7. arithmetic & number theoretic recreations a. Fibonacci numbers

P. 105: To find the least Number of Weights that will weigh from One Pound to Forty. = Badcock.

Prob. 24, p. 201. To name five weights, which, added together, make 121 pounds; by means of which may be weighed any intermediate weight, excluding fractions. 1, 3, 9, 27, 81. = New Sphinx, c1840, p. 137.

Young Man's Book. 1839. P. 242. To name five weights, .... Identical to Endless Amusement II, p. 201.

Boy's Own Book. 1843 (Paris): 346-347. "To find the least number of weights which will weigh any intermediate weight, from one pound to forty, exclusive of fractions. Indicates that one can continue the progression." = Boy's Treasury, 1844, p. 304.

Family Friend 1 (1849) 150 & 178. Problems, arithmetical puzzles, &c. -- 5. How can one divide 40 lb into four weights to weigh every value from 1 to 40? Proposer says he can't do it. Usual answer, but one solver gives weights 6, 10, 11, 13. However, the latter weights will not weigh 22, 25, 26, 31, 32, 33, 35, 36, 37, 38, 39.

Boy's Own Book. To weigh from one to forty pounds with four weights. 1855: 392; 1868: 430. 1, 3, 9, 27. No generalizations.

Magician's Own Book. 1857. The mathematical blacksmith, p. 230. 1, 3, 9, 27 and this can be continued. = Boy's Own Conjuring Book, 1860, p. 200.

Todhunter. Algebra, 5th ed. 1870. Miscellaneous Examples, no. 175, p. 559. Show that 1, 1, 5, 5, 25, 25, 125, 125 can weigh any integral amount up to 312. No solution given.

F. J. P. Riecke. Op. cit. in 4.A.1, vol. 3, 1873. Art. 3: Die Zauberkarten, p. 13. Uses balanced ternary for divination. See under 7.M.4

Mittenzwey. 1880.

Prob. 107, pp. 22 & 74; 1895?: 124, pp. 26 & 76-77; 1917: 124, pp. 24 & 74-75. Stone of weight 40 breaks into four parts which weigh up through 40. Solution is a table showing how to weigh up through 20. 1895? adds solutions for weights 1, 2, 3, 34; 5, 9, 10, 16; allowing several weighings, e.g. 4 is obtained by weighing out 2 twice.

1895?: prob. 125, pp. 26 & 77; 1917: 125, pp. 24 & 75. Five weights to get through 121.

P. A. MacMahon. Certain special partitions of numbers. Quart. J. Math. 21 (1886) 367 373. Very technical.

P. A. MacMahon. Weighing by a series of weights. Nature 43 (No.1101) (4 Dec 1890) 113 114. Less technical description of the above work.

Lucas. L'Arithmétique Amusante. 1895. Pp. 166-168. Notes that pharmacists, etc. use weights: 1, 1, 2, 5, 10, 10, 20, 50, 100, 100, 200, 500, 1000, 1000, 2000, 5000, .... Discusses ternary.

Wehman. New Book of 200 Puzzles. 1908. P. 49. 1, 3, 9, 27, 81.

Ahrens. MUS I. 1910. Pp. 88-98 discusses this and some generalizations like MacMahon's.

In 2002, Miodrag Novaković told me that a student had told him how to determine twice as many integral weights with the same number of weights. E.g., he used weights 2, 6, 18, 54 to determine integral weights 1, 2, ..., 81. One can get exact balancing for the even values: 2, 4, ..., 80. The odd values fall between two consecutive even amounts, so if a package weighs more than 6 but less than 8, we deduce it weighs 7!

7.M. BINARY SYSTEM AND BINARY RECREATIONS
The binary system has several origins.

a) Egyptian & Russian peasant multiplication.

b) Weighing -- see 7.L.2.c.

c) Binary divination -- see 7.M.4.

d) The works below.
See also: 5.E.2 for Memory Wheels; 5.F.4 for circuits on the n cube; 5.AA for an application to card-shuffling; 7.AA.1 for Negabinary.
Anton Glaser. History of Binary and Other Nondecimal Numeration. Published by the author, 1971; (2nd ed., Tomash, Los Angeles, 1981). General survey, but has numerous omissions -- see the review by Knuth at Harriot, below, and MR 84f:01126. He has no references to early Chinese material.
Shao Yung. c1060. Sung Yuan Hsüeh An, chap. 10. Fu Hsi diagram of the 64 hexagrams of the I Ching, in binary order. A version appears in Leibniz Briefe 105 (Bouvet) Bl. 27r/28r in the Niedersächsische Landesbibliothek, Hannover. Needham, vol. 2, p. 341, notes that this had only been published in Japanese and Chinese up to 1956. See Zacher & Kinzô below for reproductions. Also reproduced in: E. J. Aiton; Essay Review [of Zacher, below]; Annals of Science 31 (1974) 575 578.

Chu Hsi. Chou I Pen I Thu Shuo. 12C. Fu Hsi Liu shih ssu Kua Tzhu Hsü (Segregation Table of the symbols of the Book of Changes) -- reproduced in Hu Wei's I Thu Ming Pien. An illustration is given in Needham, vol. 2, fig. 41 = plate XVI, opp. p. 276 -- he says it is based on the original chart of Shao Yung and that Tshai Chhen (c1210) gave a simplified version. Also in Kinzô and in Aiton & Shimao, below. Shows the alternation of 0 and 1 in each binary place.

Thomas Harriot. Unpublished MS. c1604. Described by J. W. Shirley; Binary numeration before Leibniz; Amer. J. Physics 19 (1951) 452 454; and by D. E. Knuth; Review of 'History of Binary and Other Nondecimal Numeration'; HM 10 (1983)) 236 243. This shows some binary calculation. Shirley reproduces BM: Add MSS 6786, ff. 346v 347r. Knuth cites 6782, 1r, 247r; 6786, 243v, 305r, 346v, 347r, 516v; 6788, 244v.

Francis Bacon. Of the Advancement of Learning. 1605. ??NYS. Describes his binary 5 bit coding.

Francis Bacon. De augmentis scientarum. 1623. ??NYS. Full description of his coding. He does not have any arithmetic content, so he is not really part of the development of binary.

John Napier. Rabdologiae. Edinburgh, 1617. ??NYS. Describes binary as far as extracting square roots. William F. Hawkins; The Mathematical Work of John Napier (1550 1617); Ph.D. thesis, Univ. of Auckland, 1982, ??NYS, asserts this is THE invention of the binary system.

G. W. Leibniz. De Progressione Dyadica. 3pp. Latin MS of Mar 1679. Facsimile and translation into German included in: Herrn von Leibniz' Rechnung mit Null und Eins; Siemens Aktiengesellschaft, (1966), 2nd corrected printing, 1969; facsimile between pp. 20 & 21 and translation on pp. 42-47, with an essay by Hermann J. Greve: Entdeckung der binären Welt on pp. 21-31. His first, unpublished, MS on the binary system, showing all the arithmetic processes.

H. J. Zacher. Die Hauptschriften zur Dyadik von G. W. Leibniz. Klosterman, Frankfurt, 1973. Gathers almost all the Leibniz material, notably omitting the above 1679 paper. He does reproduce the Fu Hsi diagram sent by Bouvet (cf. Shao Yung above). However Leibniz's letter of 2 Jan 1697 to Herzog Rudolf Augustus, in which he gives his drawing of his plan for a medallion commemorating the binary system, is now lost, but it was published in 1734.

