Sources page biographical material

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.