Sources page biographical material

Leske. Illustriertes Spielbuch für Mädchen. 1864?

Prob. 584-11, pp. 288 & 405: Chinesisches Verwandlungsspiel. Make a square with the tangram pieces. Shows just five of the pieces, but correctly states which two to make two copies of.

Prob. 584-16, pp. 289 & 406. Make an isosceles right triangle with the tangram pieces.

Prob. 584-18/25, pp. 289-291 & 407: Hieroglyphenspiele. Form various figures from various sets of pieces, mostly tangrams, but the given shapes have bits of writing on them so the assembled figure gives a word. Only one of the shapes is as in Boy's Own Book.

Prob. 588, pp. 298 & 410: Etliche Knackmandeln. Another tangram problem like the preceding, not equal to any in Boy's Own Book.

Adams & Co., Boston. Advertisement in The Holiday Journal of Parlor Plays and Pastimes, Fall 1868. Details?? -- photocopy sent by Slocum. P. 6: Chinese Puzzle. The celebrated Puzzle with which a hundred or more symmetrical forms can be made, with book showing the designs. Though not illustrated, this seems likely to be the Tangrams -- ??

Mittenzwey. 1880. Prob. 243-252, pp. 45 & 95-96; 1895?: 272-281, pp. 49 & 97-98; 1917: 272-282, pp. 45 & 92-93. Make a funnel, kitchen knife, hammer, hat with brim being horizontal or hanging down or turned up, church, saw, dovecote, hatchet, square, two equal squares.

J. Murray (editor of the OED). Two letters to H. E. Dudeney (9 Jun 1910 & 1 Oct 1910). The first inquires about the word 'tangram', following on Dudeney's mention of it in his "World's best puzzles" (op. cit. in 2). The second says that 'tan' has no Chinese origin; is apparently mid 19C, probably of American origin; and the word 'tangram' first appears in Webster's Dictionary of 1864. Dudeney, AM, 1917, p. 44, excerpts these letters.

F. T. Wang & C. S. Hsiung. A theorem on the tangram. AMM 49 (1942) 596 599. They determine the 20 convex regions which 16 isosceles right triangles can form and hence the 13 ones which the Tangram pieces can form.

Mitsumasa Anno. Anno's Math Games. (Translation of: Hajimete deau sugaku no ehon; Fufkuinkan Shoten, Tokyo, 1982.) Philomel Books, NY, 1987. Pp. 38-43 & 95-96 show a simplified 5-piece tangram-like puzzle which I have not seen before. The pieces are: a square of side 1; three isosceles right triangles of side 1; a right trapezium with bases 1 and 2, altitude 1 and slant side 2. The trapezium can be viewed as putting together the square with a triangle. 19 problems are set, with solutions at the back.

James Dalgety. Latest news on oldest puzzles. Lecture to Second Meeting on the History of Recreational Mathematics, 1 Jun 1996. 10pp. In 1998, he extracted the two sections on tangrams and added a list of tangram books in his collection as: The origins of Tangram; © 1996/98; 10pp. (He lists about 30 books, eight up to 1850.) In 1993, he was buying tangrams in Hong Kong and asked what they called it. He thought they said 'tangram' but a slower repetition came out 'ta hau ban' and they wrote down the characters and said it translates as 'seven lucky tiles'. He has since found the characters in 19C Chinese tangram books. It is quite possible that Sam Loyd (qv under Murray, above) was told this name and wrote down 'tangram', perhaps adjusted a bit after thinking up Tan as the inventor.

At the International Congress on Mathematical Education, Seville, 1996, the Mathematical Association gave out The 3, 4, 5 Tangram, a cut card tangram, but in a 6 x 8 rectangular shape, so that the medium sized triangle was a 3-4-5 triangle. I modified this in Nov 1999, by stretching along a diagonal to form a rhombus with angles double the angles of a 3-4-5 triangle, so that four of the triangles are similar to 3-4-5 triangles. Making the small triangles be actually 3-4-5, all edges are integral. I made up 35 problems with these pieces. I later saw that Hans Wiezorke has mentioned this dissection in CFF, but with no problems. I distributed this as my present at G4G4, 2000.
6.S.1. LOCULUS OF ARCHIMEDES
See S&B 22. I recall there is some dispute as to whether the basic diagram should be a square or a double square.

