Sources in recreational mathematics

Topologie, (1884), says he saw it recently in a railway train. Hayashi (Bibl. Math., (3) 6 (1905) 323 says it is in a Japanese book by Genjun Nakane (1743) and in books by Tasoku Takeda (1844) and Riken Fukuta (1879). Can anyone supply copies or translations?

5.P.1. Shunting Puzzles. Passing with a siding or a side-line are in RM, 1883. Reversing with a turntable was patented in 1885. The Great Northern Puzzle (with two cars, one engine, with a Y off a main line, i.e. a 'delta' configuration) was patented in 1890. The Chifu Chemulpo puzzle appeared in 1903, according to Hordern. See E. Hordern, Sliding Piece Puzzles for other early versions.

5.P.2. Taquin. Can anyone supply early references? Lucas, RM, uses the term generically. Did it refer to any puzzles before the 15 puzzle?

5.Q. Number of Regions Determined by N Lines or Planes. Steiner, 1826, says the plane problem has been raised before, even in a Pestalozzi school book, but he believes he is the first to consider 3 space. He allows lines or circles and parallel families, but no three coincident. Schlafli (1901) does the general problem.

5.R.1. Peg Solitaire. Ahrens, MUS, cites the legend that a Frenchman saw American Indians doing a version of this with arrows. The earliest published references are Leibniz, 1710 & 1716, but material has been found in his MSS from c1678. Beasley has discovered an engraving from 1697 showing the game. S&B cite a 1698 engraving. Can anyone send copies of these?

5.R.2. Frogs and Toads: BBB_WWW to WWW_BBB by Moving or Jumping. Shortz reports it is in American Agriculturist (1867), NYR.

5.R.3. Fore and Aft - 3 by 3 Squares Meeting at a Corner. This is in MRE (1st ed., 1892) and Loyd's Cyclopedia. Ball, MRE (3rd ed.) says he was the first to publish it. MRE (5th ed., 1911) gives a 46 move solution (which is optimal) due to Dudeney, but I can't find it in any of Dudeney's books. I have seen an example called "The English Sixteen Puzzle" from c1895.

5.R.4. Reversing Frogs and Toads: .12...n to .n...21. There are versions in Dudeney, AM and he gives general solutions.

5.R.6. Octagram Puzzle. My earliest version is Les Amusemens, 1749.

5.R.7. Passing Over Counters. Babbage discusses this in his MSS of c1820 and seems to say the general version was posed by Roget. I have also 1826?, 1857ff.

5.S. Chain Cutting and Joining. This is in Loyd's Cyclopedia and Dudeney's MP. Dudeney, "World's best puzzles" (1908), attributes it to Loyd, though I find it in Pearson (1907).

5.S.1. Using Chain Links to Pay for a Room. My earliest source is 1939, saying he heard it in 1935.

5.T. Dividing a Cake Fairly. Steinhaus, Sur la division pragmatique, Econometrica (supplement) 17 (1949) 315 319, is cited as the source of the problems and as giving Banach & Knaster's solution, though the material occurs in English in Econometrica 16 (1948) 101 104. His Math. Snapshots discusses solutions but gives no references. I find two earlier papers by Knaster & Steinhaus in Ann. Soc. Pol. Math. 19 (1946) 228 230 & 230 231. Knaster's title is "Sur le probleme du partage pragmatique de H. Steinhaus", and says Steinhaus posed it in a letter to Knaster in 1944.

5.U. Pigeonhole Recreations. Two men have the same number of hairs, etc. is in van Etten, 1624. Fourrey refers to its occurring in c1660. Erdös is the source of the applications to n+1 numbers from first 2n have one dividing another (AMM 42 (1935) 396) and have two relatively prime (??). Erdös is also the source for n²+1 distinct numbers contain a monotonic subsequence of length n+1, which appears in the Erdös Szekeres paper of 1935.

L. Moser (AMM 55 (1948) 369) is the problem that n elements of a group of order n have a subinterval whose product is the identity. Willy Moser thinks Leo invented this.

