9. LOGICAL RECREATIONS
Many combinatorial recreations can be considered as logical.
9.A. ALL CRETANS ARE LIARS, ETC.
Diogenes Laërtius. 3C. De Clarorum Philosophorum Vitis, Dogmatibus et Apophtegmatibus, II, Life of Euclides. Ed. by C. G. Cobet; Paris, 1888, p. 108, ??NYS. Translated by C. D. Yonge; Bell, London, 1894, pp. 97‑98. Translated by R. D. Hicks; Loeb Classical Library; vol. 1, pp. 236‑237. Refers to Eubulides (c‑330) as the source of "The Lying One" or "The Liar" _ Ο Ψεθδoμεvoσ (O Pseudomenos). According to: I. M. Bochenski; Ancient Formal Logic; North Holland, Amsterdam, 1951, p. 100; Eubulides also invented: "the swindler", "the concealed", "the heap" (how many grains make a heap?), "the Electra" and "the horned" (equivalent to "Have you stopped beating your wife?").
Bochenski, Ancient Formal Logic, pp. 101‑102, says the liar paradox was unknown to Plato, but is quoted by Aristotle in his Libro de Sophisticis Elenchis 25, 180 b 2‑7, ??NYS, (See: M. Wallies, ed.; Topica cum Libro De Sophisticis Elenchis; Leipzig, 1923; ??NYS; and: Ethica Nicomachea, H3, 1146 a 21‑27; ed. by Fr. Susemihl; Leipzig, 1887; ??NYS.) It is also given in Cicero; Ac. Pr. II, 95, 96 (=? Topica, 57); ??NYS (In: G. Friedrich, ed.; Opera Rhetorica; Leipzig, 1893; ??NYS) and in many later writers.
Athenaeus Naucratica. c200. The Deipnosophists, Book 9 (end of c.64). Translated by C. D. Yonge, Bohn, London, 1854, vol. 2, p. 633. Epitaph of Philetas of Cos (c‑340/c‑285). "Traveller, I am Philetas; the argument called the Liar and deep cogitations by night, brought me to death." Sadly, there is no indication where he died or was buried. (Bochenski; Ancient Formal Logic,; p. 102 gives the Greek of Athenaeus. I. M. Bochenski; History of Formal Logic; corrected ed., Chelsea, 1970, p. 131; gives the English.)
The Stoics. c‑280. Bochenski; Ancient Formal Logic; pp. 100‑102 says they invented several paradoxes, including "the crocodile" who takes a baby and says he will return it if the mother answers his question correctly. He then asks "Will I return the baby?" She answers "No".
Anon. History of the Warring States. [The Warring States period is ‑475/‑221 and this history may be ‑2C.] The Elixir of Death. Translated in: Herbert A. Giles; Gems of Chinese Literature; op. cit. in 6.BN, p. 43. Chief Warden swallows an elixir of immortality which he was supposed to convey to the Prince. The Prince orders the Warden's execution, but the Warden argues that if the execution succeeds, then the elixir was false and he is innocent of crime. The Prince pardons him.
Tung‑Fang So (‑2C, see Giles, ibid., p. 77) is said to have been in the same situation as the Warden and argued: "If the elixir was genuine, your Majesty can do me no harm; if it was not, what harm have I done?"
St. Paul. Epistle to Titus, I, 12. c50? "One of themselves, even a prophet of their own, said, The Cretans are always liars, .... This witness is true."
M. Cervantes. Don Quixote. 1605. Book II, chap. 51. Translated by Thomas Shelton, 1612‑1620, reprinted by the Navarre Society, London, 1923, vol. 2, pp. 360‑362. Sentinel paradox _ I will be hanged.
Henri Decremps (or Descramps) [?= Jerome Sharp, pseud.]. Les Petits Aventures de Jerome Sharp. Professor de Physique Amusante; Ouvrage contenant autant de tours ingénieux que de leçons utiles, avec quelques petits portraits à la maniére noire. Brussels, 1789; also 1790, 1793. Toole Scott records an English edition, Brussels, 1793. Sentinel paradox. ??NYS. Cited by Dudeney; Some much‑discussed puzzles; op. cit. in 2; 1908; as the first appearance of this paradox.
The Sociable. 1858. Prob. 43: The Grecian paradox, pp. 299 & 317. Protagoras suing his pupil who had promised to pay for his tuition when he won his first case. = Book of 500 Puzzles, 1859, prob. 43, pp. 17 & 35. c= Magician's Own Book (UK version), 1871, pp. 26-27.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 564-21, pp. 253 & 395. Form of the sentinel paradox. To enter a garden, one must make a statement; if true, one pays 3 marks; if false, one pays 6 marks. "I will pay 6 marks."
Cesare Burali Forti. Una questione sui numeri transfiniti. Rendiconti del Circolo Matematico di Palermo 9 (1897) 154-164. ??NYS. This was the first published antinomy of modern set theory. The set of all ordinal numbers is itself an ordinal! However, Cantor had observed the paradox in 1895 and communicated it to Hilbert in a letter in 1896.
Irving Anellis. The first Russell paradox. Paper given at AMS meeting, Chicago, Mar 1985. ??NYS _ abstract given in HM 12 (1985) 380. Says it is usually believed that Russell discovered his paradox in Jun 1901, but he sent a version of it to Couturat on 8 Dec 1900 (unpublished MS in the Russell Archives, McMaster Univ.).
Gregory H. Moore. A house divided against itself: The emergence of first-order logic as the basis for mathematics. IN: Esther R. Phillips, ed.; Studies in the History of Mathematics; MAA, 1987, pp. 98-136. On pp. 114-115, he dates Russell's paradox to May 1901 and says Russell wrote about it to Frege on 16 Jun 1902. The first publications are in Russell's Principles of Mathematics and Frege's Fundamental Laws, vol. 2, both in 1903.
B. Russell. The Principles of Mathematics. CUP, 1903. ??NYS _ cited in Garciadiego. He discusses Russell's paradox and also Cantor's paradox of the greatest cardinal and Burali Forti's paradox of the greatest ordinal. I won't consider these much further, but this may have inspired the development of the more verbal paradoxes described in this section.
G. G. Berry. Letter to Russell on 21 Dec 1904. In the Russell Archives, McMaster University. Quoted in Garciadiego. "... the least ordinal which is not definable in a finite number of words. But this is absurd, for I have just defined it in thirteen words." The paradox of Jules Richard (late Jun 1905) is very similar and similar versions were found by J. König and A. C. Dixon about the same time, though these all use Zermelo's well-ordering axiom. Sometime earlier, Berry had introduced himself to Russell with a note saying "The statement on the other side of this paper is true" with the other side reading "The statement on the other side of this paper is false", and consequently is also considered the inventor of the "visiting card paradox".
B. Russell. Les paradoxes de la logique. Revue de Métaphysique et de morale 14 (1906) 627‑650. ??NYS _ cited by Garciadiego. First publication of a modified version of Berry's paradox.
B. Russell. Mathematical logic as based on the theory of types. Amer. J. Math. 30 (1908) 222‑262. On p. 223, he first gives Berry's paradox: "the least integer not nameable in fewer than nineteen syllables". He also reformulates König & Dixon as "the least indefinable ordinal".
Kurt Grelling & Leonard Nelson. Bemerkungen zu den Paradoxien von Russell und Burali‑Forti. Abhandlungen der Fries'schen Schule (NS) 2 (1908) 301‑344. ??NYS. Grelling's paradox: "Is heterological heterological?"
A. N. Whitehead & B. Russell. Principia Mathematica. CUP, 1910. Vol. 1, pp. 63‑64. Discusses several paradoxes and repeats Berry's paradox.
F. & V. Meynell. The Week‑End Book. Op. cit. in 7.E. 1924. 2nd ed., prob. five, p. 275; 5th? ed., prob. nine, p. 408, gives Russell's paradox as a problem _ and gives no solution!