E. J. Dijksterhuis. Archimedes. Munksgaard, Copenhagen, 1956; reprinted by Princeton Univ. Press, 1987. Pp. 408 412 is the best discussion of this topic and supplies most of the classical references.

Archimedes. Letter to Eratosthenes, c-250?. Greek palimpsest, c975, on MS no. 355, from the Cloister of Saint Sabba (= Mar Saba), Jerusalem, then at Metochion of the Holy Sepulchre, Constantinopole. [This MS disappeared in the confusion in Asia Minor in the 1920s but reappeared in 1998 when it was auctioned by Christie's in New York for c2M$. Hopefully, modern technology will allow a facsimile and an improved transcription in the near future.] Described by J. L. Heiberg (& H. G. Zeuthen); Eine neue Schrift des Archimedes; Bibliotheca Math. (3) 7 (1906 1907) 321 322. Heiberg describes the MS, but only mentions the loculus. The text is in Heiberg's edition of Archimedes; Opera; 2nd ed., Teubner, Leipzig, 1913, vol. II, pp. 415 424, where it has been restored using the Suter MSS below. Heath only mentions the problem in passing. Heiberg quotes Marius Victorinus, Atilius Fortunatianus and cites Ausonius and Ennodius.

H. Suter. Der Loculus Archimedius oder das Syntemachion des Archimedes. Zeitschr. für Math. u. Phys. 44 (1899) Supplement = AGM 9 (1899) 491 499. This is a collation from two 17C Arabic MSS which describe the construction of the loculus. They are different than the above MS. The German is included in Archimedes Opera II, 2nd ed., 1913, pp. 420 424.

Dijksterhuis discusses both of the above and says that they are insufficient to determine what was intended. The Greek seems to indicate that Archimedes was studying the mathematics of a known puzzle. The Arabic shows the construction by cutting a square, but the rest of the text doesn't say much.
Lucretius. De Rerum Natura. c 70. ii, 778 783. Quoted and discussed in H. J. Rose; Lucretius ii. 778 83; Classical Review (NS) 6 (1956) 6 7. Brief reference to assembling pieces into a square or rectangle.

Decimus Magnus Ausonius. c370. Works. Edited & translated by H. G. Evelyn White. Loeb Classical Library, ??date. Vol. I, Book XVII: Cento Nuptialis (A Nuptial Cento), pp. 370-393 (particularly the Preface, pp. 374-375) and Appendix, pp. 395-397. Refers to 14 little pieces of bone which form a monstrous elephant, a brutal boar, etc. The Appendix gives the construction from the Arabic version, via Heiberg, and forms the monstrous elephant.

Marius Victorinus. 4C. VI, p. 100 in the edition of Keil, ??NYS. Given in Archimedes Opera II, 2nd ed., 1913, p. 417. Calls it 'loculus Archimedes' and says it had 14 pieces which make a ship, sword, etc.

Ennodius. Carmina: De ostomachio eburneo. c500. In: Magni Felicis Ennodii Opera; ed. by F. Vogel, p. 340. In: Monumenta Germaniae Historica, VII (1885) 249. ??NYS. Refers to ivory pieces to be assembled.

Atilius Fortunatianus. 6C. ??NYS Given in Archimedes Opera II, p. 417. Same comment as for Marius Victorinus.

E. Fourrey. Curiositiés Géométriques. (1st ed., Vuibert & Nony, Paris, 1907); 4th ed., Vuibert, Paris, 1938. Pp. 106 109. Cites Suter, Ausonius, Marius Victorinus, Atilius Fortunatianus.

Collins. Book of Puzzles. 1927. The loculus of Archimedes, pp. 7-11. Pieces made from a double square.
6.S.2. OTHER SETS OF PIECES
See Hoffmann & S&B, cited at the beginning of 6.S, for general surveys.

See Bailey in 6.AS.1 for an 1858 puzzle with 10 pieces and The Sociable and Book of 500 Puzzles, prob. 10, in 6.AS.1 for an 11 piece puzzle.

There are many versions of this idea available and some are occasionally given in JRM.