O. A. Sullivan (SM 9 (1943) 116) has the problem of choosing socks and shoes in the dark.

5.V. Think A Dot. This was popular in 1967 69, but I haven't found out the inventor.

5.W. Making Three Pieces of Toast. My earliest example is 1943.

5.X.1. Counting Triangles. My earliest example is Pearson (1907).

5.X.2. Counting Rectangles or Squares. My earliest examples are 1921, 1927, 1928.

5.X.3. Counting Hexagons. My earliest example is 1939.

5.Y. Number of Routes in a Lattice. An early form was the number of ways to spell a word in an array   I have examples from 1822, 1893, 1897. Loyd Jr., 1928, gives the problem in abstract form.

5.Z. Chessboard Placing Problems. There are many forms of these, but the seem to arise c1900, except for earlier versions with queens.

5.AA. Card Shuffling. Recently added. A number of articles were in magic journals and I haven't been able to get them - particularly articles by Jordan in The Bat (Nov 1948 - Mar 1949) and by Elmsley in Ibidem (1957). The faro or riffle or perfect shuffle seems to have arisen in the late 19C, but it was not used until the late 1950s.

5.AE. Reversing Cups. New section - I only have a 1978 example.

6. GEOMETRIC RECREATIONS
6.B. Straight Line Linkages. I have located Watt's parallel motion in his 1784 patent. Sarrus was misprinted as Sarrut. I haven't yet got any of the material by Lipkin(e).

6.C. Curves of Constant Width. Euler, "De curvis triangularibus" (1778) seems to be the source for this, though he only considers the triangular case. I find a vaguely similar picture in his Intro. in Analysisin Infinitorum (1748), but he doesn't seem to study the width of the curve there.

6.D. Flexagons. Gardner (Second Book) says a hexatetraflexagon was copyrighted by Roger Montandon of the Montandon Magic. Co., Tulsa, under the name "Cherchez la Femme" in 1946, but a 1993 book attributes the invention to Gardner. Can anyone supply a copy? Margaret Joseph (1951) seems to be the first publication, although Willane's Wizardry (1947) shows the hexatetra.

6.E. Flexatube. A. H. Stone says he invented this as an outgrowth of the flexagons. Leech's note in MG (1955) seems to be the first publication. Steinhaus attributes it to the Dowkers and gives a solution different than the earlier solution.

6.F. Polyominoes, Etc. Prior to Golomb, AMM (1954), I had two references: Dudeney, CP, and Dawson & Lester, Fairy Chess Review (1937), but I have since found dozens of references in Fairy Chess Review and elsewhere, e.g. W. Stead in Fairy Chess Review 9 (1954) 2 4.. There is also a patent by Scrutchin (1908) for a polyiamond puzzle and a patent by Lester (1919) showing both polyomino and polyabolo puzzles. Grünbaum & Shephard give some other early references, NYS. Are there other pre Golomb references?

When were the number of solutions for pentomino rectangles found? Scott (1958) gives the 3 x 20, but both solutions were known in 1935; Miller (1960) gives the 6 x 10. By 1969, all the numbers were known, but they aren't in Golomb's book (1965).

I had thought Gardner (SA, Sep 1958) was the first mention of solid pentominoes, pentacubes, tetracubes. However, the solid pentominoes occur in 1948 and polycubes in 1939.

J. C. P. Miller [Eureka 23 (1960) 13 16] says van der Poel suggested assembling the 12 hexiamonds into a rhombus, but he doesn't show them or name them. Reeve & Tyrrell, MG 45 (Oct 1961) 97 99, are the first to show the 12 two sided hexiamonds, but don't name them. O'Beirne shows them in New Scientist (26 Oct 1981) and discusses them the next week. He says he devised them some time ago. He considers the 19 one sided pieces. R K Guy had already published solutions in Nabla. O'Beirne seems to be the originator of the names hexiamonds and polyiamonds.

O'Beirne (New Scientist, 21 Dec 1961) is the first use of the word 'polyabolo', but some appear in Hooper (1774) and Lester's 1919 patent, etc.

Gardner (SA, Jun 1967) is earliest mention of polyhexes??

