Sources page probability recreations

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.

Haldeman-Julius. 1937. No. 5: Noblemen-huntsmen problem, pp. 3-4 & 20. Noblemen tell the truth and huntsmen lie. Same as Rudin, no. 200. Says the problem was sent in by J. C. Furnas.

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.

Catch-My-Pal cards, issued with The Hotspur, a magazine published by D. C. Thomson, Dundee, in 1939. The cards are unnumbered, but the sixth in my set has three truthtellers or liars being asked what they are. The reply of the first is inaudible. The second says: He says he's a truthteller and he is a truthteller and so am I. The third says: I'm a truthteller and they're liars. Solution finds they are truthteller, truthteller, liar (TTL). This assumes that all parts of the statements are true or false simultaneously. If one assumes that a conjunctive statement is a lie if any one part is a lie, then there are more possibilities -- I get TTL, TLL, LLT, LLL.

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.

Leopold. At Ease! 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 photocopies from Gardner's files of letters from people who claim to have invented this problem -- only one of these seemed reasonable -- cf Goodman above. Other material ??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 - 2. 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.

Ken Weber. More Five-Minute Mysteries. Running Press, Philadelphia, 1991. No. 13, pp. 59-61 & 189-190. You have gotten lost near the border between the truthtellers and the liars and you come out in an open area where there is a border marker with a guard on each side, but you cannot tell which side is which. Further the guards are walking back and forth and exchanging positions, so you are not sure if they are on their correct sides of the border or not. You can approach one of the guards and ask one question to determine which side you are on. You ask "Are we in your country?" If he answers 'yes', you are in truthtelling country.

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 [cf 4.A.1]. 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.

Bernd Plage. Der umgestürzte Wegweiser. Verlag von R. Oldenbourg, München, 1944. Prob. 1: Der umgestürzte Wegweiser, pp. 11-12 & 67. (Kindly sent by Heinrich Hemme.)

John Paul Adams. We Dare You to Solve This!. Op. cit. in 5.C. 1957? Prob. 180: On the right track?, pp. 67 & 121.

Jonathan Always. Puzzles for Puzzlers. Tandem, London, 1971. Prob. 84: A resourceful motorist, pp. 41 & 87.

9.K. CARROLL'S BARBER PARADOX
Martin Gardner 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. This presumably refers to the now-appeared Carroll-Gardner, 1996.
Carroll-Wakeling. c1890? Prob. 16: Going out, pp. 20-21 & 67-68. This is on one of the undated sheets Carroll sent to Bartholomew Price. Wakeling notes that it is an early version of the Barber shop paradox and says that Mark Richards has pointed out a mistake in Carroll's solution.

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 in Carroll-Gardner. Gardner says the pamphlet was revised in Apr 1894.

Lewis Carroll. A logical paradox. Mind (NS) 3 (No. 11) (Jul 1894) 436-438. Gardner says Carroll reprinted this as a pamphlet.

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. Not in Lancelyn Green. Discusses the problem and seems to recognise the distinction between material and strict implication. ??NYS -- quoted in Carroll-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.

R. B. Braithwaite. Lewis Carroll as logician. MG 16 (No. 219) (Jul 1932) 174-178. Brief discussion and solution of the paradox.

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.

Carroll-Gardner. 1996. Pp. 67-71 discusses this in detail, citing a number of other references, including John Venn, but he only gives the page numbers of an article in Symbolic Logic.

Lewis Carroll discusses this 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. Bartley includes all eight known versions.

10. PHYSICAL RECREATIONS
See also 7.S and 7.Y.

I will collect here some material on physics toys in general.

Christian Ucke. Physics toys for teaching. IN: H. Kühnelt, ed.; Interdisciplinary Aspects of Physics Education; proc. of conf. at Altmünster, Austria, 1989; World Scientific, Singapore, 1990, pp. 267-273. 1: Some new and not so well known literature about physics toys gives 12 references. 2: Presenting a database about physics toys. 3. Physics toy experiments with PET bottles. 16 references at the end, some repeating those in part 1.
10.A. OVERTAKING AND MEETING PROBLEMS
See Tropfke 588.

Note. Meeting problems include two pipe Cistern Problems, 7.H. Overtaking problems include Snail in Well problems without end effect, 10.H, and Cisterns with one inlet and one outlet, 7.H. Many of the Indian versions involve gaining or losing wealth rather than covering distance. Versions going around a circle or an island are related to Conjunction of Planets, 7.P.6, and to problems of Clock Hands meeting, 10.R.

In the 17, 18 and 19 C, this problem was often discussed in relation to negative numbers as a change in the relative values leads to a negative solution -- cf: Clairaut; Manning; Hutton, 1798?; Lacroix; De Morgan, 1831? & 1836;
NOTATION. There are five types of meeting (M) and overtaking (O) problems which recur frequently with slight variations. I have recently converted all problems to this notation and I hope I have done it correctly. Tropfke 590 gives a more extended classification which includes motion on a right triangle (see 6.BF.5) and on a circle (see 7.P.6, though some occur here) and alternating motion (see 10.H), but doesn't distinguish between problems where times are given and those where rates are given.