The Richter Anchor Stone puzzles and building blocks were inspired by Friedrich Froebel (or Fröbel) (1782 1852), the educational innovator. He was the inventor of Kindergartens, advocated children's play blocks and inspired both the Richter Anchor Stone Puzzles and Milton Bradley. The stone material was invented by Otto Lilienthal (1848 1896) (possibly with his brother Gustav) better known as an aviation pioneer -- they sold the patent and their machines to F. Adolph Richter for 1000 marks. The material might better be described as a kind of fine brick which could be precisely moulded. Richter improved the stone and began production at Rudolstadt, Thüringen, in 1882; the plant closed in 1964. Anchor was the company's trademark. He made at least 36 puzzles and perhaps a dozen sets of building blocks which were popular with children, architects, engineers, etc. The Deutsches Museum in Munich has a whole room devoted to various types of building blocks and materials, including the Anchor blocks. The Speelgoed Museum 'Op Stelten' (Sint Vincentiusstraat 86, NL-4902 (or 4901) GL Oosterhout, Noord-Brabant, The Netherlands; tel: [phone]; fax: [phone]) has a room of Richter blocks and some puzzles. There was an Anker Museum in the Netherlands (Stichting Ankerhaus (= Anker Museum); Opaalstraat 2 4 (or Postf. 1061), NL-2400 BB Alphen aan den Rijn, The Netherlands; tel: [phone]) which produced replacement parts for Anker stone puzzles. Modern facsimiles of the building sets are being produced at Rudolstadt.

In 1996 I noticed the ceiling of the room to the south of the Salon of the Ambassadors in the Alcazar of Seville. This 15C? ceiling was built by workmen influenced by the Moorish tradition and has 112 square wooden panels in a wide variety of rectilineal patterns. One panel has some diagonal lines and looks like it could be used as a 10 piece tangram-like puzzle. Consider a 4 x 4 square. Draw both diagonal lines, then at two adjacent corners, draw two lines making a unit square at these corners. At the other two corners draw one of these two lines, namely the one perpendicular to their common side. This gives six isosceles right triangles of edge 1; two pentagons with three right angles and sides 1, 2, 1, 2, 2; two quadrilaterals with two right angles and sides 2, 1, 2, 22. Since geometric patterns and panelling are common features of Arabic art, I wonder if there are any instances of such patterns being used for a tangram-like puzzle?

Grand Jeu du Casse Tête Français en X. Pieces. ??NYS -- described and partly reproduced in Milano, who says it comes from Paris and dates it 1818? The figures are anthropomorphic and are most similar to those in Jeu du Casse Tete Russe.

Grande Giuocho del Rompicapo Francese. Milano presso Pietro e Giuseppe Vallardi Contrada di S. Margherita N^o 401(? my copy is small and faint). ??NYS -- described and partly reproduced in Milano, who dates it as 1818-1820. Identical problems as in the previous item, but the figures have been redrawn rather than copied exactly.

Allizeau. Les Métamorphoses ou Amusemens Géometriques Dédiée aux Amateurs Par Allizeau. A Paris chex Allizeau Quai Malaquais, N^o 15. ??NYS -- described and partly reproduced in Milano. This uses 15 pieces and the problems tend to be architectural forms, like towers.

Jackson. Rational Amusement. 1821. Geometrical Puzzles, nos. 20-27, pp. 27-29 & 88-89 & plate II, figs. 15-22. This is a set of 20 pieces of 8 shapes used to make a square, a right triangle, three squares, etc.

Crambrook. 1843. P. 4, no. 1: Pythagorean Puzzle, with Book. Though not illustrated, this is probably(??) the puzzle described in Hoffmann, below, which was a Richter Anchor puzzle No. 12 of the same name and is still occasionally seen. See S&B 28.

Edward Hordern's collection has a Circassian Puzzle, c1870, with many pieces, but I didn't record the shapes -- cf Boy's Own Book, 1843 (Paris), in section 6.S.

Mittenzwey. 1880.

Prob. 177-179, pp. 34 & 86; 1895?: 202-204, pp. 38-39 & 88; 1917: 202-204, pp. 35 & 84-85. Consider the ten piece version of dissecting 5 squares to one (6.AS.1). Use the pieces to make:

a squat octagon, a house gable-end, a church (no solution), etc.;

two dissimilar rectangles;

three dissimilar parallelograms, two dissimilar trapezoids. Solution says one can make many other shapes with these pieces, e.g. a trapezoid with parallel sides in the proportion 9 : 11.

Prob. 181-184, pp. 34-35 & 87-88; 1895?: 206-209, pp. 39 & 89-90; 1917: 206-209, pp. 36 & 85-86. Take six equilateral triangles of edge 2. Cut an equilateral triangle of edge 1 from the corner of each of them, giving 12 pieces. Make a hexagon in eight different ways [there are many more -- how many??] and three tangram-like shapes.