6.F.1. Other Chessboard Dissections. Jerry Slocum has an example of Luers' patent of 7 Sep 1880.

6.F.2. Covering Deleted Chessboard with Dominoes. Golomb's 1954 paper starts off with this. He can't remember the origin, but thinks it may go back to Dudeney?? It appears in L. A. Graham, Ingenious Math. Problems and Methods, but the original date is not given. In SM 14 (1948) 160, it is described as coming from Max Black's Critical Thinking (1946, NYS). I have seen it in the 2nd ed., (1952), but he gives no sign of having invented it. A 1969 reference claims it is due to von Neumann! The rook's tour method for the board with any two squares of opposite colour deleted is attributed to Gomory by Gardner. See also 6.U.2.

6.F.3. Dissecting a Cross into Zs and Ls. Les Amusemens (1749) has this.

6.F.4. Quadrisect an L-Tromino, etc. Les Amusemens (1749) has this, and I have three other references before 1825, though Dudeney asserts that Lord Chelmsford invented it in mid 19C.

6.G. Soma Cube. The earliest material I know is Gardner (SA, Sep 1958). There must be earlier material, perhaps in Danish by Hein, e.g. a patent?? Slocum indicates an invention date of 1936, but there is no US patent for it.

6.G.1. Other Cube Dissections.

Hoffmann (1893) is the earliest example I know.

Steinhaus, Math. Snapshots (1950) gives a cube due to Mikusinski, but with no references. Cundy & Rollett refer to Steinhaus' cube, but it is Mikusinki's. I am told that Mikusinski has a French patent on it.

A set of 6 dissections called Impuzzables has been available since 1968 (at least)   Leisure Dynamics, the US distributor, says they are due to a Robert Beck of Minneapolis. Can anyone supply more details of their origin?

Klarner's JRM article (1973) refers to Cubics, a book by J. Slothouber & W. Graatsma, Octopus Press, Holland, 1970, for the dissection of a 3 x 3 x 3 into 6 1 x 2 x 2 and 3  1 x 1 x 1. I have not yet located this book   is it in English or Dutch? Klarner also describes two dissections of the 5 x 5 x 5 cube, one due to Conway. Are these the only printed sources?

6.G.2. Dissection of 6³ into 3³, 4³ and 5³. This appears in Eureka 13 (1950). Gardner identifies the setter as John Leech. Cundy & Rollett cite this as the origin, but I have found it in MG 27 (1943) 142.

6.G.3. Dissection of a Die into Nine 1 x 1 x 3. This is in Hoffmann (1893), who says it is made by Wolff & Son.

6.H. Pick's Theorem. Peter Hajek has brought a copy of Pick's paper "Geometrisches zur Zahlentheorie", Sitzungsberichte des ... Vereines 'Lotos' in Prag 19 (1899) 311 319. Steinhaus' citation is to Ztschr. des .... Steinhaus also cites his own paper "O mierzeniu pol plaskich", Przeglad Mat Fiz. 2 (1924) 24 29. Mąkowski has kindly send me an outline translation of this and it gives a version of the theorem , but doesn't cite Pick. I can find no other references to the theorem before Steinhaus' Math. Snapshots, though C. H. Hinton has a quadrilateral version in 1904.

6.I. Sylvester's Problem of Collinear Points. I now have this sorted out. Sylvester's problem appeared in the Educational Times 46 (NS, No. 383) (1 Mar 1893) 156. Most references cite the Math. Quest. with their Sol. from the Educ. Times, which is a separate publication. Two solutions (??) appear ibid. (No. 385) (1 May 1893) 231 and are printed with the problem in Math. Quest..., but well deserve their obscurity. E. Melchior found the solution in a different context, in 1940. Erdös reopened the problem as AMM Problem 4065 in 1943. A solution by R. Steinberg appears in AMM 51 (1944) 169 171. Editorial comment outlines the solution of T. Grunwald, who later changed his name to T. Gallai. L. M. Kelly's elegant solution appears to first be published in Coxeter's article, AMM 55 (1948) 26 28.