G. W. Leibniz. Two Latin letters on the binary system, 29 Mar 1698 & 17 May 1698, recipient not identified, apparently the author of a book in 1694 which occasioned Leibniz's correspondence with him. Opera Omnia, vol. 3, 1768, pp. 183-190. Facsimile and translation into German included in: Herrn von Leibniz' Rechnung mit Null und Eins; Siemens Aktiengesellschaft, 2nd corrected printing, 1966; facsimile between pp. 40 & 41 and translation on pp. 53-60. On p. 26, it seems to say the second letter was sent to Johann Christian Schulenberg.

G. W. Leibniz. Explication de l'arithmètique binaire. Histoire de l'Academie Royale des Sciences 1703 (1705) 85-89. Facsimile and translation into German included in: Herrn von Leibniz' Rechnung mit Null und Eins; Siemens Aktiengesellschaft, 2nd corrected printing, 1966; facsimile between pp. 32 & 33 and translation on pp. 48-52. Illustrates all the arithmetic operations and discusses the Chinese trigrams of 'Fohy' and his correspondence with Father Bouvet in China.

G. W. Leibniz. Letter of 1716 to Bouvet. ??NYS -- cited in Needham, vol. 2, p. 342. Fourth section is: Des Caractères dont Fohi, Fondateur de l'Empire Chinois, s'est servi dans ses Ecrits, et de l'Arithmétique Binaire.

Gorai Kinzô. Jukyô no Doitsu seiji shisô ni oyoboseru eikyô (Influence of Confucianism on German Political Thought [in Japanese]). Waseda Univ. Press, Tokyo, 1929. ??NYS. First publication of Leibniz's correspondence with Bouvet which led to the identification of the Fu Hsi diagram with the binary numbers. Gives a redrawn Fu Hsi diagram and a segregation table.

E. J. Aiton & Eikon Shimao. Gorai Kinzô's study of Leibniz and the I Ching hexagrams. Annals of Science 38 (1981) 71 92. Describes the above work. Reproduces Kinzo's Fu Hsi diagram and segregation table.
Ahrens. MUS I. 1910. 24-104 discusses numeration systems in general and numerous properties of binary and powers of 2.

Gardner. SA (Aug 1972) c= Knotted, chap. 2. General survey of binary recreations. The material in the book is much expanded from the SA column.

7.M.1. CHINESE RINGS
See MUS I 61-72; S&B 104 107, 111 & 135.

See also 4.A.4, 11.K.1.

In various places and languages, the following names are used:

Chinese Rings

Chainese Rings [from www.tama.or.jp/~tane, via Dic Sonneveld, 13 Nov 2002]

Cardan's Rings, but Cardan called it Instrumentum ludicrum

Ryou-Kaik-Tjyo or Lau Kák Ch'a = Delay-guest-instrument

Kau Tsz' Lin Wain = Nine connected rings

Chienowa = Wisdom rings

Kyūrenkan = Nine connected rings

Lien nuan chhuan [from www.roma.unisa.edu.au, via Dic Sonneveld, 13 Nov 2002]

Tarriers or Tarriours

Tiring Irons or Tyring Irons or Tarrying Irons

The Puzzling Rings

The Devil's Needle

Complicatus Annulis [Wallis]

Baguenaudier spelled various ways, e.g. Baguenodier

Juego del ñudo Gordiano

Меледа [Meleda]

Наран-шина [Naran-shina] (stirrup ring toy)

Zauberkette

Magische Ringspiel

Nürnberger Tand

Grillenspiel

Armesünderspiel

Zankeisen

Nodi d'anelli
S. N. Afriat. The Ring of Linked Rings. Duckworth, London, 1982. This is devoted to the Chinese Rings and the Tower of Hanoi and gives much of the history.
Sun Tzu. The Art of War. c-4C. With commentary by Tao Hanzhang. Translated by Yuan Shibing. (Sterling, 1990); Wordsworth, London, 1993. In chap. 5: Posture of Army, p. 109, the translator gives: "It is like moving in a endless circle" In the commentary, p. 84, it says: "their interaction as endless as that of interlocked rings." Though unlikely to refer to the puzzle,, this and the following indicate that interlocked rings was a common image of the time.

Needham, vol. 2, pp. 189-197, describes the paradoxes of Hui Shih ( 4C). P. 191 gives HS/8: Linked rings can be sundered. On p. 193, Needham gives several explanations of this statement and a reference to the Chinese Rings in vol. III, but he does not claim this statement refers to the puzzle.

Stewart Culin. Korean Games. Op. cit. in 4.B.5. Section XX: Ryou Kaik Tjyo -- Delay Guest Instrument (Ring Puzzle), pp. 31 32. Story of Hung Ming (181 234) inventing it. (Wei-Hwa Huang says this is probably Kong Ming (= Zhuge Liang), a famous war strategist, to whom many inventions were attributed.) States the Chinese name is Lau Kák Ch'a (Delay Guest Instrument) or Kau Tsz' Lin Wain (Nine Connected Rings). Says there a great variety of ring puzzles in Japan, known as Chie No Wa (Rings of Ingenuity) and illustrates one, though it appears to be just 10 rings joined in a chain -- possibly a puzzle ring?? He says he has not found out whether the Chinese rings are known in Japan -- but see Gardner below.

Ch'ung En Yü. Ingenious Ring Puzzle Book. In Chinese: Shanghai Culture Publishing Co., Shanghai, 1958. English translation by Yenna Wu, published by Puzzles -- Jerry Slocum, Beverly Hills, Calif., 1981. P. 6. States it was well known in the Sung (960 1279). [There is a recent version, edited into simplified Chinese (with some English captions, etc.) by Lian Huan Jiu, with some commentary by Wei Zhang, giving the author's name as Yu Chong En, published by China Children's Publishing House, Beijing, 1999.]

The Stratagem of Interlocking Rings. A Chinese musical drama, first performed c1300. Cited in: Marguerite Fawdry; Chinese Childhood; Pollock's Toy Theatres, London, 1977, pp. 70-72. Otherwise, Fawdry repeats information from Culin and the story that it was used as a lock.

Needham. P. 111 describes the puzzle as known in China at the beginning of the 20C, but says the origins are quite obscure and gives no early Chinese sources. He also cites his vol. 2, p. 191, for an early possible reference -- see above.

Pacioli. De Viribus. c1500. Ff. 211v-212v, Part 2, Capitulo CVII. Do(cumento), cavare et mettere una strenghetta salda in al quanti anelli saldi. dificil caso (Remove and replace a joined string a number of joined rings - a difficult thing). = Peirani 290-292. Dario Uri says this describes the Chinese Rings. It is hard to make out, but it appears to have six rings. Uri gives several of the legends about its invention and says Cardan called it Meleda, but that word is not in Cardan's text. He lists 27 patents on the idea in five countries.

Cardan. De subtilitate. 1550. Liber XV. Instrumentum ludicrum, pp. 294 295. = Basel, 1553, pp. 408 409. = French ed., 1556, et les raisons d'icelles; Book XV, para. 2, p. 291, ??NYS. = Opera Omnia, vol. 3, p. 587. Very cryptic description, with one diagram of a ring.

In England, the Chinese Rings were known as Tarriers or Tarriours or Tiring or Tyring or Tarrying Irons. The OED entry at Tiring-irons gives 5 quotations from the 17C: 1601, 1627, 1661, 1675, 1690.