John van Heijenoort. Logical paradoxes. Encyclopedia of Philosophy 5 (1967) 45‑51. Excellent survey of the paradoxes of logic and set theory, but only a mention of pre‑19C paradoxes.
Alejandro R. Garciadiego. The emergence of some of the nonlogical paradoxes of the theory of sets, 1903-1908. HM 12 (1985) 337-351. Good survey. He has since extended this to a book: Bertrand Russell and the Origins of the Set-theoretic "Paradoxes"; Birkhäuser, 1992, ??NYS
9.B. SMITH _ JONES _ ROBINSON PROBLEM
See also 5.K.2 for a special form of these problems.
Dudeney. PCP. 1932. Prob. 49: "The Engine‑Driver's Name", pp. 24 & 132. = 536; prob. 521: "The Engineer's Name", pp. 214 & 411. The driver is Smith, but the other two names are not determined.
Phillips. Week‑End. 1932. Time tests of intelligence, no. 39, pp. 22 & 39. Same as Dudeney.
Phillips. Brush. 1936. Prob. K.2: The Engine‑driver, pp. 36 & 96. Same as Dudeney.
Rudin. 1936. No. 183, pp. 65 & 119. Similar to Dudeney, but Americanized, somewhat simplified(?) and asking for the brakeman's (= guard's) name.
James Joyce. Finnegans Wake. Viking Press, NY, 1939. P. 302, lines 23-24: "Smith-Jones-Orbison".
Irving Adler. Thinking Machines. Dobson, London, 1961. Pp. 111-116: Who is the engineer? Essentially identical to Rudin, but asks for the engineer's name. Gives a systematic solution via boolean algebra.
Doubleday - II. 1971. Flight plan, pp. 153-154. Same as Dudeney, slightly reordered.
Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. Who's who?, pp. 86 & 134-135. Spaceship version, similar to Dudeney, but more precise.
9.C. FORTY UNFAITHFUL WIVES
I now realise that this is an extension of 9.D.
See Littlewood, 1953, in 9.D.
Gamow & Stern. 1958. Forty unfaithful wives. Pp. 20‑23. (Communicated by V. Ambarzuminian.)
Michael Spivak. Calculus. 2nd ed., Publish or Perish. ??date, place. P. 35, probs. 27 & 28.
Uri Leron & Mike Eisenberg. On a knowledge-related paradox and its resolution. Int. J. Math. Educ. Sci. Technol. 18 (1987) 761-765.
Ed Barbeau. Fallacies, flaws and flimflam. CMJ 22:4 (Sep 1991) 307. Gives a brief discussion and the reference to Spivak and to Leron & Eisenberg. The paradox has to do with what information has been provided by the stranger.
9.D. SPOTS ON FOREHEADS
See also 9.C. 7.AP is somewhat related.
William Wells Newell. Games and Songs of American Children. Harper and Brothers, (1883); 2nd ed. 1903; reprinted with Editor's Note of 1883 and new Introduction and Index, Dover, 1963. Chap. IX, No. 77: Laughter games, pp. 136‑137. "In a Swiss game, .... Each child pinches his neighbor's ear; but by agreement, the players blacken their fingers, keeping two of the party in ignorance. Each of the two victims imagines it to be the other who is the object of the uproarious mirth of the company." The Notes on p. 274 indicate that this probably comes from: E. L. Rochholz; Alemannisches Kinderleid und Kinderspiel, Leipzig, 1857, ??NYS
Phillips. Week‑End. 1932. Time tests of intelligence, no. 13, pp. 14 & 188. Two boys fall down, one gets a dirty face, the other washes his own face.
Phillips. The Playtime Omnibus. Op. cit. in 6.AF. 1933. Section XVII, prob. 11: Odd, pp. 55 & 237. Identical to Week‑End.
W. E. Buker, proposer; Robert Wood and O. B. Rose, solvers. Science Question 686. SSM 35 (1935) 212 & 429. 3 persons.
A. A. Bennett, proposer; E. P. Starke and G. M. Clemence, solvers. Problem 3734. AMM 42 (1935) 256 & 44 (1937) 333‑334. n persons with smudges on foreheads. Says the 3 person case was suggested by Dr. Church of Princeton and cites the Buker problem.
Phillips. Brush. 1936.
Prob. A.2: The green crosses, pp. 1‑2 & 73. Three men with green crosses.
Prob. R.1: The roof, pp. 58 & 112. Same as in Week‑End, above.
Rudin. 1936. No. 145, pp. 51-52 & 110. 3 persons with black crosses on foreheads.
McKay. At Home Tonight. 1940. Prob. 36: The five disks, pp. 70 & 83. Three mandarins and five disks, two black and three white. Emperor puts white on each forehead. Answer argues using the number of blacks.
M. Kraitchik. Mathematical Recreations. Op. cit. in 4.A.2. 1943. Chap. 1.
Prob. 3: The problem of the three philosophers, p. 15. Three painted faces.
Prob. 4, pp. 15‑16. 3 white and 2 black discs _ three whites placed on backs. All realize simultaneously.
(Neither is in Math. des Jeux.)
Harold Hart. The World's Best Puzzles. Op. cit. in 7.AS. 1943. The problem of the marked foreheads, pp. 23 & 55. Three students with blue and green crosses.
Jules Leopold. At Ease! Op. cit. in 4.A.2. 1943. Short cut to chevrons, pp. 23-24 & 199. Three men with smudged foreheads.
A. K. Austin. A calculus for know/don't know problems. MM 49:1 (Jan 1976) 12-14. He develops a set-theoretic calculus for systematically solving problems involving spots on foreheads, etc., including problems with knowing sums, see 7.AP. His typical problem has a man with four red and three blue stamps and he sticks three on the foreheads of two boys, telling them they each have at least one red. The first sasy he doesn't know what he has; then the second says he doesn't know; then the first says he does know. What stamps did the first have?
Leeming. 1946. Chap. 3, prob. 9: The three small boys, pp. 21‑22 & 153‑154. Three boys with smudged foreheads.
Henry Cattan. The Garden of Joys. (An anthology of oriental anecdotes, fables and proverbs.) Namara Publications, London, 1979. How he won the office of Grand Vizier, pp. 107‑109 & note 89 on p. 114. Same as Kraitchik's problem 4, but with the simpler solution based on symmetry. The note says: "This story is anonymous and was heard by the author in Palestine." A letter from the author says he heard the story in this form in Palestine before 1948.
Jerome S. Meyer. Fun-to-do. Op. cit. in 5.C. 1948. Prob. 51: Red hats and green hats, pp. 46 & 190-191. 2 red hats & 3 green hats. Answer gives the symmetry solution and the logical solution without clearly recognizing the distinction.
Max Black. Critical Thinking. 1952. Op. cit. in 6.F.2. Prob. 11, pp. 12 & 432. Three sons with white marks.
J. E. Littlewood. A Mathematician's Miscellany. Op. cit. in 5.C. 1953. Pp. 3‑4 (25‑26). Three dirty faces. Mentions that this can be extended to n dirty faces, which "has not got into the books so far as I know". This may be the origin of 9.C?
The Little Puzzle Book. Op. cit. in 5.D.5. 1955. Pp. 34-35: Clean and dirty. Two men fall through a roof. Man with clean face goes to wash.
T. J. Fletcher. The n prisoners. MG 40 (No. 332) (May 1956) 98‑102. Considers Kraitchik's problem with n persons, n white discs and n‑1 black discs. Also studies various colour distributions and assignments, including < n white discs and use of three colours.
Gamow & Stern. 1958. Three soot‑smeared faces. Pp. 77‑79.
Birtwistle. Math. Puzzles & Perplexities. 1971. Hats Off!, pp. 108 & 193. A, B, C are seated in a row, so B can see A while C can see both B and A. They all know that there are three white and two red hats in a bag. A hat is taken out and put on A's head, but he can't see it. Similarly, hat are taken and put onto B and C. C is now asked it he knows what colour his hat is and replies that he does not. B, having heard C's response, is asked if he knows what colour his hat is and he replies that he does not. Is A, having heard these, able to know what colour his hat is? Extends to various other combinations and to n people. The answer is "The answer to all the questions is, yes, it is possible."