6.J. Four Bugs and Similar Pursuit Problems. The earliest source is a problem by R. Miller on the Cambridge Tripos exam (1871). His bugs are in general position, but the velocities are adjusted to make them meet. Lucas posed a problem of three dogs which was solved by Brocard in Nouv. Corr. Math 3 (1877) 175 176 & 280, NYS. (English versions in Bernhart, SM 24 (1959) 23 50.) L. A. Graham, Ingenious Math. Problems and Methods, attributes the square version to H. D. Grossman.

6.K. Dudeney's Square to Triangle Dissection. This first occurs in Dudeney's column in the Weekly Dispatch (6 Apr 1902). In CP, he says he presented it to the Royal Society at Burlington House on 17 May 1905, but I have no further details. The process extends to rectangles, but I recall an analysis finding the most unsquare rectangle for which it works, but I can't locate this.

6.L. Crossed Ladders Problem. A very simple version occurs in the Lilavati of Bhaskara II, c1150 and there are versions in Fibonacci, Pacioli's Summa and Loyd, but the first references to the standard problem are SSM problem 131 (1908) (= problem 1194 (1931)), AMM problem 2836 (1922) (special case), AMM problem 3173 (1926) (general case), SSM problem 1498 (1937) and AMM problem E433 (1940) (general case with integer solutions).

6.L.1. Ladder over Box. My earliest references are Simpson (1745), then Pearson (1907).

6.M. Spider and Fly Problems. Dudeney first gives this in the Weekly Dispatch of 14 Jun 1903, in the simpler 4 wall form. The 5 wall form is given in an interview in the Daily Mail (1 Feb 1905). In MPSL 2, Gardner says Loyd has simplified Dudeney, but probably Gardner was unaware of the earlier Dudeney version which is identical to Loyd's version. However, Shortz says Loyd gives it in 1900. I have just started to include the problem on a cylinder - my first example is Dudeney, MP, 1926.

6.N. Dissection of a 1 x 1 x 2 into a Cube. My earliest references are to AMM problem E4, solutions in 1933 and 1935.

6.O. Passing a Cube Through an Equal or Smaller One   Prince Rupert's Problem. Schrek, SM 16 (1950) 73 80 & 261 267, gives the history of this and it is apparently really due to Prince Rupert. Wallis was the first to write on it and Pieter Nieuwland found the maximal cube which will pass through a cube. This appears in J. H. van Swinden, Grondbeginsels der Meetkunde, 1816 and in the German edition of C. F. A. Jacobi, Elemente der Geometrie. Can anyone supply copies? Hennessy, Phil. Mag. (1895), says he has a model which may be the example made for Philip Ronayne (18C), another inventor(?) of the problem. Where is this model? U. Graf (1941) also had a model made   where is it?

6.P. Geometrical Vanishing.

6.P.1. Paradoxical Dissections of the Chessboard Based on Fibonacci Numbers. I have a vague reference to W. Leybourn, Pleasure with Profit, 1694, but I could find nothing in it. Loyd, Cyclopedia, claims to have presented the 8 x 8 to 5 x 13, in 1858. The first publication appears to be that signed Schl. in Z. Math. Phys. 13 (1868) 162. Weaver (AMM 45 (1938) 234 236) attributes this to Schlomilch and this seems right since he was a co editor at the time. Coxeter says Schlegel, apparently confusing this with a later article by Schlegel on the same problem in the same journal. Schlomilch gives no explanation, so the first known explanation is in Riecke (1873).

The version where 8 x 8 is converted to an area of 63 squares is in Loyd's Cyclopedia, but Gardner, Math., Magic & Mystery, attributes it to Loyd Jr. S&B show a puzzle version of c1900 and it is in Boy's Own Paper of 14 Dec 1901. Escott gives it in 1907 and White gives an extension in 1908.

Dudeney, "World's best puzzles" (1908), shows 5 x 5 to 3 x 8.

6.P.2. Other Types. The earliest is usually claimed to be Hooper, Rational Recreations (1774), but an accidental version appears in Serlio, Libro Primo d'Architettura, 1545. This was already noted by Cataneo in 1567. I have a vague reference to Ozanam (1723) (though I am not convinced there is such an ed. and I don't see it in the 1725 ed.).