Prob. 195-196, pp. 36 & 89; 1895?: 220-221, pp. 41 & 91; 1917: 220-221, pp. 37 & 87. Use four isosceles right triangles, say of leg 1, to make a square, a 1 x 4 rectangle and an isosceles right triangle.

Nicholas Mason. US Patent 232,140 - Geometrical Puzzle-Block. Applied: 13 May 1880; patented 14 Sep 1880. 1p plus 2pp diagrams. Five squares, six units square, each cut into four pieces in the same way. Start at the midpoint of a side and cut to an opposite corner. (This is the same cut used to produce the ten piece 'Five Squares to One' puzzle.) Cut again in the triangle just formed, from the same midpoint to a point one unit from the right angle corner of the piece just made. This gives a right triangle of sides 3, 1, 10 and a triangle of sides 5, 10, 345. Cut again from the same midpoint across the trapezoidal piece made by the first cut, to a point five units from the corner previously cut to. This gives a triangle of sides 5, 35, 210 and a right trapezoid with sides 2,10, 1, 6, 3. This was produced as Hill's American Geometrical Prize Puzzle in England ("Price, One Shilling.") in 1882. Harold Raizer produced a facsimile version, with facsimile box label and instructions for IPP22. The instructions have 20 problems to solve and the solutions have to be submitted by 1 May 1882.

Hoffmann. 1893. Chap. III, no. 3: The Pythagoras Puzzle, pp. 83-85 & 117-118 = Hoffmann Hordern, pp. 69-72. This has 7 pieces and is quite like the Tangram -- see comment under Crambrook. Photo on p. 71, with different version in Hordern Collection, p. 50.

C. Dudley Langford. Note 1538: Tangrams and incommensurables. MG 25 (No. 266) (Oct 1941) 233 235. Gives alternate dissections of the square and some hexagonal dissections.

C. Dudley Langford. Note 2861: A curious dissection of the square. MG 43 (No. 345) (Oct 1959) 198. There are 5 triangles whose angles are multiples of π/8 = 22½^o. He uses these to make a square.

See items at the end of 6.S.
6.T. NO THREE IN A LINE PROBLEM
See also section 6.AO.2.
Loyd. Problem 14: A crow puzzle. Tit Bits 31 (16 Jan & 6 Feb 1897) 287 & 343. = Cyclopedia, 1914, Crows in the corn, pp. 110 & 353. = MPSL1, prob. 114, pp. 113 & 163 164. 8 queens with no two attacking and no three in any line.

Dudeney. The Tribune (7 Nov 1906) 1. ??NX. = AM, prob. 317, pp. 94 & 222. Asks for a solution with two men in the centre 2 x 2 square.

Loyd. Sam Loyd's Puzzle Magazine, January 1908. ??NYS. (Given in A. C. White; Sam Loyd and His Chess Problems; 1913, op. cit. in 1; p. 100, where it is described as the only solution with 2 pieces in the 4 central squares.)

Ahrens, MUS I 227, 1910, says he first had this in a letter from E. B. Escott dated 1 Apr 1909. (W. Moser, below, refers this to the 1st ed., 1900, but this must be due to his not having seen it.)

C. H. Bullivant. Home Fun, 1910, op. cit. in 5.S. Part VI, Chap. IV: No. 2: Another draught puzzle, pp. 515 & 520. The problem says "no three men shall be in a line, either horizontally or perpendicularly". The solution says "no three are in a line in any direction" and the diagram shows this is indeed true.

Loyd. Picket posts. Cyclopedia, 1914, pp. 105 & 352. = MPSL2, prob. 48, pp. 34 & 136. 2 pieces initially placed in the 4 central squares.

Blyth. Match-Stick Magic. 1921. Matchstick board game, p. 73. 6 x 6 version phrased as putting "only two in any one line: horizontal, perpendicular, or diagonal." However, his symmetric solution has three in a row on lines of slope 2.

King. Best 100. 1927. No. 69, pp. 28 & 55. Problem on the 6 x 6 board -- gives a symmetric solution. Says "there are two coins on every row" including "diagonally across it", but he has three in a row on lines of slope 2.

