Sources in recreational mathematics

Mathematical Nuts, 1932, NYS. But the two-dimensional version is Holditch's theorem of 1858.

6.AF. What Colour Was the Bear? I find this in Leopold's At Ease! (1943); Northrop's Riddles in Mathematics (1944) and in Moulton (AMM 51 (1944) 216 & 220). Simpler North Pole problems go back to 1735. Moulton (1944) is the first to consider all solutions of the three sided problem. Perelman (1920s?) is the first to consider a closed circuit, but he has a square.

6.AG. Moving Around a Corner. I don‘t know the origin of this. My earliest version with a table is AMM 47 (1940) 569. Abraham (1933) gives the problem of taking a ladder from one street to another, of different widths. Licks (1917) gives an equivalent problem phrased as getting a stick into a circular hole in the ceiling.

6.AH. Tethered Goat. M. Fraser (MM 56 (1983) 123) cites the Ladies Diary for 1748, NYS. Dudeney (1898) has a version with two pieces of field which avoids the usual transcendental equation.

6.AI. Trick Joints. The common square version was patented in 1888. S&B cite The Woodworker of 1902 as the earliest reference, but it occurs in Tom Tit (1892). The hexagonal and triangular versions are in Abraham, 1933. I have found a pentagonal version, 1940s?, but it splits and reglues the wood so it cannot come apart.

I have a double dovetail T joint from SA (25 Apr 1896) 267. Wyatt, Wonders in Wood, cites A. B. Cutler in Industrial Arts and Vocational Education (Jan 1930) for a triple dovetail, but I couldn't find it in vols. 1 40.

6.AJ. Geometrical Illusions. This is a large field. Principally I have considered the cases below, but I have started to compile origins of other illusions.

6.AJ.1. The Two Pronged Trident. R. L. Gregory told me this was devised by an MIT draughtsman about 1950. D. H. Schuster (Amer. J. Psychol. 77 (1964) 673) refers to an ad by California Technical Industries in Aviation Week and Space Technology 80:12 (23 Mar 1964) 5. A letter to the address in the ad was returned 'insufficient address'. Gardner, SA (May 1970), says it became known in 1964. Can anyone supply a copy of Mad Magazine (Mar 1965) which had it on the cover or of the 1964 issue reprinted in Mad Power? Oscar Reutersvård developed a similar pattern in the 1930s.

6.AJ.2. Tribar and Impossible Staircase. I have now seen Oscar Reutersvård's books of the early 1980s. There appear to have been earlier exhibition catalogues. I have a reference to 150 Omojliga Figurer, Malmo, 1981. Can anyone supply a copy? The books are not very clear as to when he first exhibited his tribar   it appears to have been in the early 1960s.

Where is L. S. Penrose's original model staircase? Roger Penrose thinks it may be in Manchester?

6.AK. Polygonal Path Covering N by N Lattice of Points, Queen's Tours, etc. Loyd has 3 x 3, 7 x 7 and 8 x 8 versions in his Cyclopedia. A. C. White, Sam Loyd and His Chess Problems, shows an 8 x 8 Queen's circuit in 14 moves, by Loyd in ?Le Sphinx (Mar 1867) (or 15 Nov 1866) (NYS) and Chess Strategy (1878) no. or p. 336 (NYS). Can anyone help with the Le Sphinx? White also shows non crossing Rook's circuits in 16 segments from Chess Strategy. I have a trick version on the 6 x 6 from 1887. The 3 x 3 problem must be old, but my earliest references are an interview with Loyd in 1907 and Pearson (1907).

6.AO. Configuration Problems. (a,b,c) denotes a points in b rows of c each. Jackson, Rational Amusements for Winter Evenings, (1821) seems to be the source, but several authors, e.g. Dudeney, "World's best puzzles" (1908), say (9,10,3) is attributed to Newton.

6.AO.1. Place Four Points Equidistantly = Make Four Triangles with Six Matches. My earliest example is c1826.

6.AO.2. Place an Even Number on Each Line. My earliest examples are Mittenzwey (1879?) and Hoffmann (1893).