9.E. STRANGE FAMILIES
Complicated questions of kinship have arisen due to religious taboos on incest. Most religions have a list of kinship relations which are not permitted to marry. These get a bit more involved than I want to go into. See the items by Ascher, Stern and Turner in the first section below for some typical material.
The second section deals with marrying a deceased wife's sister.
The third section deals with general strange families riddles and puzzles, but 'That man's father ...' are collected in 9.E.1.
Ripley's Believe It or Not! books give a number of examples of strange families. I will enter these under the date of the persons involved. BION-xx denotes the xx-th series of Believe It or Not!
Problems of this type are generally put in the form of a riddle, and many of these are collected in the following.
Mark Bryant. Dictionary of Riddles. Routledge, 1990. (Based on his Riddles Ancient and Modern; Hutchinson, 1983.)
GENERAL STUDIES OF KINSHIP RELATIONS
J. Cashdan & Martin D. Stern. Forbidden marriages from a woman's angle. MG 71 (No. 456) (1987). ??NYS _ cited by Stern, 1990.
Martin D. Stern. Consanguinity of witnesses _ a mathematical analysis. Teaching Mathematics and Its Applications 6:2 (1987). ??NYS _ cited by Stern, 1990.
Martin D. Stern. Mathematical motivation through matrimony. MM 63:4 (Oct 1990) 231‑233. ??NYS _ reproduced in Robert L. Weber; Science with a Smile; Institute of Physics, Bristol, 1992, pp. 314-318. Presents a notation for kinship relations and uses it to see that the table of prohibited degrees of marriage given in the Book of Common Prayer is symmetric with respect to sex and hence there are no unexpected prohibitions. However, the Jewish restrictions on marriage and on testimony by consanguinous relatives are not symmetric _ cf the above items.
Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10. 1991. Chapter Three: The logic of kinship relations, pp. 66-83. Gives a number of folk puzzles and then analyses several complicated kinship systems. Some references.
Martin Stern. Discrete avoidance of marital indiscretion. Mathematics Review (Univ. of Warwick) 2:3 (Feb 1992) 8-11. He presents a notation for kinship relations and uses it to describe the prohibited relations in Christian, Jewish and Islamic traditions. The Jewish prohibitions are not symmetric between male and female.
DECEASED WIFE'S SISTER, ETC.
E. S. Turner. Roads to Ruin _ The shocking history of social reform. Michael Joseph, London, 1950. Chap. 5: Two wives, one mother-in-law, pp. 98-121. This surveys the British preoccupation with the legality of marrying a deceased spouse's sibling. Since a couple were considered to become 'one flesh' (Ephesians 5:31), such a marriage was considered incestuous by the Church. Leviticus 18:6 & 16 were interpreted as prohibiting such marriage, but Leviticus 18:18 was interpreted as saying that the previous verses stated that a man should not have sisters as wives at the same time [which is the Islamic interpretation], while Deuteronomy 25:5-10 not only permits, but even commands, that a man should marry his brother's widow.
The English preoccupation with the problem dates from Henry VIII's marriage to his brother's widow, Catherine of Aragon. This particular question is mentioned in Shakespeare's Henry VIII and the general question is the basis of Hamlet, whose mother marries her dead husband's brother. There was at least one execution, in early 18C Scotland, of a woman who had sex with her sister's widower.
Up to 1835, marriage to a deceased wife's sister was permitted, but it could be voided and the children declared bastards, if an action was brought. But if an action was brought and dropped, further actions were prevented. In 1835, the Duke of Beaufort, who had married his deceased wife's half-sister, persuaded the Lord Chancellor to introduce a bill to legitimize such marriages up to date. The Bishops managed to amend this to prohibit such marriages in the future. However, such a couple could go to Europe to be married and such marriages remained legal in places like Jersey, though they were not legitimate in England. The Catholic Church generally gave dispensation for such marriages. From 1841 onwards, bills to remove the prohibition were introduced in almost every Parliament. Marriage to a deceased husband's brother or to a deceased spouse's nephew/niece was not sufficiently common to be considered by the reformers. The question was mentioned by Gilbert & Sullivan (near the end of the first act of Iolanthe (1882), the Queen, referring to Strephon, says "He shall prick that annual blister, / Marriage with deceased wife's sister;"). The journal Moonshine commented: "To be able to marry two wives at the cost of but one mother-in-law is something to fight for." In 1906, the Colonial Marriages Bill legitimized such marriages made in the colonies. In 1907, the Deceased Wife's Sister Bill was passed. Canon Law was later changed to accept this. One man who had married his deceased wife's sister sued a Canon who refused him Communion and won, with his win being confirmed by the Court of Appeals and the House of Lords in 1912. However, marriage to a divorced wife's sister was not permitted while the ex-wife lived. Marriage to a deceased husband's brother was permitted in 1921. A number of other marriages were permitted in 1921 and all these acts are consolidated in the Marriage act of 1949. Turner is not clear whether marrying a divorced spouse's sibling was permitted, and I don't know the further history.
A discussion of Strawberry Hill House says marriage to a deceased husband's brother was prohibited by the 1835 act, so that Frances, the widow of John Waldegrave, had to go to Scotland to marry his half-brother George Waldegrave, 7th Earl Waldegrave.
Susan Kelz Sperling. Tenderfeet and Ladyfingers. Viking, NY, 1981, p. 98-99. She gives some details of the Hebrew view. The Hebrew law of yibbum declares that if a man dies without heir, his brother or nearest relative is obliged to marry the widow (i.e. she is marrying her dead husband's brother). However, he could decline the duty by a ceremony called halitzah, as specified in Deuteronomy, by putting on a special shoe which the widow removed and then she spat in front of him to break the contract. This takes place in Ruth, alowing her to marry Boaz.
BION-11 cites an American example where a woman succesively married the widowers of two of her sisters.
William Holman Hunt, the Victorian painter, married his deceased wife's sister in 1866, in Switzerland [Judi Culbertson & Tom Randall; Permanent Londoners; Robson Books, London, 1991, p. 140].
See Dudeney, AM, prob. 52, below, for a complication of this situation.
GENERAL FAMILY RIDDLES
The riddles which the Queen of Sheba proposed to Solomon are not recorded in the biblical account of their meeting (I Kings 10 & II Chronicles 9), which would be c-960. Bryant, p. 19, says these are given are given in the 2nd Targum to the Book of Esther and elsewhere in the rabbinical literature. The Targums are commentaries on biblical books, created after the Babylonian Captivity of ‑587/-538 and written down from 100 onwards. One of these is a strange family riddle which occurs in the Exeter Book, qv below. If this is really due to the Queen of Sheba, or even actually in the Targums, it would be by far the earliest strange families riddle known. A variant of the riddle is given by Yachya Ben Sulieman, c1430, qv below. Ms Zimmels at the library of the London School of Jewish Studies tells me that there is an 1893 German translation: Targum Shennai(?) zum Buch Esther and that the riddles occur in Ginzburg's Legends of the Jews.
The Exeter Book Riddles. 8C?? (some sources say it is 10C; Bryant, p. 26, says last quarter of 10C). Translated and edited by Kevin Crossley-Holland. (As: The Exeter Riddle Book, Folio Society, 1978, Penguin, 1979.) Revised ed., Penguin, 1993.
No. 43, pp. 47 & 103. Body and soul both have the earth as their mother and sister. Their mother because they are made from dust; their sister because all are made by the same heavenly father.
No. 46, pp. 50 & 104.
A man sat sozzled with his two wives,
his two sons and his two daughters,
darling sisters, and with their two sons,
favoured firstborn; the father of that fine
pair was in there too; and so were
an uncle and a nephew. Five people
in all sat under that same roof.
The solution is given in Genesis 19:30-38, which describes Lot and his two daughters who bore sons by him. "The first use of this incestuous story for the purpose of a riddle is attributed to the Queen of Sheba; she tried it on Solomon." Cf Queen of Sheba, above, and Yachya Ben Sulieman, c1430, below.
