QUERIES on SOURCES IN RECREATIONAL MATHEMATICS
by DAVID SINGMASTER, 1996
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?
SOURCES IN RECREATIONAL MATHEMATICS
AN ANNOTATED BIBLIOGRAPHY
by DAVID SINGMASTER, 2000
7.M.1. CHINESE RINGS
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. II, 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. 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).
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, Fawdrey 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. II, p. 191, for an early possible reference — see above.
Gardner. Knotted, chap. 2, says there are 17C Japanese haiku about it and it is used in Japanese heraldic emblems.
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 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.
J. 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).
Minguét. Engaños. 1733. Pp. 55-57 (1755: 27-28; 1822: 72-74): 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 chest 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. Cites Encyclopédie Méthodique des Jeux, p. 424+.
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.
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: G(k) = B(k) EOR B(ëk/2û). 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.] 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.
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.