Loyd Jr. SLAHP. 1928. Checkers in rows, pp. 40 & 98. Different solution than in Cyclopedia.

M. Adams. Puzzle Book. 1939. Prob. C.83: Stars in their courses, pp. 144 & 181. Same solution as King, but he says "two stars in each vertical row, two in each horizontal row, and two in each of the the two diagonals .... There must not be more than two stars in the same straight line", but he has three in a row on lines of slope 2.

W. O. J. Moser & J. Pach. No three in line problem. In: 100 Research Problems in Discrete Geometry 1986; McGill Univ., 1986. Problem 23, pp. 23.1 -- 23.4. Survey with 25 references. Solutions are known on the n x n board for n  16 and for even n  26. Solutions with the symmetries of the square are only known for n = 2, 4, 10.
6.U. TILING
6.U.1. PENROSE PIECES
R. Penrose. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10 (1974) 266 272.

M. Gardner. SA (Jan 1977). Extensively rewritten as Penrose Tiles, Chaps. 1 & 2.

R. Penrose. Pentaplexity. Eureka 39 (1978) 16 22. = Math. Intell. 2 (1979) 32 37.

D. Shechtman, I. Blech, D. Gratias & J. W. Cohn. Metallic phase with long range orientational order and no translational symmetry. Physical Rev. Letters 53:20 (12 Nov 1984) 1951 1953. Describes discovery of 'quasicrystals' having the symmetry of a Penrose like tiling with icosahedra.

David R. Nelson. Quasicrystals. SA 255:2 (Aug 1986) 32 41 & 112. Exposits the discovery of quasicrystals. First form is now called 'Shechtmanite'.

Kimberly-Clark Corporation has taken out two patents on the use of the Penrose pattern for quilted toilet paper as the non-repetition prevents the tissue from 'nesting' on the roll. In Apr 1997, Penrose issued a writ against Kimberly Clark Ltd. asserting his copyright on the pattern and demanding damages, etc.

John Kay. Top prof goes potty at loo roll 'rip-off'. The Sun (11 Apr 1997) 7.

Patrick McGowan. It could end in tears as maths boffin sues Kleenex over design. The Evening Standard (11 Apr 1997) 5.

Kleenex art that ended in tears. The Independent (12 Apr 1997) 2.

For a knight on the tiles. Independent on Sunday (13 Apr 1997) 24. Says they exclusively reported Penrose's discovery of the toilet paper on sale in Dec 1996.

D. Trull. Toilet paper plagiarism. Parascope, 1997 -- available on www.noveltynet.org/content/paranormal/www.parascope.com/arti...
6.U.2. PACKING BRICKS IN BOXES
In two dimensions, it is not hard to show that a x b packs A x B if and only if a  divides either A or B; b divides either A or B; A and B are both linear combinations of a and b. E.g. 2 x 3 bricks pack a 5 x 6 box.

See also 6.G.1.

Anon. Prob. 52. Hobbies 30 (No. 767) (25 Jun 1910) 268 & 283 & (No. 770) (16 Jul 1910) 328. Use at least one of each of 5 x 7, 5 x 10, 6 x 10 to make the smallest possible square. Solution says to use 4, 4, 1, but doesn't show how. There are lots of ways to make the assembly.

Manuel H. Greenblatt ( -1972, see JRM 6:1 (Winter 1973) 69). Mathematical Entertainments. Crowell, NY, 1965. Construction of a cube, pp. 80 81. Can 1 x 2 x 4 fill 6 x 6 x 6? He asserts this was invented by R. Milburn of Tufts Univ.

N. G. de Bruijn. Filling boxes with bricks. AMM 76 (1969) 37 40. If a₁ x ... x a_n fills A₁ x ... x A_n and b divides k of the a_i, then b divides at least k of the A_i. He previously presented the results, in Hungarian, as problems in Mat. Lapok 12, pp. 110 112, prob. 109 and 13, pp. 314 317, prob. 119. ??NYS.

D. A. Klarner. Brick packing puzzles. JRM 6 (1973) 112 117. General survey. In this he mentions a result that I gave him -- that 2 x 3 x 7 fills a 8 x 11 x 21, but that the box cannot be divided into two packable boxes. However, I gave him the case 1 x 3 x 4 in 5 x 5 x 12 which is the smallest example of this type. Tom Lensch makes fine examples of these packing puzzles.