6.AP. Dissections of a Tetrahedron. The classic bisection of a tetrahedron into two congruent pieces was patented in 1940, though I have clear picture from 1887. I have references from 1946 and 1957, but I also have a 1962 mention which indicates it was new to the author.

These pieces can be cut in half in a simple way or by another midplane of the tetrahedron to give four congruent pieces. I have an example of the first from about the 1940s, but I think the latter is more recent and my earliest source is a puzzle version of the 1970s?? There is also a trisection, made by Pussy and Tenyo. Any idea of the origins of any of these?

6.AQ. Dissection of a Cross, T or H. S&B show a 1857 dissection of a cross which trims to the common T dissection. They say that crosses date from early 19C and T's from 1903. A T puzzle has been found from 1898. Charles Babbage's MSS of c1820 show a five piece cross and I have another version of 1826?

6.AR. Quadrisection of a Square. I have a 1903 description of this and it may be in Crambrook.

6.AS.1. Twenty 1, 2, 5 Triangles Make a Square or Five Equal Squares to a Square. The 10 piece version is in Les Amusemens (1749). The Boy's Own Book (1828) has the 20 piece version.

6.AS.2. Dissect Two Adjacent Squares to a Square. This must be ancient, as it is a proof of the Theorem of Pythagoras, but my earliest reference is Les Amusemens (1749), but I have recently been told that a version occurs in Arabic, c1100.

6.AT. Polyhedra and Tessellations.

6.AT.2 Star and Stellated Polyhedra. There is a mosaic attributed to Uccello on the floor of San Marco, Venice, which shows a 'spiky' dodecahedron. Judith Field says it is not a proper stellated dodecahedron and is much later than Uccello. I'd like a good picture of it. I've now found two pictures, one very good and one not good!!

Jamnitzer (1568) shows a great dodecahedron.

The stella octangula is generally attributed to Kepler. Coxeter [The Fifty Nine Icosahedra] cites Harmonices Mundi, II, prop. XXVI, but this describes a sort of cube with ears whose faces are octagrams. I have not found anyone who can give the origin of the stella octangula.

6.AT.8. Dürer's Octahedron. New topic - I'd appreciate further references.

6.AU. Dead Dogs and Trick Ponies. I have 1849, 1857, c1859 & 1860. I have just found an early version, supposedly from Ulm, 1470, with a rotating centre section. There are many paintings showing mixtures of bodies and heads - I have c1600 Persian and several Japanese version, c1800.

6.AV. Cutting Up in Fewest Cuts. My early examples are in Perelman, 1920s??

6.AW.1. Mitre Puzzle. This is in Hanky Panky (1872) and Hoffmann (1893), but Pearson (1907) calls it Loyd's. Dudeney (1908) says he has traced it back to 1835, but gives no details. It is in Babbage's MSS of the 1820s and in Endless Amusement (1826?).

6.AW.3. Dividing a Square into Congruent Parts. Is there any way to divide a square into three congruent, connected parts, other than rectangles? I have heard that this has been proven.

6.AX. The Packer's Secret. My earliest example is Tissandier, 1888.

6.AY. Dissect 3A x 2B to Make 2A x 3B, etc. (Using a staircase cut.) This is in Cardan (1557) and a Japanese booklet of 1727. Loyd (1914) claimed that any rectangle could be converted to a square in this way, but Dudeney pointed out this error.

6.AZ. Ball Pyramid Puzzles. Len Gordon has been studying these. He & Jerry Slocum can only trace them back to the early 1970's. The first example is Pyramystery by Piet Hein, 1970, but Hein produced two versions with the same name in 1970!

6.BA. Cutting a Card so One Can Pass Through It. This is in Ozanam (1725).

6.BB. Doubling a Square without Changing Its Height or Width. This is in The Sociable (1858).

6.BC. Hoffman's Cube. Dean Hoffman presented this at a conference at Miami Univ. in 1978. Was there any publication of this Conference?