John II Tradescant (1608-1662). Musæum Tradescantianum. 1656. Op. cit. in 6.V. P. 44: "Tarriers of Wood made like our Tyring-Irons." (The following entry is: "Tarriers of Wood like Rolles to set Table-dishes on." -- I cannot figure out what this is.)

Gardner. Knotted, chap. 2, says there are 17C Japanese haiku about it and it is used in Japanese heraldic emblems.

Kozaburo Fujimura & Shigeo Takagi. Pazuru no Genryū (The Origins of Puzzles, in Japanese). Daiyamondo Sha, Tokyo, 1975. ??NYS -- information kindly sent by Takao Hayashi. Chap. 9 is on the Chinese Rings.

This says the oldest datable record in Japan is in Osaka Dokugin Shū, a book of haiku compiled in 1675. A haiku of Saikaku Ihara is "Chienowa ya shijōdōri ni nukenuran", where the first word means 'wisdom rings' and denotes the Chinese Rings puzzle.

Another poetry book, Tobiume Senku, of 1679 has "Tenjiku shintan kuguru chienowa".

The Chinese Rings was used as a family crest. The first known reference is in the description of a kimono worn by a character in Monzaemon Chikamatsu's joruri, Onna Goroshi Abura Jigoku, first staged in 1721. See next item for more details.

Another word for the puzzle is kyūrenkan, which is borrowed from Chinese and means 'nine connected rings'. It is explained in a Chinese lexicon, Meibutsu rokujō, by Choin Ito, c1725.

The mathematician Yasuaki Aida (1747-1817), in his unpublished autobiography Jizai Butsudan of 1807, says he solved the puzzle when he was nine, i.e. c1756.

Yoriyuki Arima (1714-1783), another mathematician, treats the Chinese Rings in his mathematical work Shūki Sampō, of 1769.

Gennai Hiraga (1728-1779) unlocked a bag locked with a Chinese Rings belonging to Captain Jan Crans of the Dutch factory (i.e. trading post) in c1769. This is related in Genpaku Sugita's Rangaku Kotohajime of 1815.

Chienowa is recorded in the 1777 Japanese dictionary Wakun no Shiori.

Dictionary of Representative Crests. Nihon Seishi Monshō Sōran (A Comprehensive Survey of Names and Crests in Japan), Special issue of Rekishi Dokuhon (Readings in History), Shin Jinbutsu Oraisha, Tokyo, 1989, pp. 271-484. Photocopies of relevant pages kindly sent by Takao Hayashi. Crest 3447 looks like a Chinese Rings with five rings and 3448 looks like one with four rings, but both are simplified and leave out one of the bars.

John Wallis. De Algebra Tractatus. 1685, ??NYS. = Opera Math., Oxford, 1693, vol. II, chap. CXI, De Complicatus Annulis, 472 478. Detailed description with many diagrams.

Ozanam. 1725: vol. 4. No text, but the puzzle with 7 rings is shown as an unnumbered figure on plate 14 (16). Ball, MRE, 1st ed., 1892, p. 80, says the 1723 ed., vol. 4, p. 439 alludes to it. The text there is actually dealing with Solomon's Seal (see 11.D) which is the adjacent figure on plate 14 (16).

Minguet. 1733. Pp. 55-57 (1755: 27-28; 1822: 72-74; 1864: 63-65): Juego del ñudo Gordiano, ò lazo de las sortijas enredadas. 7 ring version clearly drawn.

Alberti. 1747. No text, but the puzzle is shown as an unnumbered figure on plate XIII, opp. p. 214 (111), copied from Ozanam, 1725.

Catel. Kunst-Cabinet. 1790. Der Nürnberger Tand, p. 15 & fig. 41 on plate II. Figure shows 7 rings, text says you can have 7, 9, 11 or 13.

Bestelmeier. 1801. Item 298: Der Nürnberger Tand. Diagram shows 6 rings, but text refers to 13 rings. Text is partly copied from Catel.

Endless Amusement II. 1826? Prob. 29, pp. 204-207. Cites Cardan as being very obscure. Shows example with 5 rings and seems to imply it takes 63 moves.

The Boy's Own Book. The puzzling rings. 1828: 419 422; 1828 2: 424 427; 1829 (US): 216-218; 1855: 571 573; 1868: 673-675. Shows 10 ring version and says it takes 681 moves. Cites Cardan.

Crambrook. 1843. P. 5, no. 9: Puzzling Rings, or Tiring Irons.

Magician's Own Book. 1857. Prob. 45: The puzzling rings, pp. 279-283. Identical to Boy's Own Book, except 1st is spelled out first, etc. = Book of 500 Puzzles, 1859, pp. 93-97. = Boy's Own Conjuring Book, 1860, prob. 44, pp. 243 246.

Magician's Own Book (UK version). 1871. The tiring-irons, baguenaudier, or Cardan's rings, pp. 233-235. Quite similar to Boy's Own Book, but somewhat simplified and gives a tabular solution.

L. A. Gros. Théorie du Baguenodier. Aimé Vingtrinier, Lyon, 1872. (Copy in Radcliffe Science Library, Oxford -- cannot be located by them.) ??NYS

Lucas. Récréations scientifiques sur l'arithmétique et sur la géométrie de situation. Troisième récréation, sur le jeu du Baguenaudier, ... Revue Scientifique de la France et de l'étranger (2) 26 (1880) 36 42. c= La Jeu du Baguenaudier, RM1, 1882, pp. 164 186 (and 146 149). c= Lucas; L'Arithmétique Amusante; 1895; pp. 170-179. Exposition of history back to Cardan, Gros's work, use as a lock in Norway. He says that Dr. O.-J. Broch, former Minister and President of the Royal Norwegian Commission at the Universal Exposition of 1878, recently told him that country people still used the rings to close their chests and sacks. RM1 adds a letter from Gros.

The French term 'baguenaudier' has long mystified me. A 'bague' is a ring. My large Harrap's French English dictionary defines 'baguenaudier' as "trifler, loafer, retailer of idle talk; ring puzzle, tiring irons; bladder senna", but none of the related words indicates how 'baguenaudier' came to denote the puzzle. However, Farmer & Henley's Dictionary of Slang gives 'baguenaude' as a French synonym for 'poke', so perhaps 'baguenaudier' means a 'poker' which has enough connection to the object to account for the name?? MUS I 62-63 discusses Gros's use of 'baguenodier' as unreasonable and quotes two French dictionaries of 1863 and 1884 for 'baguenaudier' which he identifies as an ornamental garden shrub, Colutea arborescens L.

Cassell's. 1881. Pp. 91-92: The puzzling rings. = Manson, 1911, pp. 144-145: Puzzling rings. Shows 7 ring version and discusses 10 ring version, saying it takes 681 moves. Discusses the Balls and Rings puzzle.

Peck & Snyder. 1886. P. 299: The Chinese puzzling rings. 9 rings. Mentions Cardan & Wallis. Shown in Slocum's Compendium.

Ball. MRE, 1st ed., 1892, pp. 80-85. Cites Cardan, Wallis, Ozanam and Gros (via Lucas). P. 85 says: "It is said -- though a priori the fact would have seemed very improbable -- that Chinese rings are used in Norway to fasten the lids of boxes, .... I have never seen them employed for such purposes in any part of the country in which I have travelled." This whole comment is dropped in the 3rd ed.