T. H. Foregger, proposer; Michael Mather, solver. Problem E2524 -- A brick packing problem. AMM 82:3 (Mar 1975) 300 & 83:9 (Nov 1976) 741-742. Pack 41  1 x 2 x 4 bricks in a 7 x 7 x 7 box. One cannot get 42 such bricks into the box.
6.V. SILHOUETTE AND VIEWING PUZZLES
Viewing problems must be common among draughtsmen and engineers, but I haven't seen many examples. I'd be pleased to see further examples.
2 silhouettes.

Circle & triangle -- van Etten, Ozanam, Guyot, Magician's Own Book (UK version)

Circle & square -- van Etten

Circle & rhombus -- van Etten, Ozanam

Rectangle with inner rectangle & rectangle with notch -- Diagram Group.

3 silhouettes.

Circle, circle, circle -- Madachy

Circle, cross, square -- Shortz collection (c1884), Wyatt, Perelman

Circle, oval, rectangle -- van Etten, Ozanam, Guyot, Magician's Own Book (UK version)

Circle, oval, square -- van Etten, Tradescant, Ozanam, Ozanam Montucla, Badcock, Jackson, Rational Recreations, Endless  Amusement II, Young Man's Book

Circle, rhombus, rectangle -- Ozanam, Alberti

Circle, square, triangle -- Catel, Bestelmeier, Jackson, Boy's Own Book, Crambrook, Family Friend, Magician's Own Book, Book of 500 Puzzles, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury, Riecke, Elliott, Mittenzwey, Tom Tit, Handy Book, Hoffmann, Williams, Wyatt, Perelman, Madachy. But see Note below.

Square, tee, triangle -- Perelman

4 silhouettes.

Circle, square, triangle, rectangle with curved ends -- Williams

2 views.

Antilog, Ripley's, Diagram Group;

3 views.

Madachy, Ranucci,
For the classic Circle, Square, Triangle, version, the triangle cannot be not equilateral. Consider a circle, rectangle, triangle version. If D is the diameter of the circle and H is the height of the plug, then the rectangle has dimensions D x H and the triangle has base D and side S, so S = (H² + D²/4). Making the rectangle a square, i.e. H = D, makes S = D5/2, while making the triangle equilateral, i.e. S = D, makes H = D3/2.
van Etten. 1624.

Prob. 22 (misnumbered 15 in 1626) (Prob. 20), pp. 19 20 & figs. opp. p. 16 (pp. 35 36): 2 silhouettes -- one circular, the other triangular, rhomboidal or square. (English ed. omits last case.) The 1630 Examen says the author could have done better and suggests: isosceles triangle, several scalene triangles, oval or circle, which he says can be done with an elliptically cut cone and a scalene cone. I am not sure I believe these. It seems that the authors are allowing the object to fill the hole and to pass through the hole moving at an angle to the board rather than perpendicularly as usually understood. In the English edition the Examination is combined with that of the next problem.

Prob. 23 (21), pp. 20 21 & figs. opp. p. 16 (pp. 37 38): 3 silhouettes -- circle, oval and square or rectangle. The 1630 Examen suggests: square, circle, several parallelograms and several ellipses, which he says can be done with an elliptic cylinder of height equal to the major diameter of the base. The English Examination says "a solid colume ... cut Ecliptick-wise" -- ??

John II Tradescant (1608-1662). Musæum Tradescantianum: Or, A Collection of Rarities Preserved at South-Lambeth neer London By John Tradescant. Nathaniel Brooke, London, 1656. [Facsimile reprint, omitting the Garden List, Old Ashmolean Reprints I, edited by R. T. Gunther, on the occasion of the opening of the Old Ashmolean Museum as what has now become the Museum of the History of Science, Oxford. OUP, 1925.] John I & II Tradescant were gardeners to nobility and then royalty and used their connections to request naval captains to bring back new plants, curiosities and "Any thing that Is strang". These were accumulated at his house and garden in south Lambeth, becoming known as Tradescant's Ark, eventually being acquired by Elias Ashmole and becoming the foundation of the Ashmolean Museum in Oxford. This catalogue was prepared by Elias Ashmole and his friend Thomas Wharton, but they are not named anywhere in the book. It was the world's first museum catalogue.

