 1887) 83 & Solutions of puzzles; Knowledge 10 (Mar 1887) 108-109. "Puzzle XIX. Show how to cut a regular tetrahedron (equilateral triangular pyramid) so that the face cut shall be a square: also show how to plug a square hole with a tetrahedron." Solution shows the cut clearly.

Edward T. Johnson. US Patent 2,216,915 -- Puzzle. Applied: 26 Apr 1939; patented: 8 Oct 1940. 2pp + 1p diagrams. Described in S&B, p. 46.

E. M. Wyatt. Wonders in Wood. Op. cit. in 6.AI. 1946. Pp. 9 & 11: the tetrahedron or triangular pyramid. P. 9 is reproduced in S&B, p. 46.

Donovan A. Johnson. Paper Folding for the Mathematics Class. NCTM, 1957, p. 26, section 62: Pyramid puzzle. Gives instructions for making the pieces from paper.

Claude Birtwistle. Editor's footnote. MTg 21 (Winter 1962) 32. "The following interesting puzzle was given to us recently."

Birtwistle. Math. Puzzles & Perplexities. 1971. Bisected tetrahedron, pp. 157-158. Gives the net so one can make a drawing, cut it out and fold it up to make one piece.

6.AP.2. FOUR PIECES
These dissections usually also work with a tetrahedron of spheres and hence these are related to ball pyramid puzzles, 6.AZ.

The first version I had in mind dissects each of the two pieces of 6.AP.1 giving four congruent rhombic pyramids. Alternatively, imagine a tetrahedron bisected by two of its midplanes, where a midplane goes halfway between a pair of opposite edges. This puzzle has been available in various versions since at least the 1970s, including one from Stokes Publishing Co., 1292 Reamwood Avenue, Sunnyvale, California, 94089, USA., but I have no idea of the original source. The same pieces are part of a more complex dissection of a cube, PolyPackPuzzle, which was produced by Stokes in 1996. (I bought mine from Key Curriculum Press.)

In 1997, Bill Ritchie, of Binary Arts, sent a quadrisection of the tetrahedron that they are producing. Each piece is a hexahedron. The easiest way to describe it is to consider the tetrahedron as a pile of spheres with four on an edge and hence 20 altogether. Consider a planar triangle of six of these spheres with three on an edge and remove one vertex sphere to produce a trapezium (or trapezoid) shape. Four of these assemble to make the tetrahedron. Writing this has made me realise that Ray Bathke has made and sold these 5-sphere pieces as Pyramid 4 for a few years. However, the solid pieces used by Binary Arts are distinctly more deceptive.

Len Gordon produced another quadrisection of the 20 sphere tetrahedron 0 0

using the planar shape at the right. This was c1980?? 0 0 0  
David Singmaster. Sums of squares and pyramidal numbers. MG 66 (No. 436) (Jun 1982) 100-104. Consider a tetrahedron of spheres with 2n on an edge. The quadrisection described above gives four pyramids whose layers are the squares 1, 4, ..., n². Hence four times the sum of the first n squares is the tetrahedral number for 2n, i.e. 4 [1 + 4 + ... + n²]  =  BC(n+2, 3).
6.AQ. DISSECTIONS OF A CROSS, T OR H
The usual dissection of a cross has two diagonal cuts at 45^o to the sides and passing through two of the reflex corners of the cross and yielding five pieces. The central piece is six-sided, looking like a rectangle with its ends pushed in and being symmetric. Depending on the relative lengths of the arms, head and upright of the cross, the other pieces may be isosceles right triangles or right trapeziums. Removing the head of the cross gives the usual dissection of the T into four pieces -- then the central piece is five-sided. Sometimes the central piece is split in halves. Occasionally the angle of the cuts is different than 45^o. Dissections of an H have the same basic idea of using cuts at 45^o -- the result can be a bit like two Ts with overlapping stems and the number of pieces depends on the relative size and positioning of the crossbar of the H -- see: Rohrbough.
S&B, pp. 20 21, show several versions. They say that crosses date from early 19C. They show a 6 piece Druid's Cross, by Edwards & Sons, London, c1855. They show several T puzzles -- they say the first is an 1903 advertisement for White Rose Ceylon Tea, NY -- but see 1898 below. They also show some H puzzles.
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 a cross cut into 5 pieces in the usual way.

Endless Amusement II. 1826? Prob. 30, p. 207. Usual five piece cross. The three small pieces are equal. = New Sphinx, c1840, pp. 139-140.

Crambrook. 1843. P. 4.

No. 10: Five pieces to form a Cross.

No. 11: The new dissected Cross.

Without pictures, I cannot tell what dissections are used??

Boy's Own Book. 1843 (Paris): 435 & 440, no. 2. Usual five piece cross, very similar to Endless Amusement. One has to make three pieces of fig. 2. = Boy's Treasury, 1844, pp. 424 & 428. = de Savigny, 1846, pp. 353 & 357, no. 1.

Family Friend 2 (1850) 58 & 89. Practical Puzzle -- No. II. = Illustrated Boy's Own Treasury, 1860, No. 32, pp. 401 & 440. Usual five piece cross to "form that which, viewed mentally, comforts the afflicted." Three pieces of fig. 1.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Mechanical puzzles, no. 7, p. 178-179 (1868: 189). Five piece dissection of a cross, but the statement of the problem doesn't say which piece to make multiple copies of.

Magician's Own Book. 1857. Prob. 17: The cross puzzle, pp. 272 & 295. Usual 5 piece cross, essentially identical to Family Friend, except this says to "form a cross." = Book of 500 Puzzles, 1859, prob. 17, pp. 86 & 109. = Boy's Own Conjuring Book, 1860, prob. 16, pp. 234 & 258.