Hoffmann. 1893. Chap. X, no. 5: Cardan's rings, pp. 334 335 & 364 367 = Hoffmann Hordern, pp. 222-225, with photo. Cites Encyclopédie Méthodique des Jeux, p. 424+. Photo on p. 223 shows The Puzzling Rings, by Jaques & Son, 1855 1895, with instructions, and Baguenaudier, with box, 1880-1895. Hordern Collection, p. 92, shows the Jaques example, an ivory example with an elaborate handle and another of ivory or bone, all dated 1850-1900. I now have an example of the Jaques version which has rings coloured red, white and blue.

H. F. Hobden. Wire puzzles and how to make them. The Boy's Own Paper 19 (No. 945) (13 Feb 1896) 332-333. Magic rings (= Chinese rings) with 10 rings, requiring 681 moves. (I think it should be 682.)

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... it is said to be used by the Norwegians as a form of lock for boxes and bags ..."

Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904. Note 60, p. 1091, reports that a Norwegian professor of Ethnography says the story of its use as a lock in Norway is erroneous. He repeats this in MUS I 63.

M. Adams. Indoor Games. 1912. Pp. 337 341 includes The magic rings.

Bartl. c1920. P. 309, no. 80: Armesünderspiel oder Zankeisen. Seven ring version for sale.

Collins. Book of Puzzles. 1927. The great seven-ring puzzle, pp. 49-52. Cites Cardan and Wallis. Says it is known as Chinese rings, puzzling rings, Cardan's rings, tiring irons, etc. Says 3 rings takes 5 moves, 5 rings takes 21 and 7 rings takes 85.

Rohrbough. Puzzle Craft. 1932. The Devil's Needle, p. 7 (= p. 9 of 1940s?). Cites Boy's Own Book of 1863.

R. S. Scorer, P. M. Grundy & C. A. B. Smith. Some binary games. MG 28 (No. 280) (Jul 1944) 96 103. Studies the binary representations of the Chinese Rings and the Tower of Hanoi. Gives a triangular coordinate system representation for the Tower of Hanoi. Studies Tower of Hanoi when pegs are in a line and you cannot move between end pegs. Defines an n th order Chinese Rings and gives its solution.

E. H. Lockwood. An old puzzle. With Editorial Note by H. M. Cundy. MG 53 (No. 386) (Dec 1969) 362 364. Derives number of moves by use of a second order non homogeneous recurrence. Cundy mentions the connection with the Gray code and indicates how the Gray value at step k, G(k), is derived from the binary representation of k, B(k). [But he doesn't give the simplest expression, given by Lagasse, qv in 7.M.3.] This easily gives the number of steps.

Marvin H. Allison Jr. The Brain. This is a version of the Chinese Rings made by Mag-Nif since the 1970s. [Gardner, Knotted.]

William Keister. US Patent 3,637,215 -- Locking Disc Puzzle. Filed: 22 Dec 1970; patented: 25 Jan 1972. Abstract + 3pp + 1p diagrams. This is a version of the Chinese Rings, with discs on a sliding rod producing the interaction of one ring with the next. Described on the package. Keister worked on puzzles of this sort since the 1930s. It was first produced by Binary Arts in 1986 under the name Spin Out.

Ed Barbeau. After Math. Wall & Emerson, Toronto, 1995. Protocol, pp. 163-166. Gives seating and standing problems which lead to the same sequence of moves as for the Chinese rings, but one is in reverse order.

Anatoli Kalinin says that the Chinese Rings are a old folk puzzle called Меледа [Meleda], especially popular among the Kalmyks near the Caspian Sea, where it is called Наран-шина [Naran-shina] (stirrup ring toy). The name Меледа is derived from a verb which is no longer in Russian.

7.M.2. TOWER OF HANOI
See MUS I 52-61, S&B 135.

See also 5.F.4 for connection with Hamiltonian circuits on the n cube and 5.A.4 for the Panex Puzzle.

All the following have three pegs unless specified otherwise.
The Conservatoire National des Arts et Métiers -- Musée National des Techniques, 292 Rue St. Martin, Paris, has two examples -- No. 11271 & 11272 -- presented by Edouard Lucas, professeur de mathématiques au Lycée Saint-Louis à Paris, in 1888. The second is a 'grand modèle pour les cours publics' 1.05 m high! Elisabeth Lefevre has kindly sent details and photocopies of the box, instruction sheet (one sheet printed on both sides) and an article. Both versions have 8 discs. I am extremely grateful to Jean Brette, at the Palais de la Decouverte, who told me of these examples in 1992.

The box is 157 x 180 mm and has an elaborate picture with the following text:

La Tour d'Hanoï / Veritable casse Téte Annamite / Jeu / rapporté du Tonkin / par le Professeur N. Claus (de Siam) / Mandarin / du College / Li Sou Stian / Brevete / S. G. D. G.

This cover is shown in Claus [Lucas] (1884) and Héraud (1903). I will refer to this as the original cover. (The Museum does not know of any patent -- they have looked in 1880-1890. S. G. D. G. stands for Sans garantie du gouvernement.) The bottom of the box has an ink inscription: Hommage del'auteur Ed Lucas Paris 1888 -- but the date is not clearly legible on the photocopy. Inside the cover, apparently in the same hand (that is, in Lucas's writing), is an ink inscription:

La tour d'Hanoï, --

Jeu de combinaison pour

expliquer le systeme de la numération

binaire, inventé par M. Edouard Lucas,

(novembre 1883). -- donné par l'auteur.

The Museum describes the puzzle as 15 cm long by 14.5 cm wide by 10 cm high. There is no photo available, but the examples are shown in catalogues of 1906 and 1943.

The instruction sheet reads as follows.
La Tour d'Hanoï

Véritable casse tête annamite

Jeu rapporté du Tonkin

par le Professeur N. Claus (de Siam)

Mandarin du Collège Li Sou Stian!

....

Paris, Pékin, Yédo et Saïgon

....

1883

The sheet mentions the Temple of Bénarès where there are 64 discs. A prize of a million (= a thousand thousand) francs is offered for a demonstration of the solution with 64 discs! The second sheet of the instructions gives the rules and the number of moves for 2, 3, ..., 8 discs and the general rule. It also refers to RM.

Edward Hordern's collection has an example with the original instruction sheet, but in a simple box with just 'La Tour d'Hanoï' in Chinese style lettering, not like the box described above. Also, my recollection is that it is much smaller than the example above. It has 8 discs.

G. de Longchamps. Variétés. Journal de Mathématiques Spéciales (2) 2 (1883) 286-287. (The article is only signed G. L., but the author is further identified in the index on p. 290. Copy provided by Hinz.) Solves the recurrence relation u_n = 2u_n-1 + 1, u₀ = 0. Says he was 'inspired by a letter which we have recently received from professor N. Claus.' Describes the Tower of Hanoi briefly and says the above solution gives the number of moves when there are n discs.

Henri de Parville. Column: Revue des sciences. Journal des Débats Politiques et Littéraires (27 Dec 1883) 1-2. On p. 2, he reports receiving an example in the post with a cover like that of the original. Gives the Benares story. Wonders who the mandarin could be and notes the anagrams on Lucas d'Amiens and Saint-Louis.