Get Off The Earth. Gardner (SA, Nov 1971) says this had predecessors, e.g. an 1880 premium by Wemple and Company, NY, called the Magical Eggs. I have seen an example from c1890, by R. March, London. Has anyone got an example of this or other early forms? I have found that Loyd has a US patent 563,778   Transformation Picture (14 Jul 1896) but this is a simple version. Loyd published 'Get Off the Earth' in the Brooklyn Daily Eagle during April & May, 1896. Readers submitting clippings of of all the Chinamen received a coloured version of the puzzle. I would like to get an example of this.

6.Q. Knotting a Strip of Paper to Make a Pentagon. This is in Lucas, RM and in Tom Tit. Lucas (1895) and E. Fourrey (1924) attribute it to Urbano d'Aviso; Trattato della Sfera ..., (or Traite de la Sphere); Rome, 1682, NYS. Can anyone supply this??

6.R.1&2. Geometric Fallacies. Rouse Ball, MRE (1st ed., 1892) shows every triangle is isosceles and an obtuse angle is acute and says he thinks this is their first publication, though the latter may be in Mittenzwey (1879), NYR.

6.R.3. Lines Approaching but Not Meeting. This is in Proclus (5C), van Etten et al. (1630), Ozanam-Montucla (1778).

6.S. Tangrams. Needham refers to early 19C books in China. E. Scott had one with covers probably from 18C and one with Empire fashions. Nob Yoshigahara has sent a recent reprint of two booklets from Japan, attributed to Sei Shonagon (1742) & by anon. (1837), but the first is a different dissection than the tangram and the second seems to use this same dissection. The history and bibliography by Jan van der Waals in Joost Elffers' Tangram is the most thorough that I have seen. Has anyone got copies of the early works mentioned   have any been reissued? I have a facsimile of the 1881 Japanese translation of an 1803 Chinese book recognised as the earliest known Tangram book. Van der Waals refers to a woodcut of Utamaro (1780) which I haven't seen. In Dudeney's books of bound articles (in the Strens Collection at Calgary) are two letters from J. Murray about the word 'Tangram' (excerpted in Dudeney's AM). He says that 'tan' is not a Chinese syllable and that 'tangram' appears to be a mid 19C word. Was it invented by Loyd?

6.S.1. Loculus of Archimedes. Dijksterhuis' Archimedes has a thorough survey of the material.

6.T. No Three in a Line Problem. Ahrens, MUS, says he had it in a letter from Escott in 1909. It appears in Sam Loyd's Puzzle Magazine (Jan 1908). Dudeney gives it in The Tribune (7 Nov 1906) NX (at Colindale). Can anyone supply copies?

6.U.2. Packing Bricks in Boxes. Many people have discovered the conditions for an a x b brick to fill a p x q box and it was folklore by about 1965. But I don't know any published proof nor the originator of the idea. Greenblatt, Mathematical Entertainments (1968 or 1965?) asserts that filling a 6 x 6 x 6 with 1 x 2 x 4 was invented by R. Milburn of Tufts Univ. De Bruijn seems to have discovered the basic necessary condition for higher dimensions, c1969.

6.V. Silhouette and Viewing Puzzles. Van Etten (1653 ed.) has four silhouette versions. Ozanam (1725) appears to have copied van Etten and added an extra figure. Alberti copied the latter. None are the classic circle, triangle, square problem, which first(?) appears in Catel's 1790 catalogue. What is the origin of the puzzles where three or two orthogonal views are shown?

6.W. Burr Puzzles. Slocum says Wyatt (1928) is the first to use the word 'burr'. The Crambrook catalogue of 1843 mentions several items which may be these puzzles, but pictures of the objects are not known??

6.W.1. Three Piece Burr. The earliest I know is Hoffmann (1893), but Crambrook (1843) may have the same puzzle.