Alcuin. 9C.
Prob. 11: Proposito de duobus hominibus singulas sorores accipientibus. Two men marry each other's sister.
Prob. 11a (in the Bede text): Propositio de duobus hominibus singulas matres accipientibus. Two men each marrying the other's mother. This is the classic "I'm my own grandfather" situation
Prob. 11b (in the Bede text): Propositio de patre et filio et vidua ejusque filia. Father and son marrying daughter and mother. This is like 11a.
Abbot Albert. c1240. P. 335.
Prob. 11. Two widows and sons marry. This is the same as Alcuin/Bede 11a. He says the sons of the unions are each other's paternal uncles.
Prob. 12. Two widowers and daughters marry. This is the same as the previous except for a sex change. Latin distinguishes paternal uncle from maternal uncle.
Prob. 13. Complex situation with a man and three wives.
Dialogue of Salomon and Saturnus. 14C. ??NYS. Given in Bryant, p. 12. "Tell me, who was he that was never born, was then buried in his mother's womb, and after death was baptised?" Answer: Adam. Cf: Adevineaux Amoureux, 1478; Vyse, 1771?, prob. 2.
Yachya Ben Sulieman. Hebrew text, c1430. ??NYS. Quoted in Folk‑Lore (1890) ??NYS. Quoted in Tony Augarde, op. cit. in 5.B, p. 3. A riddle attributed to the Queen of Sheba. "A woman said to her son, thy father is my father, and thy grandfather my husband; thou art my son, and I am thy sister." "Assuredly," said he [Solomon], "it was the daughter of Lot who spake thus to her son." Bryant, no. 1116, pp. 259 & 346 gives the same wording, with an extra level of quotation marks, and attributes it to the Queen of Sheba with no further details. Cf Queen of Sheba and Exeter Book above.
Adevineaux Amoureux. Bruges, 1478. ??NYS _ quoted by Bryant, no. 6, pp. 67-68 & 333. "Je fus nez avant mon pere / Et engendré avant ma mere, / Et ay occis le quart du monde, / Ainsi qu'il gist a la reonde, / Et si despucelay ma taye. / Or pensez se c'est chose vraie." (Bryant's translation: "I was born before my father, begotten before my mother and have slain a quarter of the world's population. How can this be?" Answer: Cain. Cf: Dialogue of Salomon and Saturnus, 14C; Vyse, 1771?, prob. 2.
Chuquet. 1484. Prob. 166. Same as Alcuin/Bede 11a. FHM 233 mentions it briefly without giving the relationships.
In 1491, the 14 year-old Duchess Anne of Brittany married Charles VIII, King of France in 1491. This was slightly complicated because both of them were married already, indeed Charles was married to the daughter of Anne's husband, the future Emperor Maximilian of Austria, so he was marrying his own step-mother-in-law. Fortunately, as was often the case in those days, both marriages were unconsummated _ indeed the couples had probably not yet seen each other and such proxy marriages were more like engagements _ so a little influence at Rome got both marriages dissolved. Somewhat surprisingly, as the marriage was more or less forced by Charles' siege of Rennes, the couple got on very well and developed a definite affection.
Ivan Morris. Foul Play and Other Puzzles of all Kinds. (Bodley Head, London, 1972); Vintage (Random House), NY, 1974. Prob. 21: No incest, pp. 39 & 93. Quotes Dudeney (??NYS) who gave an authentic 1538 epitaph describing the situation of Alcuin/Bede 11a with each couple having a child.
Tartaglia. General Trattato, 1556, art. 135, p. 256r. 2 fathers and 2 sons make only 3 people.
16C(?) riddle in Mantuan dialect. Given in: Franco Agostini & Nicola Alberto De Carlo; Intelligence Games; (As: Giochi della Intelligenza; Mondadori, Milan, 1985); Simon & Schuster, NY, 1987; p. 69. Two fathers and two sons make three people. The discussion is a bit unclear as to the date of this riddle.
Tombstone in the church at Martham, Norfolk. 1730. ??NYS _ quoted in a letter from Judith Havens (Norwich, Norfolk) in Challenging Centipede; The Guardian, section 2, (1 Dec 1994) 6.
"Here Lyeth / The Body of Christ. / Burraway, who departed / this life ye 18 day / of October, Anno Domini 1730 / Aged 59 years / And their Lyes / Alice, who by hir Life / was my Sister, my Mistress, / My Mother, and my Wife. / Dyed Feb ye 12, 1729 / Aged 76 years."
This was a response to a vague description of the epitaph in: Centipede; Famous last words; The Guardian, section 2 (24 Nov 1994) 4. There it is stated that Burraway was the result of an incestuous union between a man and his daughter. The baby was sent away to be brought up and years later happened to return to his native village where he met a older woman and became her lover, then her husband! (Shades of Oedipus!) Consequently he was his own uncle, step-father and brother-in-law!
Vyse. Tutor's Guide. 1771?
Prob. 1, p. 318 & Key p. 360. "Suppose two Women, and each a Son, were walking together, and were met by another Person, who asked the Boys in what Relation they stood to each other? They replied, We are Sons and Grandsons by the Fathers; Brothers and first Cousins by the Mothers; who also are Aunts to each of us. This Combination of Kinship once happened, but in what Manner? See Gen. xix. ver. 31." This is situation in the Exeter Book Riddles, no. 46, above.
Prob. 2, p. 318 & Key p. 360. "Who was he that was begot before his Father, born before his Mother, and had the Maidenhead of his Grandmother?" The answer notes that Adam was made "of the Dust of the Ground", etc. and then runs: "Now Abel ... was murdered by his Brother Cain; therefore he got the Maidenhead of his Grandmother (the Earth); and was got before his Father (Adam, who was made of the Earth, therefore was not begotten; and was born before his Mother (Eve), who was made of Man, therefore was not born." I think the meaning is that Abel was buried so he was the first man in the Earth. Cf: Dialogue of Salomon and Saturnus, 14C; Adevineaux Amoureux, 1478.
Jackson. Rational Amusement. 1821. Arithmetical Puzzles. No. 6, pp. 2 & 52. Father, mother, son, grandson, brother and daughter comprise only 3 people in the situation of Alcuin/Bede 11a when a couple has a son.
Curiosities for the Ingenious selected from The most authentic Treasures of Nature, Science and Art, Biography, History, and General Literature. 2nd ed., Thomas Boys, London, 1822. Singular intermarriage, p. 100. Man and daughter marry daughter and father. "My father is my son, and I am my mother's mother; / My sister is my daughter, and I'm grandmother to my brother."
Judge Leicester King (1789-1856) of Akron, Ohio, and his son married sisters, so he was his son's brother-in-law. BION-11.
Illustrated Boy's Own Treasury. 1860. Arithmetical and Geometrical Problems, No. 32, pp. 430 & 434. Will involving 6 relatives who turn out to be just 3 due to the situation of Alcuin/Bede 11a.
Charades, Enigmas, and Riddles. 1862?: prob. 584, pp. 110 & 157-158. "How can a man be his own grandfather?" Mother and daughter marry son and father. Mother and son produce a son, Tom. Notes that perhaps Tom is only his own grandfather-in-law.
Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 564-12, pp. 253 & 395. 25 relationships among only 7 people. Answer is a couple, with their son and his wife, with their son and two daughters. Dudeney, AM 54, gives the same grouping, but with a different list of 23 relationships. However neither counts the relationships properly _ e.g. both count 4 children and 2 sons and 2 daughters. I find 23 reasonable relationships _ Dudeney's 23 less 4 repeated children plus 2 husbands and 2 wives that he omitted. Leske has the husbands and wives, but omits the grandmother and a grandchild.