N. Claus (de Siam) [= Lucas (d'Amiens)]. La tour d'Hanoï. Jeu de calcul. Science et Nature 1:8 (19 Jan 1884) 127-128. Says it takes 2ⁿ - 1 moves "que M. de Longchamps l'a démontré (1)." "(1) Journal de mathématiques spéciales.". Observes that each of the discs always moves in the same cycle of pegs and hence gives the standard rule for doing the solution, which is attributed to the nephew of the inventor, M. Raoul Olive, student at the Lycée Charlemagne. Asks for the minimum number of moves to restore an arbitrary distribution of discs to a Start position. Says this is a complex problem in general, depending on binary and refers to RM 1 for this idea.

(This paper is not in Harkin's bibliography (op. cit. in 1). Hinz, 1989, cites it.)

Henri de Parville. Récréations Mathématiques: La Tour d'Hanoï et la question du Tonkin. La Nature (Paris) 12 (No. 565, part 1) (29 Mar 1884) 285 286. Illustration by Poyet. Asserts Lucas is the inventor.

R. E. Allardice & A. Y. Fraser. La Tour d'Hanoï. Proc. Edin. Math. Soc. 2 (1883 1884) 50 53. Includes de Parville from J. des Débats. Then derives number of moves.

Anton Ohlert. US Patent 303,946 -- Toy. Applied: 24 Jul 1884; patented: 19 Aug 1884. 1p + 1p diagrams. 8 discs. Ohlert is a resident of Berlin.

Edward A. Filene, No. 4 Winter St., Boston Mass. Eight Puzzle. Copyrighted in 1887. ??NYS -- described and illustrated in S&B, p. 135. 8 disc advertising version.

Tissandier. Récréations Scientifiques. 5th ed., 1888, La tour d'Hanoï et la question du Tonkin, pp. 223-228. Not in the 2nd ed. of 1881 nor the 3rd ed. of 1883. Essentially de Parville's article with same illustration, but introduces it with comments saying it has had a great success and comes in a box labelled: "la Tour d'Hanoï, véritable casse-tête annamite, rapporté du Tonkin par le professeur N. Claus (de Siam), mandarin du collège Li-Sou-Stian". This would seem to be the original box.

= Popular Scientific Recreations; [c1890]; Supplement: The tower of Hanoï and the question of Tonquin, pp. 852 856.

Anon. Jeux, Calculs et Divertissements. Récréations Mathématiques. La Tour d'Hanoï. Liberté (9 Dec 1888) no page number on clipping. Gives the story of N. Claus and says it was invented by Lucas and comes in a box decorated with annamite illustrations. This would seem to be the original box.

Lucas. Nouveaux jeux scientifiques de M. Édouard Lucas. La Nature 17 (1889) 301 303. Describes a series of games under the title of the next item, so the next may refer to the game or its instruction booklet. A later version of the Tower of Hanoi is described on pp. 302 303, having 5 pegs, and is shown on p. 301. He says you can have 3, 4 or 5 pegs.

Lucas. Jeux scientifiques pour servir à l'histoire, à l'enseignement et à la pratique du calcul et du dessin. Première série: No. 3: La Tour d'Hanoï. Brochure, Paris, 1889. ??NYS. (Not listed in BNC, but listed in Harkin, op. cit. in Section 1. I wonder if these were booklets that accompanied the actual games?? Hinz says he has found some of these, but not the one on the Tower of Hanoi. The booklet for La Pipopipette (= Dots and Boxes) is reproduced in his L'Arithmétique Amusante of 1895 -- see 4.B.3.)

Jeux Scientifiques de Ed. Lucas. Advertisement by Chambon & Baye (14 rue Etienne-Marcel, Paris) for the 1^re Serie of six games. Cosmos. Revue des Sciences et Leurs Applications 39 (NS No. 254) (7 Dec 1889) no page number on my photocopy.

B. Bailly [name not given, but supplied by Hinz]. Article on Lucas's puzzles. Cosmos. Revue des Sciences et Leurs Applications. NS, 39 (No. 259) (11 Jan 1890) 156-159. Shows 'La nouvelle Tour d'Hanoï', which has five pegs and 15? discs. I need pp. 156 157.

Alfred Gartner & George Talcott. UK Patent 20,672 -- Improvements in Games and Puzzles. Applied: 18 Dec 1890; patented: 21 Feb 1891. 2pp + 1p diagrams. Shunting puzzle equivalent to Tower of Hanoi with 7 discs.

Ball. MRE, 1st ed., 1892, pp. 78-79. "... described by M. Tissandier as being common in France but which I have never seen on sale in England." Gives the Benares story from De Parville in La Nature.

Hoffmann. 1893. Chap. X, no. 4: The Brahmin's Rings, pp. 333-334 & 361-364 = Hoffmann/Hordern, pp. 220-222, with photo. Gives Benares story and then gives the problem with 8 discs, noting that it is made by Messrs. Perry & Co. Photo on p. 221 shows an example, with box with instructions on the top, by Perry & Co., 1880-1900. Hordern Collection, p. 90, shows a version with box, by R. Journet, 1905-1920.

Lucas. L'Arithmétique Amusante. 1895. La tour d'Hanoï, pp. 179-183. Description, including Benares story. Says the nephew of the inventor, Raoul Olive, has noted that the smallest disc always moves in the same direction. Says the whole idea of the mandarin and his story was invented a dozen years ago at 56 rue Monge, which was built on the site where Pascal died. [I visited this site recently -- it is a hotel and they knew nothing about Pascal. Another source says Pascal died at 67 Rue Cardinal Lemoine, which is several blocks away, but also is not an old building.]

Ball. MRE, 3rd ed., 1896, pp. 99-101. Omits the Tissandier reference and says: "It was brought out in 1883 by M. Claus (Lucas)."

A. Héraud. Jeux et Récréations Scientifiques -- Chimie, Histoire Naturelle, Mathématiques. Not in the 1884 ed. Baillière et Fils, Paris, 1903. Pp. 300 301 shows the original cover.

Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909. The five shields, pp. 20-24 & 59. Five discs.

Tom Tit??. In Knott, 1918, but I can't find it in Tom Tit. No. 165: The tower of Hanoi, pp. 382 383. Describes it with cards A   10 on piles labelled with J, Q, K.

Robert Ripley. Believe It Or Not! Book 2. (Simon & Schuster, 1931); Pocket Books, NY, 1948, pp. 52 53. = Believe It or Not! Two volumes in one; (Simon & Schuster, 1934); Garden City Books, 1946, pp. 222-223. = Omnibus Believe It Or Not!; Stanley Paul, London, nd [c1935?], pp. 256 257. The Brahma Pyramid. Outlines the Benares story, says he didn't locate the temple when he was in Benares, but it 'really exists', and that it will take 2⁶⁴ moves, but he then writes out 2⁶⁴   1. He says the Brahmins have been at it for 3000 years!!

B. D. Price. Pyramid patience. Eureka 8 (Feb 1944) 5-7. Straightforward development of basic properties.

Scorer, Grundy & Smith. 1944. Op. cit. in 7.M.1. They develop the graph of all positions of the Tower of Hanoi.

Donald W. Crowe. The n-dimensional cube and the tower of Hanoi. AMM 63 (1956) 29-30. Describes the connection with Hamiltonian circuits and binary ruler markings.

