Sources page probability recreations

1897, no. 3. The beggar. "A beggar had a brother, the brother died and the man who died had no brother."

1897, no. 16. The man in jail. "Brothers and sisters have I none, but that man's father is my father's son."

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."

Hummerston. Fun, Mirth & Mystery. 1924. Grandfather's problems: The portrait, p. 68. "Sisters and brothers have I none, But that man's father is my father's son."

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.'

Haldeman-Julius. 1937. No. 44: Portrait problem, pp. 7 & 22. Woman points to a man's portrait and says to her brother: "The man's mother was my mother's mother-in-law." Answer is that she is his daughter, but she might be his step-daughter.

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 children 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.

Ripley's Believe It or Not!, 8th series. Pocket Books, 1962, p. 26. Jimmy Burnthet, of Emmerdale, England, courted a girl for 40 years. She broke off the engagement and married his nephew, whereupon he married her niece.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 16, pp. 25 & 88. "The father of the person in the portrait is my father's son, but I have no brothers or sons."

Ripley's Puzzles and Games. 1966.

P. 12. "What relation is your mother's brother's brother-in-law to you. Answer: Your father". There are several more answers. My mother's brother is my maternal uncle. He could have several sisters whose husbands are his brothers-in-law -- one is my father, the others are brothers-in-law of my mother. Also, he could be married and any of his wife's brothers are also his brothers-in-law.

P. 13. "Moab and Ben-am-mi were brothers yet cousins and their father was their grandfather." These are the sons of Lot by his daughters.

P. 13. Bible puzzle. "Two people died who were never born" -- Adam and Eve. "Two people were born who never died" -- Enoch and Elias both just disappeared! "The oldest man who ever lived died before his father did" -- Methusaleh, the son of Enoch.

Pat Fairon. Irish Riddles. Appletree Press, Belfast, 1992; Chronicle Books, San Francisco, 1992; pp. 27 & 60. "Brothers and sisters have I none But this man's father Was my father's son." Answer is "Oneself".
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 defence 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.

Find correct answer in one question from a truthteller or liar: Goodman; Gardner; Harbin; Rice; Doubleday - 2; Eldin. See: Nozaki for a generalization.

Problem with three truthtellers or liars and first one mumbles: Rudin; Haldeman-Julius; Depew; Catch-My-Pal; 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.

Lewis Carroll. Diary entry for 27 May 1894. "I have worked out in the last few days some curious problems on the plan of 'lying' dilemma. E.g. 'A says B lies; B says C lies; C says A and B lie.' Answer: 'A and C lie; B speaks truly'. The problem is quoted in Carroll-Gardner with his discussion of the result, pp. 22-23. Gardner says this was printed as an anonymous leaflet in 1894.

Carroll-Wakeling. Prob. 9: Who's telling the truth?, pp. 11 & 65. Wakeling says "This is a puzzle based on a piece of logic that appears in his diary.

The Dodo says the Hatter tells lies.

The Hatter says that the March Hare tells Lies.

The March Hare says that both the Dodo and the Hatter tell lies.

Who is telling the truth?"

In a letter of 28 May 2003, Wakeling quotes the text from the Diary as above, but continues with it: "And today 'A says B says C says D lies; D says two lie and one speaks true.' Answer: 'D lies; the rest speak truly.' Wakeling adds that the Diary entry for 2 Jun 1896 (not given in the Lancelyn Green edition) says: "Finished the solution of the hardest 'Truth-Problem' I have yet done", but Carroll gives no indication what it was.

Lewis Carroll. The problem of the five liars. In his unpublished Symbolic Logic, Part II. He was working on this after Part I appeared in 1896 and he had some galley proofs when he died in 1898. Published in Lewis Carroll's Symbolic Logic, ed. by William Warren Bartley III; Clarkson N. Potter, NY, 1977, pp. 352 361, including facsimiles of several letters of 1896 to John Cook Wilson (not in Cohen). Each of five people make two statements, e.g. A says "Either B or D tells a truth and a lie; either C or E tells two lies." When analysed, one gets contradictions because a form of the Liar Paradox is embedded.

Carroll-Wakeling II. c1890? Prob. 32: Bag containing tickets, pp. 50 & 73. This is one of the problems on undated sheets of paper that Carroll sent to Bartholomew Price. Wakeling reproduces the MS.

A bag contains 12 tickets, 3 marked 'A', 4 'B', 5 'C'. One is drawn in the presence of 12 witnesses of equal credibility: three say it was 'A', four 'B', five 'C'. What is the chance that it was 'A'?

There is no answer on the Carroll MS. Wakeling gives an answer.

Let the credibility of a witness be "a" when telling the truth. Hence, the credibility of a witness when telling a lie is "1 - a".

If it was A, then 3 tell the truth, and 9 lie; hence the credibility is 3 in 12, or 1 in 4.

Therefore the chance that it is A, and no other, is:

3/12 x 1/4 + 4/12 x 3/4 + 5/12 x 3/4 = 5/8

I cannot see how this last formula arises. Wakeling writes that he was assisted in this problem by a friend who has since died, so he does not know how the formula was obtained.

Assuming a is the probability of a person telling the truth, this probability depends on what the chosen ticket is and is not really determined by the information given -- e.g., if the ticket was A, then any value of a between 0 and 1 is possible. The value 3/12 is an estimate of a, indeed the maximum likelihood estimate. If k of the 12 people are telling the truth, I would take the situation as a binomial distribution. There are BC (12, k) ways to select them and the probability of having k liars is then BC (12, k) a^k (1-a)^12-k. Now it seems that Bayes' Theorem is the most appropriate way to proceed. Our basic events can be denoted A, B, C and their a priori probabilities are 3/12, 4/12, 5/12. Taking a = k/12, the a posteriori probabilities are proportional to

k/12 x BC (12, k) (k/12)^k ({12-k}/12)^12-k, for k = 3, 4, 5. Dropping the common denominator of 12¹³, these expressions are 6.904, 8.504, 10.1913 times 10¹². Dividing by the total gives the a posteriori probabilities of A, B, C as 27.0%, 33.2%, 39.8%.

This is really a probability problem rather than a logical problem, but it illustrates that the logical problem seems to have grown out of this sort of probability problem.

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.

Nelson Goodman. The problem of the truth-tellers and liars. Anonymous contribution to the Brainteasers column, The Boston Post (Jun 1929). ??NYS -- described in an undated letter from Goodman to Martin Gardner, 1960s?, where he says he 'made it up out of whole logical cloth' and submitted it to the paper.

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 double-cross 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.