In theory, n people can have n(n-1) possible relationships, but not all of these relationships have distinct names. E.g. in the above the son of the first couple is a son to both parents, so two distinct relationships are both denoted by 'son'. However, in the classic problem of man, son and grandson, we actually have 2 fathers, 2 sons, 1 grandfather and 1 grandson, giving a full 6 relationships among just 3 people. Extending this to a string of n generations gives the full n(n-1) relationships among n people. One might ask if one can compact this a bit by using fewer generations for the n people. E.g., Leske's problem has 3 generations. So I pose the following problem: for n people in g generations, how many of the n(n-1) relationships are distinctly named in English (where 'distinctly named' is a bit vague!).
E. S. Turner. Op. cit. above, p. 109. [Retold in his: Amazing Grace; Michael Joseph, London, 1975; pp. 279-280.] The 7th Duke of Marlborough described to the House of Lords, c1880, the supposedly real case of a father and son marrying a daughter and mother. The son was his own grandfather and he became so confused that he committed suicide. In a footnote, Turner quotes a letter in the Welwyn News-Chronicle of 1949 from a man who married his step-mother's sister, i.e. widower and son married sisters.
J. M. Letter: Genealogical puzzle. Knowledge 3 (6 Jul 1883) 13, item 865. + Answer to genealogical puzzle in our last. Ibid. (13 Jul 1883) 29. Two unrelated persons have the same brother. Editorial note to the Answer says there are several ways to solve the puzzle _ how many?
E. W. Cole. Cole's Fun Doctor. The Funniest Book in the World. Routledge, London & E. W. Cole, Melbourne, nd [HPL dates it 1886 and gives the author's name as Arthur C. Cole]. P. 57: A smart cut-out & Genealogy are two stories of widow and daughter marrying son and father. = Alcuin 11b.
Lemon. 1890. How is this?, no. 725, pp. 83 & 123. 33 relatives who are only 8 people.
Hoffmann. 1893. Chap. IX, no. 42: The family party, pp. 321 & 328‑329. A man is his father's brother‑in‑law, his brother's father‑in‑law, his father‑in‑law's brother‑in‑law and his brother‑in‑law's father‑in‑law!
C. C. Bombaugh. Facts and Fancies for the Curious. Lippincott, 1905, ??NYS. A Mr. Harwood and John Cosick, both widowers, married each other's daughter, at Durham in eastern Canada. Quoted in: George Milburn; A Book of Puzzles and Brainteasers; Little Blue Book No. 1103, Haldeman‑Julius, Girard, Kansas, nd [1920s?], pp. 33‑34.
Pearson. 1907.
Part I, no. 25: A family party, pp. 120 & 184. Two widows and sons marry and each couple has a daughter, causing 24 apparent people to be only 6.
Part I, no. 36: Quite a family party, pp. 123 & 186. Same as Hoffmann.
Part I: A table of affinity, p. 127. Reports that M. de Lesseps and his son were to marry sisters and discusses the complications that would ensue.
Dudeney. AM. 1917. Several examples, including the following.
Prob. 52: Queer relationships, pp. 8 & 153. Discusses two brothers who married two sisters. One man and one woman died and the survivors married and had a child. The man married his deceased wife's sister which was legal, so he is married to the woman and his child is legitimate. But the woman married her deceased husband's brother, which was not legal at the time, so she is not married to the man and her child is illegitimate!! Mentions a man who married his widow's sister and a man who has a nephew, but the nephew is not the nephew of his sister.
Prob. 54: A family party, pp. 8 & 153. 23 apparent people are only 7. See Leske.
Prob. 55: A mixed pedigree, pp. 8-9 & 153. A is B's father's brother-in-law, brother's father-in-law and father-in-law's brother.
Prob. 56: Wilson's poser, pp. 9 & 153. A is B's uncle and nephew. This is due to two men marrying the mother of the other and both couples producing _ the results are A and B.
Ahrens. A&N. 1918. Pp. 105‑122, esp. 111‑122. Describes an 11C report of a person with only three grandparents, but it turns out to be erroneous. However, Cleopatra had only three grandparents, since her parents were half‑siblings.
For convenience in the following, let n‑parents denote one's ancestors n generations back, so 1‑parents are parents, 2‑parents are grandparents, 3‑parents are great‑grandparents, etc. Normally one has 2n n‑parents.
Prince Don Carlos of Spain (1545‑1568) had only 4 3‑parents, 6 4‑parents, 12 5‑parents and 20 6‑parents. Ahrens also gives more extended examples, e.g. the 12 generations of ancestors of Kaiser Wilhelm II comprise only 1549 people instead of the expected 8190, and one person occurs in 70 places.
Smith. Number Stories. 1919. Pp. 114‑116 & 142. Tartaglia's problem. Also a more complex version where 27 apparent people are only 7.
Ackermann. 1925. Pp. 92‑93. Three mothers, each with two daughters, require only 7 beds.
Loyd Jr. SLAHP. 1928. A puzzling estate, pp. 68 & 111. Three fathers and three sons are only four people.
Collins. Fun with Figures. 1928. He was his own grandfather, pp. 231-232. Claims to quote a Pittsburgh newspaper story of a resident who committed suicide when he found he was his own grandfather. Son and father married widow and daughter.
Dr. Th. Wolff. Die lächelnde Sphinx. Academia Verlagsbuchhandlung, Prague, 1937.
Prob. 4, pp. 188 & 197. 'My grandfather is only five years older than my father.'
Prob. 5, pp. 188 & 197. 'My father and my grandfather are twins.'
Depew. Cokesbury Game Book. 1939.
Three ducks, p. 201. Two fathers and two sons make three people.
Separate Room, p. 216. Three mothers, each with two daughters, make seven people.
Ripley's Believe It Or Not! 6th series, Pocket Books, NY, 1958. P. 145. Jacob van Nissen, of Zwolle, Holland, and his son married a girl and her mother.
W. Leslie Prout. Think Again. Frederick Warne & Co., London, 1958. Catch Quiz, No. 6, pp. 12 & 115. "Said one boy to another: "My mother's sister is your sister's mother." What relation were the two boys?"
Scot Morris. The Book of Strange Facts and Useless Information. Doubleday, NY, 1979, p. 98. In order to give his divorced mother some benefit from his father's estate, Robert Berston adopted his mother in 1967.
Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10. 1991. Chapter Three: The logic of kinship relations, pp. 66-83. Gives a number of folk puzzles and references.
Two mothers and two daughters are three people (Brazil).
Two brothers say "My brother's son is buried there"; third brother says "My brother's son is not buried there" (Ireland).
Who is the sister of my aunt, who is not my aunt? (Puerto Rico).
His mother is my mother's mother-in-law (Russia).
Who is my mother's brother's brother-in-law? (Wales).
"Smallweed" diary column. The Guardian (10 Apr 1993) 18. Bill Wyman of the Rolling Stones married and later divorced Mandy Smith. Mandy's mother Patsy is about to marry Stephen Wyman, Bill's son from his first marriage.
Item of front of Society section of The Guardian (2 Apr 1997) 1. "A 46-year-old Bedfordshire woman has married the father of an 18-year-old girl who eloped with her husband."
9.E.1. THAT MAN'S FATHER IS MY FATHER'S SON, ETC.
New section _ I have just found the 17C example, but there must be other and older examples?? But see Proctor, 1883.
For the 'Blind beggar' type of problem see: Boy's Own Book, 1828; Rowley, 1866; Rowley, 1875; Neil, 1880s; Lemon, 1890; Home Book, 1941; Charlot, c1950s. Again one thinks this should be older.
Tomé Pinheiro da Veiga. Fastiginia o fastos geniales. (This work is a chronicle of courtly life in Castilla, from 1601 to 1606. Translated & edited by Narciso Alonso Cortés. Imprenta del Colegio de Santiago, Valladolid, 1916, pp. 155b-156a, ??NYS.) French translation in: Augustin Redondo; Le jeu de l'énigme dans l'Espagne du XVIIe siècle. Aspect ludique et subversion; IN: Les Jeux à la Renaissance; Actes du XXIIIe Colloque International d'Études Humanistes (Tours, 1980); J. Vrin, Paris, 1982; pp. 445-458, with the problem being on p. 453. There are two brothers, born of the same father and mother. One is my uncle, but the other isn't. How is this possible?