M. Gardner. The Icosian Game and the Tower of Hanoi. (SA (May 1957)) = 1st Book, chap. 6. Describes Crowe's work.

A. J. McIntosh. Binary and the Tower of Hanoi. MTg 59 (1972) 15. He sees the connection between binary and which disc is to be moved, but he wonders how to know which pegs are involved. [This is a valid query -- though we know each disc moves cyclically, alternate ones in alternate directions, I don't know any easy way to translate a particular step number into the positions of all the discs -- Hinz (1989) gives a method which may be as simple as possible, but I have a feeling it ought to be easier. Actually, I have now seen a fairly easy way to do this.]

Andy Liu & Steve Newman, proposers and solvers. Problem 1169 (ii) -- The two towers. CM 12 (1986) 179 & 13 (1987) 328 332. Three pegs, two identical piles of size n on two of them. The object is to interchange the bottom discs and reform the piles (though the smaller discs may or may not be interchanged). They find it takes 7*2ⁿ⁺¹   3n   (10 or 11)/3 steps, depending on whether n is odd or even.

Andreas M. Hinz. The Tower of Hanoi. L'Enseignement Math. 35 (1989) 289-321. Surveys history and current work. 50 references. Finds many properties, particularly the average distance from having all discs on a given peg and the average distance between legal positions. He also studies illegal positions. Uses the Grundy, Scorer & Smith graph. Gives general results, such as Schwenk's below. He asserts that the minimality of the classic solution was not proven until 1981, but I think the classic method clearly implies the proof of its minimality.

Hugh Noland, proposer; Norman F. Lindquist, David G. Poole & Allen J. Schwenk, solvers. Prob. 1350 -- Variation on the Tower of Hanoi. MM 63:3 (1990) 189 & 64:3 (1991) 199-203. Three pegs, 2n discs, initially with the evens on one peg and the odds on another. How many moves to get all onto the empty peg? Answer is (5/7) 4ⁿ. Schwenk gives a solution for any starting position of N discs and shows the average number of moves to get to a single pile is (2/3)(2^N-1).

Andreas M. Hinz. Pascal's triangle and the Tower of Hanoi. AMM 99 (1992) 538-544. Shows the Grundy, Scorer & Smith graph is equivalent to the pattern of odd binomial coefficients in the first 2ⁿ rows and hence to Sierpiński's fractal triangle. Gives some life of Lucas.

David Poole. The bottleneck Towers of Hanoi problem. JRM 24:3 (1992) 203-207. Studies the problem when big discs can go on smaller discs, but not too much smaller ones. 11 references to recent work on variations of the classical problem.

Ian Stewart. Four encounters with Sierpiński's gasket. Math. Intell. 17:1 (1995) 52-64. This discusses the connections between the graph of the Tower of Hanoi, the pattern of odd binomial coefficients, Sierpiński's gasket and Barnsley's iterated fractal systems. Lots of references, including 11 on the Tower of Hanoi.

Vladimir Dubrovsky. Nesting Puzzles -- Part I: Moving oriental towers. Quantum 6:3 (Jan/Feb 1996) 53-59 & 49-51. Outlines the history and theory of the Tower of Hanoi. Misha Fyodorov, a Russian high school student, observed that the peg not used in a move always moves in the same direction. Discusses Kotani's modification which prevents placing some discs on a particular peg. Also discusses the Panex Puzzle -- cf Section 5.A.4.

Jagannath V. Badami. Musings on Arithmetical Numbers Plus Delightful Magic Squares. Published by the author, Bangalore, India, nd [Preface dated 9 Sep 1999]. Section 4.15: The Tower of Brahma, pp. 123-124. "The author has lived in Banaras for a number of years and does not find any basis for this legend."

David Singmaster. The history of some combinatorial recreational problems. Draft of a chapter for History of Combinatorics, ed. Robin J. Wilson. Jan 2001. This gives a detailed development of the distances d_i of a position from the three perfect positions and the complementary distances d'_i = (2ⁿ-1) - d_i, leading to THEOREM 4. The Scorer, Grundy & Smith graph of positions in the Tower of Hanoi with n discs and with adjacency between positions one move apart, is isomorphic to the graph of triples of binary n tuples (d'₀, d'₁, d'₂) satisfying (*') Σ_i D_k(d'_i) = 1 for each k, considered as triangular coordinates in a triangle of edge length 2^n-1 and with adjacency being adjacency in the lattice.
7.M.2.a. TOWER OF HANOI WITH MORE PEGS
Lucas. Nouveaux jeux scientifiques ..., op. cit. in 4.B.3, 1889. (See discussion in 7.M.2.)

Dudeney. The Reve's Puzzle. The Canterbury Puzzles. London Mag. 8 (No. 46) (May 1902) 367 368 & 8 (No. 47) (Jun 1902) 480. = CP, prob. 1, pp. 24 25 & 163 164. 4 pegs, 8, 10 or 21 discs.

Dudeney. Problem 447. Weekly Dispatch (25 May, 15 Jun, 1902) both p. 13. 4 pegs, 36  discs.

Dudeney. Problem 494. Weekly Dispatch (15 Mar, 29 Mar, 5 Apr, 1903) all p. 13. 5 pegs, 35 discs.

B. M. Stewart, proposer; J. S. Frame & B. M. Stewart, solvers. Problem 3918. AMM 48 (1941) 216 219. k pegs, n discs. General solution, but editorial note implies there is a gap in each solver's work.

Scorer, Grundy & Smith. Op. cit. in 7.M.1. 1944. They give some variations on the Tower of Hanoi with four pegs.

Doubleday - 2. 1971. Keep count, pp. 91-92. 15 discs, 6 pegs -- solved in 49 moves.

Ted Roth. The tower of Brahma revisited. JRM 7 (1974) 116 119. Considers 4 pegs.

Brother Alfred Brousseau. Tower of Hanoi with more pegs. JRM 8 (1975/76) 169 176. Extension of Roth, with results for 4 and 5 pegs.

The Diagram Group. Baffle Puzzles -- 3: Practical Puzzles. Sphere, 1983. No. 10. 6 pegs, 15 discs. Gives a solution in 49 moves.

Joe Celko. Puzzle Column: Mutants of Hanoi. Abacus 1:3 (1984) 54 57. Discusses variants: where a disc can only move to an adjacent peg in a linear arrangement; with two or three colours of discs; with several piles of discs; where a disc can only move forward in a circular arrangement.

Grame Williams. In: Joe Celko; Puzzle column replies; Abacus 5:2 (1988) 70 72. Table of minimum numbers of moves for k pegs, k = 3, ..., 8 and n discs, n = 1, ..., 10.

Andreas Hinz. An iterative algorithm for the Tower of Hanoi with four pegs. Computing 42 (1989) 133-140. Studies the problem carefully. 17 references.

A. D. Forbes. Problem 163.2 -- The Tower of Saigon. M500 163 (Aug 1998) 18-19. This is the Tower of Hanoi with four pegs. Quotes an Internet posting by Bill Taylor giving an algorithm and its number of moves up to 12 discs. Asks if this is optimal.

7.M.3. GRAY CODE
See Gardner under 7.M.
L. A. Gros. Op. cit. in 7.M.1, 1872. ??NYS. (Afriat.)

J. Émile Baudot. c1878. ??NYS. Used Gray code in his printing telegraph. (Described by F. G. Heath; Origins of the binary code; SA (Aug 1972) 76 83.)

Anon. Télégraphe multiple imprimeur de M. Baudot. Annales Télégraphiques (3) 6 (1879) 354 389. Says the device was presented at the 1878 Exposition and has been in use on the Paris Bordeaux line for several months. See pp. 361 362 for diagrams and p. 383 for discussion.