6.BD. Bridge a Moat with Planks. This is in Mittenzwey (1879?), Lucas (1883), Hoffmann (1893). The first example with a circular moat seems to be Always (1969).

6.BE. Reverse a Triangular Array of Ten Circles. My earliest example is 1939.

6.BF.4. Rail Buckling. My earliest examples are 1943 and 1956.

6.BG. Quadrisect a Paper Square with One Cut. I reinvented this. Gardner says it is well known, but I have only found it once in print.

6.BH. Moiré Patterns. I haven't found any really good history of this.

6.BL. Tan^-1 ⅓ + Tan^-1 ½ = Tan^-1 1, etc. This problem is usually presented with three squares in a row with lines drawn from one corner to the opposite corners of the squares. Euler has a general version, but the particular puzzle version seems to be mid-20C.

6.BN. Round Peg in Square Hole or Vice Versa. I am collecting quotations based on this simile. The fact that the round peg fits better was the basis of my first paper in 1964, but the 2-D case had already been done in 1944.

6.BR. What is a General Triangle? More references would be useful.

6.BS. Form Six Coins into a Hexagon. My earliest example is 1963.

6.BT. Placing Objects in Contact. My earliest example is 1926 putting five coins in contact. I know there is more recent work, but I don't know where.

7. ARITHMETICAL RECREATIONS
7.A. Fibonacci Numbers. D. Bernoulli knew the Binet formula for F_n in 1732. This is about a century before Binet and is much clearer!

7.B. Josephus or Survivor Problem. Josephus is very vague, though the Slavonic text implies that he definitely cheated. Ahrens, MUS II, cites Codex Einsidelensis No. 326 (early 10C) and other items. Murphy's article in Béaloideas - The Journal of the Folklore of Ireland Society 12 (1942) 3-28 describes early versions and variants from northern Europe and Ireland - he believes it originates in c800 Ireland! Apparently Cardan (1539) is the first to associate it with Josephus. Most of the early forms involve 15 & 15 counted out by 9 or 10, but Meermanische Codex (10C), Munich Codex 14836 (11C) and Chuquet give rules for counting out by other values. Murphy and some 19C articles associate the problem with St. Peter. Pacioli, De Viribus, describes other numbers, e.g. 2 Christians and 30 Jews. Ahrens is not clear as to the origin of the idea of counting out till one is left. Any information on early versions would be appreciated.

The Japanese variant of the problem, with 15 & 15 counted by 10s, goes back at least to Muramatsu (1665) and Ahrens believes it may have an independent origin as early as 11C?, though I find this hard to believe. Can anyone supply copies of the early Japanese versions?

7.C. Egyptian Fractions. Was Sylvester the first to show that an expansion terminated?

7.E. Monkey and Coconuts Problems. Mahavira gives several versions, including two with indeterminate final result. Versions with final result specified occur in Chiu Chang Suan Ching, Zhang Qiujian, Ananias of Shirak, Bakhshali MS, etc. There are coconuts versions in Pearson (1907) and there are other indeterminate versions in Ozanam (1725) and Dudeney (1903). I have a reference to al-Tabari - can anyone supply details?

7.E.1. Versions with All Getting the Same. These have the i-th person getting a*i + b plus r of the rest or r of the remainder plus a*i + b more and all getting the same. These occur in Fibonacci, Maximus Planudes, Gherardi, dell'Abbaco, etc.

7.F. Illegal Operations Giving Correct Result. I have an article by Witting (1910) which gives some of the examples. Ahrens, A&N, gives it in 1918. Lietzmann, Lustiges und Murkwürdiges uber Zahlen und Figuren, gives 16/64 = 1/4 in 1922?? (my edition is 1923), citing Witting and Ahrens.

The illegal cancellation of a   b into a²   b² is in SM 12 (1946) 111.

7.G. Inheritance Problems.