The Book of Merry Riddles. London, 1629. ??NYS _ Santi 235 gives this and says it is reprinted in J. O. Halliwell, The literature of the sixteenth and seventeenth centuries, London, 1851, pp. 67‑102, ??NYS, and as pp. 7-29 in Alois Brandl, Shakespeares Book of Merry Riddles und die anderen Räthselbücher seiner Zeit, Jahrbuch der deutschen Shakespeare-Gesellschaft 42 (1906) 1-64, ??NYS. Bryant, pp. 100-102, quotes from: A Booke of Merrie Riddles, Robert Bird, London, 1631 and says it is also known as Prettie Riddles. Santi 237 gives Booke of Merrie Riddles, London, 1631, and says it is reprinted as pp. 53-63 in Brandl, ??NYS. Santi 307 gives The Booke of Merry Riddles, London, 1660, reprinted by J. O. Halliwell in 1866 in an edition of 25 copies, of which 15 were destroyed!, ??NYS. In Bryant, no. 275, pp. 102 & 336: "I know a child borne by my mother, / naturall borne as other children be, / that is neither my sister nor my brother. / Answer me shortly: what is he?"
Boy's Own Book. Conundrums.
"What kin is that child to its own father who is not its father's own son?" 1828: No. 70, pp. 432 & 440. 1828‑2: No. 70, pp. 436 & 444; 1829 (US): No. 70, pp. 238 & 258; 1855: No. 102, pp. 583 & 595. (I think this is the forerunner of the Blind Beggar problem.)
"What relation is your uncle's brother to you who is not your uncle?" 1828: No. 77, pp. 433 & 440. 1828‑2: No. 77, pp. 437 & 444; 1829 (US): No. 77, pp. 238 & 258; 1855: No. 109, pp. 584 & 595.
The Riddler. 1835. Conundrums Nos. 66 and 73, p. 16, no answers in my copy. These are identical to Nos. 70 and 77 in Boy's Own Book.
Boy's Treasury. 1844. Puzzles and paradoxes, no. 11, pp. 425 & 429. Woman says that a man's mother was her mother's only daughter.
Child. Girl's Own Book. 1848: Enigma 46, p. 236; 1876: Enigma 37, p. 199. "His mother was my mother's only child."
Charades, Enigmas, and Riddles. 1859?. Prob. 176 & 177, pp. 21 & 44.
Prob. 176. "If your uncle's sister is not your aunt, what relationship does she bear to you?"
Prob. 177.
My mother had a child, my very own mother,
It was not my sister nor yet was it my brother;
If you are as clever as I fancy you to be,
Pray tell me what relation that child was to me.
Boy's Own Conjuring Book. 1860. P. 381. "I've no sister or brother; / You may think I am wild; / But that man's mother / Was my mother's child."
Hugh Rowley. Puniana or Thoughts Wise and Other-wise A New Collection of the Best Riddles, Conundrums, Jokes, Sells, etc, etc. Chatto & Windus, London, 1866.
P. 36. Brothers A and B were walking. ""I must speak to those children," said A; "they are my nephews and nieces." "Ah!" said B, "as I have no nephews and nieces, I shall walk on." How was this?"
P. 88. "What kin is that child to his own father who is not his own father's son?"
P. 88. "If Dick's father is Tom's son, what relation is Dick to Tom?"
P. 88. "Who were your grandfather's first cousin's sister's son's brother's forefathers? Why, his aunt's sisters, of course." (This is non-logical, being a pun on ancestors, but it illustrates that the idea of such problems must have been well known.)
P. 137. A fiddler said his brother played the double-bass, but the double-bass player denied having a brother.
Hugh Rowley. More Puniana; or, Thoughts Wise and Other-Why's. Chatto & Windus, London, 1875.
P. 28. "A blind beggar had a brother, who died and went to heaven. What relation was the blind beggar to the person who went to heaven?" Answer is 'sister', but the posing of the question is defective _ it should include the assertion that the person who went to heaven had no brother.
P. 134. "What relation is that child to its own father, who is not its own father's son?"
P. 217. "That gentleman's mother is my mother's only child."
P. 231. "What relation is your father's only brother's sister-in-law to you?"
P. 231. "Brothers and sisters have I none, but this man's father is my father's son."
[Richard A. Proctor] Letters received and short answers. Knowledge 3 (26 Oct 1883) 264. Answer to Harry. "Sisters and brothers have I none But that man's father is my father's son." Implies that the puzzle is not well known. Recalls it being posed on a ship and distracting all the passengers for a day.
E. W. Cole. Cole's Fun Doctor. The Funniest Book in the World. Routledge, London & E. W. Cole, Melbourne, nd [HPL dates it 1886 and gives the author's name as Arthur C. Cole]. P. 329: A riddle. "His mother was my mother's only child."
James Neil [= "A Literary Clergyman"]. Riddles: And Something New About Them. (Lang Neil & Co., London, nd [1880s?]; Simpkin Marshall & Co., London); Village Games, London, 1993. General Riddles: Relationship, p. 28.
"A blind beggar had a brother, the brother died, deceased had no brother. What relation was the blind beggar to deceased?"
"What relation is that child to its own father who is not its own father's own son?"
"If your uncle's sister is not your aunt, what relation is she to you?"
Lemon. 1890.
Do you see it?, unnumbered section after no. 80, pp. 15: "That gentleman's mother is my mother's only child."
Conundrums, no. 142(a), pp. 23 & 102 (= Sphinx, no. 470(a), pp. 65 & 113): child "who is not his own father's son."
Fireside Amusements _ A Book of Indoor Games. Op. cit. in 7.L.1. 1890? P. 99, no. 21. "His mother was my mother's only child."
William Crompton. The odd half-hour. The Boy's Own Paper 13 (No. 657) (15 Aug 1891) 731-732. A true friend. "If your uncle's sister is not your aunt, what relationship does she bear to you?"
Bennett Coll. Prove it! The Idler 2 (1892-1893, probably Dec 1892) 510-517. Man in front of a portrait says "Sisters and brothers have I none; That man's father is my father's son." Says the portrait is himself! Observes that this leads to his father being his own son and being the father of his father. Describes the difficulties people have in trying to see this answer [not surprisingly]. Various other solutions given: grandfather, brother, uncle on the mother's side.
Hoffmann. 1893. Chap. IX, no. 25: The portrait, pp. 318 & 326. "Uncles and brothers have I none, But that man's father is my father's son." He notes "This venerable puzzle forms the subject of a humorous article, entitled "Prove It," in a recent number of the Idler. Its most amusing feature is that the writer has himself gone astray, ...." [I'm not sure whether Coll has gone astray or is using the error to generate humour??]
W. H. Howe. Everybody's Book of Epitaphs Being for the Most Part What the Living Think of the Dead. Saxon & Co., London, nd [c1895] (reprinted by Pryor Publications, Whitstable, 1995). P. 165 has the following entry.
"In Llanidan Churchyard, Anglesea:_
Here lies the world's mother,
By nature my Aunt _ sister to my mother,
My grandmother _ mother to my mother,
My great grandmother _ mother to my grandmother,
My grandmother's daughter and her mother."
Could this be a real case of 'I'm my own grandmother'??
Somerville Gibney. So simple! The Boy's Own Paper 20 (No. 992) (15 Jan 1898) 252. 'That very old catch _ "If Dick's father is Tom's son, What relation is Dick to Tom?"'
Dudeney. "The Captain" puzzle corner. The Captain 3:1 (Apr 1900) 1 & 90 & 3:3 (Jun 1900) 193 & 279. No. 3: Overheard in an omnibus. "Was that your father." "No, that gentleman's mother was my mother's mother-in-law." Essentially the same as: AM; 1917; Prob. 53: Heard on the tube railway, pp. 8 & 153; "That gentleman's mother was my mother's mother-in-law, but he is not on speaking terms with my papa."