P. 37, last entry: "A Hollow cut in wood, that will fit a round, square and ovall figure."

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. He says square, circle and triangle is in a book in front of him dated 1674. I suspect this must be the 1674 English edition of van Etten, but I don't find the problem in the English editions I have examined. Perhaps Dudeney just meant that the idea was given in the 1674 book, though he is specifically referring to the square, circle, triangle version.

Ozanam. 1725. Vol. II, prob. 58 & 59, pp. 455 458 & plate 25* (53 (note there is a second plate with the same number)). Circle and triangle; circle and rhombus; circle, oval, rectangle; circle, oval, square. Figures are very like van Etten. See Ozanam-Montucla, 1778.

Ozanam. 1725. Vol. IV. No text, but shown as an unnumbered figure on plate 15 (17). 3 silhouettes: circle, rhombus, rectangle.

Simpson. Algebra. 1745. Section XVIII, prob. XXIX, pp. 279-281. (1790: prob. XXXVII, pp. 306-307. Computes the volume of an ungula obtained by cutting a cone with a plane. Cf Riecke, 1867.

Alberti. 1747. No text, but shown as an unnumbered figure on plate XIIII, opp. p. 218 (112), copied from Ozanam, 1725, vol IV. 3 silhouettes: circle, rhombus, rectangle.

Ozanam-Montucla. 1778. Faire passer un même corps par un trou quarré, rond & elliptique. Prob. 46, 1778: 347-348; 1803: 345-346; 1814: 293. Prob. 45, 1840: 149-150. Circle, ellipse, square.

Catel. Kunst-Cabinet. 1790. Die mathematischen Löcher, p. 16 & fig. 42 on plate II. Circle, square, triangle.

E. C. Guyot. Nouvelles Récréations Physiques et Mathématiques. Op. cit. in 6.P.2. 1799. Vol. 2, Quatrième récréation, p. 45 & figs. 1 4, plate 7, opp. p. 45. 2 silhouettes: circle & triangle; 3 silhouettes: circle, oval, rectangle.

Bestelmeier.

1801. Item 536: Die 3 mathematischen Löcher. (See also the picture of Item 275, but that text is for another item.) Square, triangle and circle.

1807. Item 1126: Tricks includes the square, triangle and circle.

Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. P. 14, no. 23: How to make a Peg that will exactly fit three different kinds of Holes. "Let one of the holes be circular, the other square, and the third an oval; ...." Solution is a cylinder whose height equals its diameter.

Jackson. Rational Amusement. 1821. Geometrical Puzzles.

No. 16, pp. 26 & 86. Circle, square, triangle, with discussion of the dimensions: "a wedge, except that its base must be a circle".

No. 29, pp. 30 & 89-90. Circle, oval, square.

Rational Recreations. 1824. Feat 19, p. 66. Circle, oval, square.

Endless Amusement II. 1826? P. 62: "To make a Peg that will exactly fit three different kinds of Holes." Circle, oval, square. c= Badcock.

The Boy's Own Book. The triple accommodation. 1828: 419; 1828-2: 424; 1829 (US): 215; 1855: 570; 1868: 677. Circle, square and triangle.

Young Man's Book. 1839. Pp. 294-295. Circle, oval, square. Identical to Badcock.

Crambrook. 1843. P. 5, no. 16: The Mathematical Paradox -- the Circle, Triangle, and Square. Check??

Family Friend 3 (1850) 60 & 91. Practical puzzle -- No. XII. Circle, square, triangle. This is repeated as Puzzle 16 -- Cylinder puzzle in (1855) 339 with solution in (1856) 28.

Magician's Own Book. 1857. Prob. 21: The cylinder puzzle, pp. 273 & 296. Circle, square, triangle. = Book of 500 Puzzles, 1859, prob. 21, pp. 87 & 110. = Boy's Own Conjuring Book, 1860, prob. 20, pp. 235 & 260.

Illustrated Boy's Own Treasury. 1860. Practical Puzzles, No. 42, pp. 403 & 442. Identical to Magician's Own Book, with diagram inverted.

F. J. P. Riecke. Op. cit. in 4.A.1, vol. 1, 1867. Art. 33: Die Ungula, pp. 58 61. Take a cylinder with equal height and diameter. A cut from the diameter of one base which just touches the other base cuts off a piece called an ungula (Latin for claw). He computes the volume as 4r³/3. He then makes the symmetric cut to produce the circle, square, triangle shape, which thus has volume (2π   8/3) r³. Says he has seen such a shape and a board with the three holes as a child's toy. Cf Simpson, 1745.