M-(a, b). Two travellers can cover a route in a, b (usually days). They start at opposite ends at the same time toward each other. When do they meet? This is identical to the cistern problem (a, b) of 7.H. Sometimes, the distance apart is given and the point of meeting is also wanted. If the distance is, say 100, I will then say "with D = 100". This corresponds to asking how much each pipe contributes to a cistern of capacity D. Sometimes, one starts later than the other. If, say the first starts 2 later, I will say "with the first delayed by 2". This corresponds to opening one pipe later than the other. This is the version of Tropfke's I B a where times are given.

MR-(a, b; D). The same problem except that a, b are the actual rates of the two travellers and hence D must be given. This corresponds to a simple form of cistern problem which does not have the characteristic feature of giving the times required to do the entire task. This is the version of Tropfke's I B a with rates given.

MR-(a, b; c, d; D). The same problem except that the travellers travel in arithmetic progressions, so this gives:

a + a+b + ... + a+(n-1)b + c + c+d + ... + c+(n-1)d = D.

Hence MR-(a, 0; c, 0; D) = MR-(a, c; D). The value of d is often 0. One can interpret this as a cistern problem as for MR-(a, b; D), but it is even harder to imagine a pipe increasing its rate in arithmetic progression that to imagine a traveller doing so. (An additional difficulty is that the traveller is usually viewed discretely while a pipe ought to be viewed continuously.) This is the version of Tropfke's I B b with an arithmetic progression specified.

O-(a, b). Two travellers start from the same point at rates a, b, with the slower starting some time T before the faster, or they start at the same time at rates a, b, with the slower starting some distance D ahead of the other. I.e. the slower has a headstart of time T or distance D. When does the faster overtake the slower? This corresponds to a cistern with rates given, so that O-(a, b) with headstart D, which is the same as MR-(a, -b; D), is a cistern problem with one inlet and one outlet. When a > b, then this corresponds to a full cistern of size D, inlet rate a and outlet rate b. The case a < b is most easily viewed by negating the amount done, which interchanges inlet and outlet, and taking an empty cistern. Hound and hare problems are basically of this form, with a headstart of some distance, but usually with rates and distances complicatedly expressed. This is Tropfke's I C a. Sometimes the rates are not given explicitly, so I assume the first has the headstart.

O-(a, b; c, d). Two travellers start from the same point at the same time, but in arithmetic progressions. When do they meet again? This gives us:

a + a+b + ... + a+(n-1)b = c + c+d + ... + c+(n-1)d. Some versions of this are in 7.AF.

If the first has a headstart of time T, then we either increase the first n by T or decrease the second n by T, depending on which number of days is wanted. This is Tropfke's I C b with an arithmetic progression specified. Sometimes the first has a headstart of distance D. Occasionally it is the second that has the headstart which is denoted by negative values of T or D.
Snail in the well problems without end effect (see 10.H) are special cases of meeting problems, usually MR-(a, 0; D) (Tropfke's I A c). When there are approaching animals, then it may be MR(a, b; D) or O (a, b) with headstart D (Tropfke's I B c).
Hound and hare problems. Here one is often only given the ratio of speeds, r, so one can determine where the hare is caught, but not when. In this case, the problem is O-(a, ra) with some headstart and one asks for the distance to overtaking, but not the time. See: Chiu Chang Suan Ching; Zhang Qiujian; Alcuin; Fibonacci; Yang Hui; BR; Bartoli; Pseudo-dell'Abbaco; AR; The Treviso Arithmetic; Ulrich Wagner; HB.XI.22; Calandri, c1485; Calandri, 1491; Pacioli; Tagliente; Riese; Apianus; van Varenbraken; Cardan; Buteo; Gori; Wingate/Kersey; Lauremberger; Les Amusemens; Euler; Vyse; Hutton, c1780?; Bonnycastle (= Euler); King; Hutton, 1798?; D. Adams, 1801; De Morgan, 1831?; Bourdon; D. Adams, 1835; Hutton Rutherford; Family Friend (& Illustrated Boy's Own Treasury); Anon: Treatise (1850); Brooks; Clark; Haldeman-Julius (in verse).

Cases where leaps differ in both time and distance: Pacioli?; Apian; Cardan; Wingate/Kersey; Lauremberger; Euler; Bonnycastle; King; Hutton, 1798? (= Lauremberger); De Morgan, 1831?; Bourdon (= Lauremberger); Brooks; Todhunter; Mittenzwey; Clark (= Lauremberger);

General versions. Newton.

Versions with geometric progressions. Chiu Chang Suan Ching; della Francesca; Chuquet; Pacioli; Cardan. See also 7.L.

Versions with sum of squares. Simpson.

Circular versions. Aryabhata(?); AR; Wingate/Kersey; Vyse; Pike; Anon: Treatise (1850); Todhunter; Perelman. See also 7.P.6, where problems with more than two travellers in a circle occur.

Versions using negatives. Clairaut; Manning.
Chiu Chang Suan Ching. c 150. (See also Vogel's notes on pp. 126 127.)

Chap. VI.