James Joyce. Ulysses. (Dijon, 1922); Modern Library (Random House), NY, 1934, apparently printed 1946. P. 692 (Gardner says the 1961 ed. has p. 708; this is about 4/5 of the way between the start of Part III and Molly's soliloquy). "Brothers and sisters had he none, Yet that man's father was his grandfather's son." This is given as a quotation, while Bloom is looking in a mirror _ otherwise it could be a cousin. [Given in Bryant, no. 782, pp. 194 & 342.]
Streeter & Hoehn. Op. cit. in 7.AE. Vol. 2, 1933, p. 16, no. 10: "Brain twister". "My son's father is your father's only child. What relative of yours am I?"
Dr. Th. Wolff. Die lächelnde Sphinx. Academia Verlagsbuchhandlung, Prague, 1937. Prob. 33, pp. 194 & 204. 'This man's mother is my mother's only child.'
McKay. Party Night. 1940. No. 7, p. 175. "Brothers and sisters have I none; yet this man's father was my father's son."
Meyer. Big Fun Book. 1940. No. 5, pp. 175 & 756. "My father is the brother of your sister. What relative am I of yours?" Answer is nephew, but son is also possible.
The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941.
P. 149, prob. 8. "Sisters and brothers I have none, but that man's father is my father's son."
P. 149, prob. 10. "A beggar's brother died. But the man who died had no brother."
John Henry Cutler. Dr. Quizzler's Mind Teasers. Greenberg, NY, 1944. ??NYS _ excerpted in: Dr. Quizzler's mind teasers; Games Magazine 16:3 (No. 109) (Jun 1992) 47 & 43, prob. 14, with additional comments in Ibid 16:4 (No. 110) (Aug 1992) 4 and 16:6 (No. 112) (Dec 1992) 4. "What relation is a man to his mother's only brother's only niece?" Answer is her brother, but comments point out that she could be his cousin, i.e. his mother's sister's daughter, or even a kind of cousin-in-law, i.e. his mother's brother's wife's sibling's daughter.
Yvonne B. Charlot. Conundrums of All Kinds. Universal, London, nd [c1950?].
P. 77: "If your aunt's brother is not your uncle, who is he?"
P. 82: "What kin are those chldren to their own father who are not their own father's sons?"
Hubert Phillips. Party Games. Witherby, London, 1952. Chap. XIII, prob. 3: Photograph, pp. 204 & 252‑253.
"Though sons and brothers have I none,
Your father was my father's son."
Solution says this "is my own invention".
See Ascher in 9.E for some examples.
Iona & Peter Opie. I Saw Esau: The Schoolchild's Pocket Book. (Williams & Norgate, London, 1947.) Revised edition, Walker Books, London, 1992, ??NX. No. 42, p. 45, just gives the rhyme; illustration on p. 44; answer on p. 144 just gives the answer, with no historical comments.
9.E.2. IDENTICAL SIBLINGS WHO ARE NOT TWINS
Two siblings are born on the same day to the same parents but are not twins. New section. This must be older than the example below.
Harold Hart. The World's Best Puzzles. Op. cit. in 7.AS. 1943. The problem of the two students, pp. 4 & 50
9.F. THE UNEXPECTED HANGING
Nicholas Falletta. The Paradoxicon. Op. cit. in 8.H.1. 1983. Pp. 162‑163 relates that during World War II, Swedish Radio announced there would be an unexpected civil defense exercise next week. Lennart Ekbom, a Swedish professor of mathematics, noted the paradoxical nature of this and discussed it with his students.
D. J. O'Connor. Pragmatic paradoxes. Mind 57 (1948) 358‑359. Discusses several other paradoxes, e.g. "I remember nothing", then the unexpected blackout exercise.
L. Jonathan Cohen. Mr. O'Connor's "Pragmatic paradoxes". Mind 59 (1950) 85‑87. Doesn't deal much with the unexpected blackout.
Peter Alexander. Pragmatic paradoxes. Mind 59 (1950) 536‑538. Also doesn't deal much with the unexpected blackout.
Michael Scriven. Paradoxical announcements. Mind 60 (1951) 403‑407. "A new and powerful paradox has come to light." Entirely concerned with the unexpected blackout and considers the case of only two possible dates.
Max Black. Critical Thinking. 1952. Op. cit. in 6.F.2. Prob. 1, pp. 156 & 433.
Gamow & Stern. 1958. The date of the hanging. Pp. 23‑27.
M. Gardner. SA (Mar 63) = Unexpected, chap. 1, with an extensive historical addendum and references.
Joseph S. Fulda. The paradox of the surprise test. MG 75 (No. 474) (Dec 1991) 419-421.
9.G. TRUTHTELLERS AND LIARS
I have just started to consider problems where a number of statements are given and we know at least or at most some number of them are lies. There must be earlier versions _ e.g. Lewis Carroll??
Find correct answer in one question from a truthteller or liar: Gardner; Harbin; Rice; Doubleday-II; Eldin. See: Nozaki for a generalization.
Problem with three truthtellers or liars and first one mumbles: Rudin; Depew; Kraitchik; Hart; Leopold; Wickelgren.
Magician's Own Book. 1857. P. 216. A lies 1/4 of the time; B lies 1/5 of the time; C lies 1/7 of the time. "What is the probability of an event which A and B assert, and C denies?" Answer is 140/143, but I get 2/3. = Book of 500 Puzzles, 1859, p. 54.
Chas. G. Shaw. Letter: The doctrine of chances. Knowledge 7 (27 Feb 1885) 181, item 1620. Says Whitaker's Almanac for this year, under The Doctrine of Chances, gives the following problem with a wrong answer. A lies 1/4 of the time; B lies 1/5 of the time; C lies 1/6 of the time. What is the chance of an event which A and B assert, but C denies? Whitaker and I get (3/4)(4/5)(1/6) / [(3/4)(4/5)(1/6) + (1/4)(1/5)(5/6)] = 12/17, but Shaw claims 19/24 by asserting that the probability of an event when A and B testify to it ought to be 1 ‑ (1/4)(1/5) = 19/20 instead of (3/4)(4/5) = 3/5, He then says this leads to 19/24 by modifying the above formula, but I can't see how this can be done.
A. C. D. Crommelin. Problem given in an after-dinner speech, reported by Arthur Eddington in 1919. ??where, ??NYS _ quoted in: Philip Carter & Ken Russell; Classic Puzzles; Sphere, London, 1990, pp. 50 & 120-121. Four persons who tell the truth once with probability 1/3. If "A affirms that B denies that C declares that D is a liar, what is the probability that D was speaking the truth?"
Collins. Fun with Figures. 1928. The evidence you now give, etc., etc., pp. 22-23. Three witnesses who tell the truth 1/3, 1/5, 1/10 of the time. First two assert something which the third denies. What is the probability the assertion is true? Asserts it is 9 to 8, which I also get.
H. A. Ripley. How Good a Detective Are You? Frederick A. Stokes, NY, 1934, prob. 22: An old spanish custom. King will present the Princess's suitor a choice of two slips, one marked 'win', the other 'lose'. The king is determined to doublecross the suitor so he has both marked 'lose'. But the suitor realises this, so when he picks a slip, he drops it in the fire and then asks the King to reveal the other slip!
Rudin. 1936.
No. 43, pp. 14-15 & 84. 9 statements with only 3 correct.
No. 85, pp. 29 & 92-93. 11 statements with at least 7 lies. Makes a table to solve it.
No. 200, pp. 72 & 122. Three truthtellers or liars. First one is inaudible. Second says the first claims to be a truthteller. Third says the second is lying. Author adds that there is just one liar and determines which each is.
Hubert Phillips. Question Time, op. cit. in 5.U. 1937.
Prob. 10: Red and blue, pp. 6 & 178. Involves truthtellers, liars and alternators.
Prob. 25: Tom, Dick, and Harry, pp. 14 & 181. Involves truthteller, liar and alternator.
Depew. Cokesbury Game Book. 1939. Mixed blood, p. 202. Same as Rudin 200.