George R. Stibitz. US Patent 2,307,868 -- Binary Counter. Applied: 26 Nov 1941; granted: 12 Jan 1943. 3pp + 1p diagrams. Has an electromechanical binary counter using the Gray code with no comment or claims on it.

Frank Gray. US Patent 2,632,058 -- Pulse Code Communication. Applied: 13 Nov 1947, patented: 17 Mar 1953. 9pp + 4pp diagrams. Systematic development of the idea and its uses.

A. J. Cole. Cyclic progressive number systems. MG 50 (No. 372) (May 1966) 122 131. These systems are Gray codes to arbitrary bases -- e.g. in base 4, the sequence begins: 0, 1, 2, 3, 13, 12, 11, 10, 20, 21, 22, 23, 33, 32, 31, 30, 130, 131, .... For odd bases, the sequence is harder. He gives conversion rules and rules for arithmetic.

J. Lagasse. Logique Combinatoire et Séquentielle. Maîtrise d'E.E.A. C3 -- Automatique. Dunod, Paris, 1969. Pp. 14-18 discuss the Gray code (code réfléchi) stating that the Gray value at step k, G(k), is given by G(k) = B(k) EOR B(k/2). [I noted this a few years ago and am surprised that it does not appear to be old. Gardner's 1972 article describes it but not so simply.] My thanks to Jean Brette for this reference.

William Keister. US Patent 3,637,216 -- Pattern-Matching Puzzle. Filed: 11 Dec 1970; patented: 25 Jan 1972. Abstract + 4pp + 2pp diagrams. This has a bar to remove from a frame -- one has to move various bits in the pattern of the Gray code (or similar codes) to extract the bar. Made by Binary Arts since about 1986.

7.M.4. BINARY DIVINATION
The classic cards for this process have the numbers 1 - 2ⁿ on them, the i-th card containing those numbers whose binary expression has a 1 in its (i-1)-st place -- e.g. the first card contains all the odd numbers. Then one adds up the smallest numbers, i.e. 2^i-1 on the i th card, on the chosen cards to get the number thought of. If one replaces the numbers by holes in the corresponding positions, one can overlay the cards to read off the answer. This takes a little more work though -- one has to have each card containing its set of holes to be used if it is chosen and also containing the complementary holes to be used if it is not chosen. This can be achieved if the holes are centrally located -- then turning the card around produces the complementary set of holes.

A very similar principle is used as a kind of logical device. See: Martin Gardner; Logic Machines and Diagrams; McGraw Hill, NY, 1958, pp. 117-124; slightly extended in the 2nd ed., Univ. of Chicago Press, 1982, and Harvester Press, Brighton, 1983, pp. 117-124; for discussion of this idea and references to other articles.

A simple form of binary division is used to divine a card among sixteen cards arranged in two columns, but it is surprisingly poorly described. This is related to the 21 card trick which is listed in 7.M.4.b. I have only recently added this topic and may not have noticed many versions.
Pacioli. De Viribus. c1500. Ff. 114r - 116r. C(apitolo). LXIX. a trovare una moneta fra 16 pensata (To find a coin thought of among 16). = Peirani 161-162. Divides 16 coins in half 4 times, corresponding to the value of the binary digits. Pacioli doesn't describe the second stage clearly, but Agostini makes it clear.

Bachet. Problemes. 1612. Prob. XVI, 1612: 87-92. Prob. 18; 1624: 143-151; 1884: 72-83. 15 card trick. His Avertissement mentions that other versions are possible and describes divining from sixteen cards in two columns and in four columns, but with no diagrams!

Yoshida (Shichibei) Kōyū (= Mitsuyoshi Yoshida) (1598-1672). Jinkō ki. 2nd ed., 1634 or 1641??. Op. cit. in 5.D.1. ??NYS Shimodaira (see entry in 5.D.1) discusses this on pp. 2-12 since there are several Japanese versions of the idea. The Japanese call these Metsukeji (Magic Cards). The binary version is discussed on pp. 4-7 where it is said that they are known since the 14C or even earlier. The Japanese magic card shown on p. 6 has 1, 2, 4, 8, 16 associated with branchings on a picture of a tree used to divine one of the 21 characters written on the flowers and leaves. The other kinds of magic cards are more complex, not involving binary, but just memorisation. The recent transcription of part of Yoshida into modern Japanese does not include this problem.

Ozanam. 1725. Prob. 39, 1725: 231-233. Prob. 15, 1778: 164-165; 1803: 165-166; 1814: ??NYS. Prob. 14, 1840: 74-75. Sixteen counters being disposed in two rows, to find that which a person has thought of. Similar to Bachet, but with some diagrams.

Ozanam-Hutton. 1814: 124-126; 1840: 64. (This is an addition which was not in the 1803 ed.) Six cards to divine up through 63.

Endless Amusement II. 1826? Pp. 180-181. Sixteen Cards being disposed in Two Rows, to tell the Card which a Person has thought of. c= Ozanam, with 'counter' replaced by 'card'.

Young Man's Book. 1839. Pp. 202-203. Identical to Endless Amusement II.

Crambrook. 1843. P. 7, no. 5: A pack of cards by which you may ascertain any person's age. Not illustrated, but seems likely to be binary divination -- ??

Magician's Own Book. 1857. The mathematical fortune teller, pp. 241 242. Six cards each having 30 numbers used to divine a number up through 60. Some cards have duplicate numbers in the 30th position. = Boy's Own Conjuring Book, 1860, pp. 211-212. = Illustrated Boy's Own Treasury, 1860, prob. 38, pp. 402 & 442.

Book of 500 Puzzles. 1859. The mathematical fortune teller, pp. 55-56. Identical to Magician's Own Book. However, my example of the book has 61 in the last cell of the last card, but a photocopy sent by Sol Bobroff has this as a 41, as do all other versions of this problem.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 563-VI, pp. 249-250: Überraschungen mittels sieben Zauberkarten. Seven cards used to divine up to 100.

Adams & Co., Boston. Advertisement in The Holiday Journal of Parlor Plays and Pastimes, Fall 1868. Details?? -- photocopy sent by Slocum. P. 5: Magic Divination Cards. For telling any number thought of, or a person's age. Amusing, curious, and sometimes "provoking." Not illustrated, but seems likely to be binary divination -- ??

Magician's Own Book (UK version). 1871. The numerical fortune teller, pp. 89-90. Very similar to Magician's Own Book, pp. 241-242, with the same cards, but different text.

Bachet-Labosne. Problemes. 3rd ed., 1874. Supp. prob. X, 1884: 196-197. Divine a number up to 100 with 7 cards.

F. J. P. Riecke. Op. cit. in 4.A.1, vol. 3, 1873. Art. 3: Die Zauberkarten, pp. 11 13. Describes 5 cards giving values up to 31, with explanation. Describes how to use balanced ternary to construct 7 cards giving values up to 22. The limitation to 22 is due to the size of the cards -- the method works up to 40.

Mittenzwey. 1880. Prob. 37, pp. 6-8; 1895?: 43, pp. 12-13; 1917: 43, pp. 11 13. Seven cards. Calls them "Boscos Zauberkarten" -- Bosco (1793- ) was a noted conjurer of the early 19C.