7.G.1. Half + Third + Ninth, Etc. It is in Dudeney, MP, and Loyd, Cyclopedia. The French ed. of MRE says it is Arabic. Kraitchik, Math. des Jeux, says it is a Hindu problem. Dudeney and Loyd both give it in a rephrased manner that makes the 18th camel unnecessary. Sanford, Short History of Math., says Tartaglia was one of the first to suggest the 18th camel, but I haven't found it yet. Tartaglia does discuss the problem and says people claim the problem is impossible or illegal, but he simply divides proportionally. Hanky Panky (1872) and Cassell's Book ... (1881) give it clearly. H&S says it is a modern problem in that previously it was considered as a proportion   1/2 : 1/3 : 1/9 = 9 : 6 : 2. Ahrens, A&N, gives it clearly and cites the French edition of MRE - he adds that the problem has been in German oral tradition for a long time. Dell'Abbaco has division into half and third. I recall a paper which found all solutions of 1/a + 1/b + 1/c = n/(n+1), but I can't find it.

7.G.2. Posthumous Twins, Etc. I have references for the posthumous twins problem to the Lex Falcidia ( 1C), Juventius Celsus (1C), and others, but don't yet know if they are easily found.

7.H. Division and Sharing Problems   Cistern Problems. This is generally attributed to Hero(n), but it is in a dubious work and his solutions are very confused and wrong. (The dating of Hero seems to have recently changed - I have c150 - what is the new date??) Smith, History II 538, quotes a cistern problem from Bachet's Diophantos, but it is in a section taken from The Greek Anthology. H&S says a cistern problem is in Alcuin but I can only find a simple problem (8) which mentions a cistern, but otherwise is unrelated. Datta & Singh cite Brahmagupta, but it is actually in Prthudakasvami's commentary of 860. The Chiu Chang Suan Ching (c 150) gives a five pipe cistern problem, which Vogel says is the first example. However, the Chiu Chang Suan Ching also gives a number of 'assembly problems' which lead to the identical mathematics as cistern problems, but all the rates are inverted. Eleanor Robson has told me that there are several of these problems in Old Babylonian and I now have some details of these. The Bakhshali MS (c7C??) has several variants of the cistern problem, including one with seven rates.

7.H.2. Division of Casks. Alcuin divides 10 full, 10 half full and 10 empty casks among three people. Abbot Albert divides 9 casks containing 1, 2, ..., 9 among 3 people.

7.H.3. Sharing Unequal Resources   Problem of the Pandects. E.g. one man has 5 loaves and another has 3 which they share with a third. He pays them. How do they split the money? It is in Fibonacci, 1202, and in Kazwini's Cosmographia, 1262. Kraitchik gives a Roman(?) version, taken from Unterrichtsblatter fur Math. & Naturw. 11, pp. 81 85, (NYS), and this is my only source for the term 'Pandects'.

7.H.4. Each Doubles Other's Money to Make All Equal. Diophantos gives general formulations for 3 and 4 people and it is in Mahavira and Fibonacci.

7.H.5. Sharing Cost of Stairs, Etc. Mahavira and Sridhara have versions where carriers are paid from the goods carried. This really leads to an exponential function and is a bit like the Explorer's problem (5.N). Mahavira and Sridhara also have problems of sharing payment among carriers who carry for different parts of the journey or spectators who watch different parts of a performance. Dell'Abbaco has a house being shared.

7.H.6. Sharing a Grindstone. I have just added this and my only example is 1928.

7.H.7. Digging Part of a Well. Qazwini and dell'Abbaco have versions where a well is being dug.

7.I. Four Fours. The earliest reference to four fours is a pseudonymous letter to Knowledge (1881), but Dilworth, c1744, asks for 12 in four identical figures and for 34 as four threes. I have now two 18C references and a number of examples of the idea before 1880.

7.I.1. Largest Number Using Four Ones, etc. My earliest version is Perelman, 1920s?

7.J. Salary Puzzle. That is, it is better to get one quarter of the raise twice as often. This is in Ball (MRE, 3rd ed., 1896), Cunnington (1904) and the Daily Mail (30 Jan 1905). The puzzle seems to be a fairly direct evolution from more straightforward salary problems.