M. Kraitchik. Mathematical Recreations, 1943, op. cit. in 4.A.2, chap. 1, prob. 2, pp. 14‑15. Truthtellers and liars. (Not in Math. des Jeux.) Same as Rudin 200, but doesn't give number of liars so only determines which the second and third are.
Harold Hart. The World's Best Puzzles. Op. cit. in 7.AS. 1943. The problem of the nobles and the slaves, pp. 11 & 51. Similar to Rudin, but doesn't say how many liars, but the statements are more elaborate so all can be determined.
Jules Leopold. At Ease! Op. cit. in 4.A.2. 1943. Simpletons and liars, pp. 6-7 & 194. Similar to Rudin.
Hubert Phillips. Something to Think About, op. cit. in 7.AD, 1945.
Prob. 83‑85: Crazy island problems, pp. 51‑54 & 110‑112.
Prob. 83 involves three truthtellers or liars, not like Rudin.
Prob. 84 involves three truthtellers or liars or alternators. (Not the same as either problem in his Question Time, above.)
Prob. 85 involves three truthtellers or liars or Minimums, who tell the truth at most a third of the time.
Hubert Phillips. Hubert Phillips's Heptameron. Eyre & Spottiswoode, London, 1945. Day 1, prob. 25: Crazy island, pp. 18 & 231. Same as prob. 83 in Something to Think About.
Leeming. 1945. Chap. 3, prob. 17: Which was the officer?, pp. 25‑26 & 155‑156. Two truthtellers and a liar.
Gardner. SA (Feb 1957) c= 1st Book, chap. 3, problem 4: The fork in the road. Truthteller or liar. The book version includes a number of letters and comments. I have xeroxes from Gardner's files of letters from people who claim to have invented this problem _ one of these seemed reasonable as he said he did it in a puzzle column in Boston in 1919 _ ??NYR _ DO.
N. A. Longmore, proposer; editorial solution. The oracle of three gods. RMM 4 (Aug 1961) 47 & 5 (Oct 1961) 59. Truthteller, liar and alternator.
Robert Harbin. Party Lines. Op. cit. in 5.B.1. 1963. The road to freedom, p. 30. Truthteller or liar.
Charlie Rice. Challenge! Op. cit. in 5.C. 1968. Prob. 8, pp. 22-23 & 55. Truthteller or liar.
F. W. Sinden. Logic puzzles. In: R. P. Dilworth, et al., eds.; Puzzle Problems and Games Project _ Final Report; Studies in Mathematics, vol. XVIII; School Mathematics Study Group, Stanford, Calif., 1968; pp. 197‑201. The District Attorney, pp. 200‑201. Two truthtellers and a liar _ determine which in two questions.
Doubleday - II. 1971. Truth will out, pp. 151-152. Truthteller or liar.
Peter Eldin. Amaze and Amuse Your Friends. Piccolo (Pan), London, 1973. No. 34: Where am I?, pp. 79 & 106. You are on an island of truthtellers or an island of liars. Determine which in one question.
Wickelgren. How to Solve Problems. Op. cit. in 5.O. 1974. Pp. 36‑37. He uses 'truar' for truthteller. From statements by three truars or liars, you can deduce the number of each, though you can't tell which is which!!
Rowan Barnes-Murphy. Monstrous Mysteries. Piccolo, 1982. Tollimarsh Tower, pp. 14 & 57. Two monster guards, one truar and one liar, and you have one question. You discover one is asleep and the other says: "It doesn't matter that he's asleep, he always tells people to do A." Do you do A or not A?
Shari Lewis. Abracadabra! Magic and Other Tricks. (World Almanac Publications, NY, 1984); Puffin, 1985. Free choice, p. 22. Truthteller and liar have distributed items A and B. You want to determine who has which item with one question. You ask "Did the liar take B?" If the person answers 'yes', he has item A; if 'no', he has item B.
Akihiro Nozaki. How to get three answers from a single yes‑no question. JRM 20:1 (1988) 59‑60. You have to ask a truthteller or a liar which of three roads is correct. The author's question results in neither being able to answer in the third case. He suggests extensions.
9.H. PRISONER'S DILEMMA
Charles Babbage. On the Economy of Machinery and Manufactures. (1832, ??NYS); 4th ed., (1835), reprinted by Augustus M. Kelley, NY, 1971. Section 348, p. 289. "... both parties are often led to adopt arrangements ... at variance ... with the true interests of both."
Frederick Winslow Taylor. The Principles of Scientific Management. (1911); Harper & Brothers, NY, 1923. P. 10. Speaking of employers and employés, he says "that perhaps the majority on either side do not believe that it is possible so to arrange their mutual relations that their interests become identical."
Merrill M. Flood & Melvin Dresher. c1950. ??NYS _ details. They identified the paradox, but I have no reference to any publication.
Robert Axelrod. The Evolution of Cooperation. Basic Books, NY, 1986. p. 216, ??NYS. "The Prisoner's Dilemma game was invented in about 1950 by Merrill Flood and Melvin Dresher, and formalised by A. W. Tucker, shortly thereafter."
Keith Devlin. It's only a game. The Guardian, second section (17 Nov 1994) 12-13. Says Tucker invented the dilemma in 1950.
Sylvia Nasar. Albert W. Tucker, 89, pioneering mathematician. New York Times (27 Jan 1995) ?? Asserts Tucker invented the dilemma when teaching game theory to psychology students at Stanford in 1950.
9.I. HEMPEL'S RAVEN PARADOX
Carl G. Hempel. Studies in the logic of confirmation. (Mid 1940s?). Reproduced in: M. H. Foster & M. L. Martin, eds.; Probability, Confirmation and Simplicity; Odyssey Press, NY, 1966, pp. 145‑183. ??NYS.
9.J. FALLEN SIGNPOST
How do you use a fallen signpost to find your way?
Mr. X. His Pages. The Royal Magazine 9:5 (Mar 1903) 490-491 & 10:1 (May 1903) 50-51. The sense of direction.
King. Best 100. 1927. No. 22, pp. 14 & 44.
H. A. Ripley. How Good a Detective Are You? Frederick A. Stokes, NY, 1934, prob. 30: Class day.
John Paul Adams. We Dare You to Solve This!. Op cit. in 5.C. 1957? Prob. 180: On the right track?, pp. 67 & 121.
9.K. CARROLL'S BARBER PARADOX
Martin Gardner recently asked me to look up some of these items as he is doing a section on it in a book about Carroll which will be much more detailed than the following, citing numerous other discussions.
Lewis Carroll. Diary entry for 31 Mar 1894. Says he has just had a leaflet "A Disputed Point in Logic" printed containing the problem that he and John Cook Wilson "have been arguing so long." ??NYS _ quoted by Gardner.
Lewis Carroll. A logical paradox. Mind (NS) 3 (No. 11) (Jul 1894) 436-438.
Alfred Sidgwick. "A logical paradox". Mind (NS) 3 (No. 12) (Oct 1894) 582.
W. E. Johnson. A logical paradox. Mind (NS) 3 (No. 12) (Oct 1894) 583.
Lewis Carroll. Diary entry for 21 Dec 1894. Discusses the problem and seems to recognise the distinction between material and strict implication. ??NYS _ quoted by Gardner.
Alfred Sidgwick & W. E. Johnson. "Hypotheticals in a context". Mind (NS) 4 (No. 13) (Jan 1895) 143-144.
E. E. C. Jones. Lewis Carroll's logical paradox (Mind, N.S., 3). Mind (NS) 14 (No. 53) (Jan 1905) 146-148.
W. [= John Cook Wilson, according to Gardner]. Lewis Carroll's logical paradox (Mind, N.S., 3 and 53, P. 146). Mind (NS) 14 (No. 54) (Apr 1905) 292-293. He admits that Carroll had been right all along.
Warren Weaver. Lewis Carroll: Mathematician. Op. cit. in 1. 1956. Discusses the paradox. Alexander B. Morris's letter says the paradox is not real. Weaver's response discusses this and other unpublished letters, saying he is not sure if the paradox is resolved.