6.W.2. Six Piece Burr = Chinese Cross. This appears in an 1790 catalogue of the Berlin firm of Catel, called the Small Devil's Hoof. Minguet é Irol (1822) diagrams the pieces. The Magician's Own Book (1857) shows a different set of pieces. The Illustrated Boy's Own Treasury (1860) diagrams the pieces, shows how to assemble it and calls it 'The Chinese Cross'. Is there any Chinese connection??

6.W.3. Three Piece Burr With Identical Pieces. This appears in SA (1 Apr 1899), but Crambrook (1843) looks like it has it.

6.W.4. Diagonal Six Piece Burr = Trick Star. A spherical version with a key piece was patented in 1904. A version with six identical pieces was patented in 1905, but it doesn't show the 3 + 3 assembly. Other versions go back to 1875 and possibly to Crambrook (1843)

6.W.5. Six Piece Burr With Identical Pieces. A version with 6 U shaped cards appears in 1891, and there may be a version in Crambrook (1843). The Bonbon nut is in Hoffmann (1893).

6.X. Rotating Rings of Polyhedra. Rouse Ball, MRE (11th ed., 1939) attributes the tetrahedral idea to J. M. Andreas and R. M. Stalker. Coxeter says neither ever published on the subject. Pedersen tells me that it appears in Fedorov, late 19C, NYS. Bruckner (1900) considers rings of tetrahedra. What about the cubical versions? The 'Shinsei Mystery' version is attributed to Naoki Yoshimoto, 1972, but may be in Slothouber & Graatsma, Cubics, 1970, NYS. Was Engel the first to consider using the 'Jacob's ladder' hinge?

6.Y. Rope Round the Earth. That is, if it is moved out 1 m all around, it gains 6.28 m in length. The geometric idea is clear in Lucca 1754 (c1390); Muscarello (1478); Pacioli's Summa (1494). "A Lover of the Mathematicks"; A Mathematical Miscellany; Dublin, 1735; describes travellers whose heads go 12 yards more than their feet. Dudeney gives it in Strand Mag. (1909).

6.Z. Langley's Adventitious Angles. (Let ABC be an isosceles triangle with  A = 20^o, and draw BD, CE making angles 50^o and 60^o with the base. Then  CED = 30^o.) This appeared a few years ago in MG and they cite Langley, MG (1922) and 13 solutions in 1923. The latter cites the Peterhouse and Sidney entrance scholarship examination, Jan 1916. JRM 15 (1982-83) 150 cites Math. Quest. Educ. Times 17 (1910) 75. Can anyone supply copies of these?

6.AA. Nets of Polyhedra. (I.e. unfoldings into a planar figure.) Dürer, 1525, gives a net for each regular polyhedron and some Archimedean ones. Panofsky's biography of Dürer asserts that Dürer invented the concept of a net. Cardan shows the nets of the regular polyhedra in 1557. Dudeney, MP, gives all 11 nets of the cube. Perelman [1920s?] gives the problem and finds 10 nets. The 11 nets of the octahedron are in Math. Teaching 40 (Aut 1967) 48 52 (where 13 are given), corrected Ibid. 41 (Wint 1967) 29. I believe the dodecahedron and the icosahedron were unsolved until my student Peter Light and I found (1984) that there are 43380 nets on these.

Gardner mentions the 4 cube in his Foreword to Steinhaus's One Hundred Problems (1964). He posed it again in SA (Nov 1966) and the Addendum to this in Carnival says that he received several answers, no two agreeing! Peter Turney, JRM 17 (1984/5) finds 261 nets on the 4 cube.

6.AB. Self Rising Polyhedra. Kac (AMM 81 (1974)) says the dodecahedron is due to Steinhaus. I have recently obtained a 1939 ed. of Steinhaus's Mathematical Snapshots with the original example in the pocket at the back. I have recently seen several pop up octahedrons. Does anyone know the source of the other versions?

6.AC. Conway's Life. Gardner, SA (Oct 1970), seems to be the first printed material.

6.AD.1. Largest Parcel One Can Post. I have this from 1883, complete with the cylindrical solution.

6.AE. 6" Hole Through a Sphere Leaves Constant Volume. Gardner cites S. I. Jones,