7.K.1. Casting Out Nines. This is mentioned by St. Hippolytus, Philosphumena, c200, NYS. A special case is in Iamblichus. Al-Khowarizmi, c820, describes it. A fairly general use of 9s is in Aryabhata II's Mahasiddhanta, c950, and Narayana's Ganita kaumudi, c1356, allows any modulus. Have either of these ever been translated into a western language?? There are also Arabic references from 952/953, c1000 and c1020 (Avicenna, who attributes it to the Hindus).

7.L.1. Geometric Progressions. I am looking for the source of the story of Sessa and the chessboard, i.e. 1 + 2 + 4 + ...+ 2⁶³. Murray feels it is of Indian origin, but his earliest version is Al-Yaqubi, c875. Mas'udi's Prairies d'Or (10C) refers to the summation without reference to Sessa. Al Biruni computed 2⁶⁴ by repeated squaring but doesn't cite Sessa. It appears in Fibonacci, etc.. J. Wallis gives an Arabic and Latin version. The earliest horseshoe nails version seems to be AR (c1450).

I am interested in occurrences of (1 + )7 + 49 + 343 + ... which appears in Papyrus Rhind and in Fibonacci and in Munich 14684. Buddha is said to have been asked to compute 7¹⁷. (Cajori, History of Mathematics, p. 90.) Are there earlier nursery-rhyme versions than c1730? Are there other examples going up by 9s?

Fibonacci also gives the use of binary weights (1, 2, 4, 8, 15) to get to 30 and Bachet's weights (1, 3, 9, 27, ...). Al-Tabari is said to have used the latter - can anyone supply details?

7.M. Binary System and Binary Recreations. I have just added material on the origins of the binary system, but the history seems contorted. Binary coding is present in the Chinese arrangement of the I Ching hexagrams, c1060. But it is already implicit in Egyptian multiplication and the Chinese rings. Binary arithmetic seems due to Harriot (unpublished, c1604) and Napier (Rabdologia, 1617), before Leibniz (1679).

7.M.1. Chinese Rings. Cardan's 1550 description is brief! Ch'ung En Yu's Ingenious Ring Puzzle Book calls it the Nine Interlocked Rings Puzzle and says it was well known in the Sung Dynasty (960 1279). Jerry Slocum and Needham give older Chinese references, but these are pretty vague. Culin, Games of the Orient, gives a legend that it was invented by Hung Ming (181 234). There is a Chinese musical drama, The Stratagem of Interlocking Rings, c1300 - but I have no information about it. Gardner says there are 17C Japanese haiku about the puzzle and it occurs in Japanese heraldry - can anyone supply details? Afriat says he had a copy of Gros's 1872 pamphlet obtained by the Radcliffe Science Library at Oxford, but they could not find it for me. Can anyone help with this?

7.M.2. Tower of Hanoi. I have now seen photocopies of the original versions deposited at the Conservatoire National des Arts et Métiers by Lucas himself, with his inscriptions, including the date of invention - Nov 1883 - and the assertion that he invented it. The commercial version had the fancy cover and the instructions are dated 1883. Edward Hordern has an example with the same instructions, but in a different box. The earliest article seems to be: G. de Longchamps; "Variétés"; Journal de Mathématiques Spéciales (2) 2 (1883) 286-287, but the British Library says there are no copies in the UK - can any French reader send a copy? De Parville has a note on the puzzle on 27 Dec 1883. There is an 1889 booklet (Brochure) by Lucas on the Tower of Hanoi, NYS (not in British Library or Bibliothèque Nationale catalogues and A. Hinz has not located any example). But Lucas' article in La Nature (1889) indicates that this last may refer to the game itself or its instructions. Dudeney, World's best puzzles, dates it to 1883. The original cover says 'Brevete', but the Conservatoire could not locate a patent in 1880-1890.

7.M.2.a. Tower of Hanoi with More Pegs. Lucas' La Nature article of 1889 mentions the use of 4 or 5 pegs and illustrates such a game. Dudeney gives a 4 peg version in London Magazine (May 1902), a 4 peg version in Weekly Dispatch (25 May 1902) and a 5 peg version in Weekly Dispatch (15 Mar 1903). The general version is unsolved.