Hoffmann. 1893. Chap. IV, no. 68: The magic cards, pp. 160 161 & 216 217 = Hoffmann Hordern, pp. 141-142, with photo. Seven cards. Photo on p. 141 shows an ivory set of six cards, 1850-1900, and two German sets of seven cards from a box of puzzles called Hokus Pokus: Zauber Karten and Ich weiss wie alt du bist, both 1870 1890. These have their own boxes or wrappers. Hordern Collection, p. 73, shows the same(?) Zauber-Karten, with its box or wrapper, dated 1860 1890, and fully spread out so the instruction card is legible.

Lucas. L'Arithmétique Amusante. 1895. L'éventail mystérieux, pp. 168 170. Shows five cards for divining 1 through 31 and notes it is based on binary.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:2 (Jun 1903) 140-141. To tell a lady's age. Six cards to divine up through 63.

Ahrens. MUS I. 1910. Pp. 39-40 does 5 cards. Pp. 43-48 develops a set of four see-through templates.

M. Adams. Indoor Games. 1912. How to divine ages, pp. 349-350.

The Quaker Oats Age Prediction Cards. (I have a facsimile of a set, made by Dave Rosetti for G4G5. He says it was invented in 1921.) 8 cards with various holes and numbers for divining from 1 to 99. You turn the card round if the number is not on it but put it on the pile. At the end, you turn the pile over and the age and the slogan 'Never Known to Fail' can be read through the holes.

Card 1 is just a viewing window with four windows that are five digits wide. Cards 6, 7, 8 have numbers ingeniously arranged so that groups of five digits are in the window positions. Reading two digits at a time, these groups give four possible values, hence 16 for each configuration of cards 6, 7, 8 -- except that when both 6 and 7 are up, then card 8 is not visible, and when cards 6 and 7 are turned, there are only four different values -- this results in just the 100 values: 00, 01, ..., 99. Cards 2 and 3 select which window is open. Cards 4 and 5 select which pair of digits in the window can be seen.

William P. Keasby. The Big Trick and Puzzle Book. Whitman Publishing, Racine, Wisconsin, 1929. Magic columns, pp. 176-177. Applies the idea with letters, so you add the column heads to get the number of the letter.

Rohrbough. Brain Resters and Testers. c1935. How to Mystify People, pp. 10-11. = Keasby, whom he cites elsewhere.

John Fisher. John Fisher's Magic Book. Muller, London, 1968. Think a Drink, pp. 27-29. 5  cards with cut out holes to divine 32 types of drink. The last card seems to need one side reversed.

7.M.4.a. TERNARY DIVINATION
This is much less common than binary divination and I have just added it. Gardner, op. cit. at beginning of 7.M.4, describes one set of triangular cards.
Riecke. 1873. See in 7.M.4.

Martin Hansen. Mind probe. MiS 21:1 (Jan 1992) cover & 2-6. Adapts binary divination to locate a number among 1 - 80 with four cards -- but each card has its numbers half in black and half in white. Describes how to make 4 x 4 and 8 x 8 binary cards with holes so the chosen number will appear in the hole. Adapts to ternary to produce triangular cards with holes so that the chosen number appears in the hole. Hansen has kindly given me a set of these: 'The Kingswood Mathemagic Club's Window Cards'. He also describes 'logic cards' which display the truth values of three basic quantities which are consistent with various statements -- these were previously described by Gardner and Cundy & Rollett -- see the Gardner item at the beginning of 7.M.4 for details and references.

7.M.4.b. OTHER DIVINATIONS USING BINARY OR TERNARY
New section. Discussion with Bill Kalush has revealed that the classic 21-card trick is based on the ternary system. Another trick involving twice taking 3/2 of a number, which goes back to at least Pseudo-Bede, makes a little use of binary.

The 21-card trick can also be done with 15 or 27 cards and it is easiest to explain for 27 cards. One deals 27 cards out, face up, into three columns and asks the spectator to mentally choose one card and tell you which column it is in. You pick up the three columns, carefully placing the chosen column in the middle of the other two, then deal out the deck into three columns again. Ask the spectator to tell you what column the card is now in and pick up the cards again with the chosen column in the middle and deal them out again. Repeat the whole process a third time, then deal out the cards face down and ask the spectator to turn over the middle card, which will be his chosen card. The first process puts the chosen card among the positions whose initial ternary digit is a 1. The second process puts it among the position whose ternary representation begins 11. The third process puts it at the position whose ternary representation is 111. For any number of cards which is a multiple of three, each process puts the chosen card in the 'middle third' of the previously determined portion of the deck and the trick works, though the patterns are less systematic than with 27 cards. When the trick is done with, e.g. 24 cards, there is no middle card and you have to expose the chosen card yourself or develop some other way to do it. One can also adapt the idea to other numbers of columns -- the two column case will usually be connected to binary and may be listed in 7.M.

In the 3/2 method, you ask a person to think of a number, x, then take 3/2 of it (or itself plus half of itself). If there is a half present, he is to round it up and let you know. Do this again on the result. Now ask how many nines are contained in his result. You then tell the number thought of. The process actually converts x = a + 2b + 4c to 3a + 5b + 9c, where a, b  =  0 or  1. We have that a = 1 iff there is a rounding at the first stage and b = 1 iff there is a rounding at the second stage and the person then tells you c. We will denote this as PB2.

Sometimes the second stage is omitted or the division by two in it is omitted, which makes things simpler. The latter case takes a + 2b to 6a + 9b and the person gives you b and there is a remainder if and only if a = 1. We will denote this by PB1.

I'm including some early examples of simple algebraic divination here.
Pseudo-Bede. De Arithmeticis propositionibus. c8C, though the earliest MS is c9C.

IN: Venerabilis Bedae, Anglo Saxonis Presbyteri. Opera Omnia: Pars Prima, Sectio II -- Dubia et Spuria: De Arithmeticis propositionibus. Tomus 1, Joannes Herwagen (Hervagius), Basel, 1563, Columns 133-135, ??NYS. Folkerts says Hervagius introduced the title De Arithmeticis propositionibus.

Revised and republished by J. P. Migne as: Patrologiae Cursus Completus: Patrologiae Latinae, Tomus 90, Paris, 1904, columns 665 668.

Critical edition by Menso Folkerts. Pseudo-Beda: De arithmeticis propositionibus. Eine mathematische Schrift aus der Karolingerzeit. Sudhoffs Archiv 56 (1972) 22-43. A friend of Bill Kalush has made an English translation of the German text, 1998?, 11pp.

See also: Charles W. Jones; Bedae Pseudepigrapha: Scientific writings falsely attributed to Bede. Cornell Univ. Press & Humphrey Milford, OUP, 1939, esp. pp. 50 53.

This has three divination problems and Folkerts says these are the first known western examples.

1. Triple and halve, then triple and tell quotient when divided by nine and whether there is a remainder in this. I.e. PB1.

2. 3/2 twice and tell if there is a rounding up at each stage as well as the quotient when divided by 9. I.e. PB2.

3. Divine a digit a from a * 2 + 5 * 5 * 10, where the operations are performed sequentially from left to right. The result is 100a + 250. This is not of the type considered in this section, but is the prototype of most later divination methods.

Folkerts mentions several later occurrences of these methods.

(The fourth and last part of the text is probably slightly later in the 9C and describes adding positive and negative numbers in a way not repeated in the west until the 15C.)

Fibonacci. 1202. Pp. 303-304 (S: 427-428). Take 3/2 twice, i.e. PB2.

Folkerts. Aufgabensammlungen. 13-15C.