Charades, Enigmas, and Riddles. 1860: prob. 33, pp. 60 & 66; 1862: prob. 33, pp. 136 & 143; 1865: prob. 577, pp. 108 & 156. Usual five piece cross, showing all five pieces.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 584-12, pp. 288 & 406: Ein Kreuz. Begins as the usual five piece cross, but the central piece is then bisected into two mitres and the base has two bits cut off to give an eight piece puzzle.

Frank Bellew. The Art of Amusing. 1866. Op. cit. in 5.E. 1866: pp. 239-240; 1870: pp. 236 238. Usual five piece cross.

Elliott. Within Doors. Op. cit. in 6.V. 1872. Chap. 1, no. 1: The cross puzzle, pp. 27 & 30. Usual five piece cross, but instructions say to cut three copies of the wrong piece.

Mittenzwey. 1880. Prob. 188, pp. 35 & 88; 1895?: 213, pp. 40 & 91; 1917: 213, pp. 37 & 87. This is supposed to be a 10 piece dissection of a cross obtained by further dissecting the usual five pieces. However, pieces 3 & 4 are drawn as trapezoids in the problem and triangles in the solution and piece 2 in the solution is half the size given in the problem. Further, pieces 1 & 2 appear equilateral in the problem, but are isosceles right triangles in the solution. One could modify this to get a 9 piece version where eight of the pieces are right trapezoids -- four having edges 1, 1, 2, 2 and four having edges 2, 2, 22, 2, but the arms would be twice as long as they are wide. Or one can make the second four pieces be 2, 2, 2 isosceles right triangles. In either case, the ninth piece would be a rectangle.

Lemon. 1890. A card board puzzle, no. 33, pp. 8 & 98. Usual five piece cross.

Hoffmann. 1893. Chap. III, no. 12: The Latin cross puzzle, pp. 93 & 126 = Hoffmann Hordern, pp. 82-83, with photo. As in Indoor & Outdoor. Photo on p. 83 shows two versions: one in metal by Jaques & Sons, 1870-1895; the other in ivory, 1850-1900. Hordern Collection, p. 59, shows a Druid's Cross Puzzle.

Lash, Inc. -- Clifton, N.J. -- Chicago, Ill. -- Anaheim, Calif. T Puzzle. Copyright Sept. 1898. 4 piece T puzzle to be cut out from a paper card, but the angle of the cuts is about 35^o instead of 45^o which makes it less symmetric and less confusing than the more common version. The resulting T is somewhat wider than usual, being about 16% wider than it is tall. It advertises: Lash's Bitters The Original Tonic Laxative. Photocopy sent by Slocum.

Benson. 1904. The cross puzzle, pp. 191 192. Usual 5 piece version.

Wehman. New Book of 200 Puzzles. 1908. The cross puzzle, p. 17. Usual 5 piece version.

A. Neely Hall. Op. cit. in 6.F.5. 1918. The T puzzle, pp. 19 20. "A famous old puzzle ...." Usual 4 piece version, but with long arms.

Western Puzzle Works, 1926 Catalogue. No. 1394: Four pieces to form Letter T. The notched piece is less symmetric than usual.

Collins. Book of Puzzles. 1927. The crusader's cross puzzle, pp. 1-2. The three small pieces are equal.

Arthur Mee's Children's Encyclopedia 'Wonder Box'. The Children's Encyclopedia appeared in 1908, but versions continued until the 1950s. This looks like 1930s?? Usual 5 piece cross.

A. F. Starkey. The T puzzle. Industrial Arts and Vocational Education 37 (1938) 442. "An interesting novelty ...."

Rohrbough. Puzzle Craft. 1932. The "H" Puzzle, p. 23. Very square H -- consider a 3 x 3 board with the top and bottom middle cells removed. Make a cut along the main diagonal and two shorter cuts parallel to this to produce four congruent isosceles right triangles and two odd pentagons.

See Rohrbough in 6.AS.1 for a very different T puzzle.
6.AR. QUADRISECTED SQUARE PUZZLE
This is usually done by two perpendicular cuts through the centre. A dissection proof of the Theorem of Pythagoras described by Henry Perigal (Messenger of Mathematics 2 (1873) 104) uses the same shapes -- cf 6.AS.2.

The pieces make a number of other different shapes.

Crambrook. 1843. P. 4, no. 17: Four pieces to form a Square. This might be the dissection being considered here??

A. Héraud. Jeux et Récréations Scientifiques -- Chimie, Histoire Naturelle, Mathématiques. (1884); Baillière, Paris, 1903. Pp. 303 304: Casse tête. Uses two cuts which are perpendicular but are not through the centre. He claims there are 120 ways to try to assemble it, but his mathematics is shaky -- he adds the numbers of ways at each stage rather than multiplying! Also, as Strens notes in the margin of his copy (now at Calgary), if the crossing is off-centre, then many of the edges have different lengths and the number of ways to try is really only one. Actually, I'm not at all sure what the number of ways to try is -- Héraud seems to assume one tries each orientation of each piece, but some intelligence sees that a piece can only fit one way beside another.

Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. P. 14: The divided square puzzle. Crossing is off-centre.

Tom Tit, vol 3. 1893. Carré casse-tête, pp. 179-180. = K, no. 26: Puzzle squares, pp. 68 69. = R&A, Puzzling squares, p. 99. Not illustrated, but described: cut a square into four parts by two perpendicular cuts, not necessarily through the centre.

A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton & Co., London, nd [1927 -- BMC]. No. 77: Pattern making, pp. 69-70 & 109. Make five other shapes.

M. Adams. Puzzle Book. 1939. Prob. C.12: The broken square, pp. 125 & 173. As above, but notes that the pieces also make a square with a square hole.

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.