7.M.3. Gray Code. Baudot used this c1878, but I have no contemporary reference other than Annales Télégraphiques of 1879. Was there a patent?

7.M.4. Binary Divination. Pacioli's De Viribus has a clear version for finding one thing from 16.

7.M.6. Binary Button Games. I have recently added this. The earliest material I have is Berlekamp's switching game at Bell Labs, c1970, and my analysis of the XL25 in 1985.

7.N. Magic Squares. This is getting sorted out?. I am back to c 650 in the Shu Ching and c 5C in the Confucian Analects, though these are cryptic references which may be irrelevant or later interpretations. Theon of Smyrna is often cited but his square is not magic. Ho Peng Yoke says Xu Yiu (Hsu Yo), c190, is the first to give the order 3 square, but there is doubt about the date and authorship - some say it was written by Zhen Luan in c570. Cammann, Lam and Hayashi refer to the Ta Tai Li Chi, c80, as the earliest example.

Ahrens' survey in Der Islam (1917) has clarified most of the Arabic references. Jabir ibn Hayyan and Tabit ibn Korra give the first Arabic magic squares. The relevant works of Korra and ibn al Haitham seem to have perished.

I have vague references to Ts'ai Yuan Ting (c1160) and Najm al Din al Lubudi (c1250).

The 1C Hindu version remains mysterious - I have only a brief reference to this - A. N. Singh, "Hist. of magic squares in India", Proc. ICM, 1936. The 4 x 4 magic square seems likely to have originated in India. I'm not sure if the square at Khajuraho is the same as the one at Dudhai, Jhansi?? Narayana Pandita (1356) gives oddly even squares and the editor cites some earlier Hindu sources. (I believe someone has studied Narayana's work on magic squares??) Can anyone help with these?

7.N.1. Magic Cubes. Fermat seems to be the first to construct one. However, it fails along 8 of the 24 2 agonals as well as along all 4 3 agonals. Benson & Jacoby cite G. Frankenstein, in the Commercial Traveller (Cincinnati) (11 Mar 1875), for a perfect 8³ (NYS). Maxey Brooke cites Joseph Sauvier (1710) for the first magic cube (NYS) and Schlegel (1892) for the first magic 3⁴ (NYS). In W. S. Andrews' Magic Squares and Cubes, C. Planck cites a book of his, Theory of Path Nasiks, NYS, which seems to be the first approach to a theory. Gardner, SA (Feb 1976) says there is an unpublished MS of Rosser & Walker at Cornell and that there are reports by Schroeppel and Beeler on this. Can anyone help find these?

7.N.2. Magic Triangles. Frenicle (1640) and Scheffler (1882) give versions.

7.N.3. Anti Magic Squares and Triangles. When do these begin? Loyd Jr (1928) gives an antimagic 3 x 3 square. Gardner mentions anti magic squares in SA (Jan 1961), citing MM, 1951. Fults' book asserts Trigg considered these in 1951, but he gives no reference and I see this is a corruption of Gardner's reference.

7.N.4. Magic Knight's Tour. New section. Beverley, 1848, seems to be the first to consider this. De Jaenisch has a lot on it.

7.N.5. Other Magic Shapes. I have a magic cross from 1893 & 1907 and a number of odd magical shapes from late 19C.

7.O. Magic Hexagons. Trigg, RMM 14 (1964) 40 43 reports that Adams saw the idea in an issue of The Pathfinder, c1910. Can anyone find this? Gardner, Puzzles from Other Worlds (1984), reports that there are 1896 patents in the UK and the US by W. Radcliffe. Heinrich Hemme has kindly sent material on the first version by von Haselberg, 1888-89.

7.O.1. Other Magic Hexagons. Frenicle gives one form in 1640. Another form with constant 26 on a Star of David appears in 1895, 1908 and 1918.

7.P.1. Hundred Fowls and Other Linear Problems. This seems due to Zhang Qiujian (Chang Chhiu Chin) (c475). The Indian