Magician's Own Book (UK version). 1871. The round peg in the square hole: To pass a cylinder through three different holes, yet to fill them entirely, pp. 49-50. Circle, oval, rectangle; circle & (isosceles) triangle.

Alfred Elliott. Within Doors. A Book of Games and Pastimes for the Drawing Room. Nelson, 1872. [Toole Stott 251. Toole Stott 1030 is a 1873 ed.] No. 4: The cylinder puzzle, pp. 27 28 & 30 31. Circle, square, triangle.

Mittenzwey. 1880. Prob. 257, pp. 46 & 97; 1895?: 286, pp. 50 & 99-100; 1917: 286, pp. 45 & 94-95. Circle, square, triangle.

Will Shortz has a puzzle trade card with the circle, cross, square problem, c1884.

Tom Tit, vol. 2. 1892. La cheville universelle, pp. 161-162. = K, no. 28: The universal plug, pp. 72 73. = R&A, A versatile peg, p. 106. Circle, square, triangle.

Handy Book for Boys and Girls. Op. cit. in 6.F.3. 1892. Pp. 238-242: Captain S's peg puzzle. Circle, square, triangle.

Hoffmann. 1893. Chap. X, no. 20: One peg to fit three holes, pp. 344 & 381 382 = Hoffmann-Hordern, pp. 238-239, with photo. Circle, square, triangle. Photo on p. 239 shows two examples: one simply a wood board and pieces; the other labelled The Holes and Peg Puzzle, from Clark's Cabinet of Puzzles, 1880-1900, but this seems to be just a card box with the holes.

Williams. Home Entertainments. 1914. The plug puzzle, pp. 103-104. Circle, square, triangle and rectangle with curved ends. This is the only example of this four-fold form that I have seen. Nice drawing of a board with the plug shown in each hole, except the curve on the sloping faces is not always drawn down to the bottom.

E. M. Wyatt. Puzzles in Wood, 1928, op. cit. in 5.H.1.

The "cross" plug puzzle, p. 17. Square, circle and cross.

The "wedge" plug puzzle, p. 18. Square, circle and triangle.

Perelman. FMP. c1935? One plug for three holes; Further "plug" puzzles, pp. 339 340 & 346. 6 simple versions; 3 harder versions: square, triangle, circle; circle, square, cross; triangle, square, tee. The three harder versions are also in FFF, 1957: probs. 69-71, pp. 112 & 118-119; 1979: probs. 73 75, pp. 137 & 144 = MCBF: probs. 73-75, pp. 134-135 & 142-143.

Anonymous [Antilog]. An elevation puzzle. Eureka 19 (Mar 1957) 11 & 19. Front and top views are a square with a square inside it. What is the side view? Gives two solutions.

Anonymous. An elevation puzzle. Eureka 21 (Oct 1958) 7 & 29. Front is the lower half of a circle. Plan (= top view) is a circle. What is the side view? Solution is a V shape, but it ought to be the other way up! Nowadays, one can buy potato crisps (= potato chips) in this shape.

Joseph S. Madachy. 3 D in 2 D. RMM 2 (Apr 1961) 51 53 & 3 (Jun 1961) 47. Discusses 3 view and 3 silhouette problems.

3 circular silhouettes, but not a sphere.

Square, circle, triangle.

Ernest R. Ranucci. Non unique orthographic projections. RMM 14 (Jan Feb 1964) 50. 3 views such that there are 10 different objects with these views.

Ripley's Puzzles and Games. 1966. Pp. 18-19, item 1. Same problem as Antilog, 1957. Gives one solution.

Cedric A. B. Smith. Simple projections. MG 62 (No. 419) (Mar 1978) 19-25. This is about how different projections affect one's recognition of what an object is. He starts with an example with two views and the isometric projection which is very difficult to interpret. He gives three other views, each of which is easily interpreted.

The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex, 1984. Problem 114, with Solution at the back of the book. Front view is a rectangle with an interior rectangle. Side view is a rectangle with a rectangular notch on front side. Solution is a short cylinder with a straight notch in it. This is a fairly classic problem for engineers but I haven't seen it in print elsewhere.

Marek Penszko. Polish your wits -- 3: Loop the loop. Games 11:2 (Feb/Mar 1987) 28 & 58. Draw lines on a glass cube to produce three given projections. Problem asks for all three projections to be the same.