Sources page probability recreations

Proverbs 11:1. "A false balance is abomination to the Lord: But a just weight is his delight."

Proverbs 16:11. "A just weight and balance are the Lord's: All the weights of the bag are his work."

Proverbs 20:10. "Divers weights, and divers measures, Both of them are alike abomination to the Lord."

Proverbs 20:23. "Divers weights are an abomination unto the Lord: And a false balance is not good."

Hosea 12:7. "He is a chapman, the balances of deceit are in his hand."

Aristotle. Mechanical Questions. c-340. ??NYS -- cited by van Etten.

Pappus. Collection. Book 3. c320. ??NYS - cited by W. Leybourn.

Muhammed. Koran. c630. Translated by J. M. Rodwell, Everyman's Library, J. M. Dent, 1909. Sura LXXXIII -- Those who stint: 1-3. "Woe to those who STINT the measure: Who when they take by measure from others, exact the full; But when they mete to them or weigh to them, minish --". (I saw the following version on a British Museum label, erroneously attributed to Sura LXXX: "Woe be unto those who give short measure or weight.")

Cardan. De Subtilitate. 1550, Liber I, ??NYS. = Opera Omnia, vol. IV, p. 370: Modus faciendi librā, que pondera rerum maiora quàm sunt[?? -- nearly obliterated in the text I have seen] ostendat. Describes a scale with arm divided 11 : 12.

John Wecker. Op. cit. in 7.L.3. (1582), 1660. Book XVI -- Of the Secrets of Sciences: Chap. 20 -- Of Secrets in Arithmetick: Fraud in Balances where things heavier shall seem to be lighter, p. 293. Says such fraud is mentioned by Aristotle.

van Etten. 1624. Prob. 54 (49), pp. 49 50 (73 74). Mentions Aristotle's mechanical questions and cites Archimedes' law of the lever. Discusses example with arm lengths 12 : 11

W. Leybourn. Pleasure with Profit. 1694. Tract. IV, pp. 2-3. Cites Solomon and Pappus' Collections, Book 3. Discusses arm lengths 11 : 10.

Ozanam. 1694. Prob. 4, 1696: 275-276 & fig. 131, plate 46. Prob. 4 & fig. 26, plate 14, 1708: 351 352. Vol. II, prob. 7, 1725: 339 340 & fig. 131, plate 46. Vol. II, prob. 3, 1778: 4-5; 1803: 4-6; 1814: 3-5; 1840: 196-197. Construct a balance which is correct when empty, but gives dishonest weight. This can be detected by interchanging the contents of the two pans. Hutton notes that the true weight is the geometric mean of the two weights so obtained, and that this is close to the average of these two values. Illustrates with weights 16 and 14. The figure is just a picture of a balance and is not informative -- the same figure is also cited for various sets of weights.

Vyse. Tutor's Guide. 1771? Prob. 1, 1793: p. 303; 1799: p. 316 & Key p. 356. Cheese weighs 76 in one pan and 56 in the other. States the general rule with no explanation.

Jackson. Rational Amusement. 1821. Curious Arithmetical Questions. No. 8, pp. 16 & 72. Cheese weighs 16 on one side and 9 on the other. Says the answer is the mean proportional. = Illustrated Boy's Own Treasury, 1860, prob. 12, pp. 428 & 431.

Julia de Fontenelle. Nouveau Manuel Complet de Physique Amusante ou Nouvelles Récréations Physiques .... Nouvelle Édition, Revue, ..., Par M. F. Malepeyre. Librairie Encyclopédique de Roret, Paris, 1850. Pp. 407-408 & fig. 146 on plate 4 (text erroneously says V): Balance trompeuse. A bit like Ozanam, but doesn't indicate the true weight is the geometric mean. Figure copied from Ozanam, 1725.

Magician's Own Book. 1857. The false scales, p. 251. Cheese weighs 9 on one side and 16 on the other. Says the true weight is the mean proportional, hence 12 here. = Book of 500 Puzzles, 1859, p. 65. = Boy's Own Conjuring Book, 1860, p. 223. Almost identical to Jackson.

Hoffmann. 1893. Chap. IV, no. 94: The false scales, pp. 169 & 229 = Hoffmann-Hordern, p. 154. On one side a cheese weighs 9 and on the other it weighs 16. Answer notes that the true weight is always the geometric mean. Almost identical to Jackson and Magician's Own Book.

Clark. Mental Nuts. 1904, no. 66; 1916, no. 84. The grocer puzzled. Weights of 8 and 18. Answer says to solve 8 : x :: x : 18.

Briggs & Bryan. The Tutorial Algebra -- Part II. Op. cit. in 7.H. 1898. Exercises X, pp. 125 & 580. Weighing one way gains an extra 11% profit, but weighing the other way gives no profit at all. What is the legitimate profit?

Pearson. 1907. Part II, no. 72, pp. 128 & 205. Same problem as Hoffmann. Answer says to take the square root of 9 x 16.

Ernest K. Chapin. Loc. cit. in 5.D.1. 1927. Prob. 1, p. 98 & Answers p. 10: The druggist's balance. One arm is longer than the other, but the shorter is weighted to give balance when the scales are empty. He uses the two sides equally -- does he gain or lose (or come out even)? If the lengths are L and l, then balancing against weight W will result in WL/l and Wl/L equally often and the arithmetic mean of these is greater than their geometric mean of W, so the druggist is losing. Answer only does the example with L = 2l.

Loyd Jr. SLAHP. 1928. The jeweler's puzzle, pp. 21 22 & 90. More complex version.

Kraitchik. La Mathématique des Jeux. Op. cit. in 4.A.2. 1930. Chap. II, prob. 26, p. 34. = Mathematical Recreations; op. cit. in 4.A.2; 1943; Chap. 2, prob. 54, pp. 41 42. Merchant has a false balance. He weighs out two lots by using first one side, then the other. Is this fair on average?

10.Q. PUSH A BICYCLE PEDAL
Holding a bicycle upright, with the pedals vertical, push the bottom pedal backward. What happens?
Pearson. 1907. Part II, no. 17: A cycle surprise, pp. 14 & 189.

Ernest K. Chapin. Loc. cit. in 5.D.1. 1927. P. 99 & Answers p. 11. Pull the bottom pedal forward. What happens? What is the locus of the pedal in ordinary travel? Answer says it is a cycloid but it is actually a curtate cycloid.

W. A. Bagley. Paradox Pie. Op. cit. in 6.BN. 1944. No. 60: Another cycling problem, p. 46. Uses a tricycle -- which simplifies the experimental process.

David E. Daykin. The bicycle problem. MM 45:1 (Jan 1972) 1. Short analysis of this 'old problem'. No references.

10.R. CLOCK HAND PROBLEMS
These are questions as to when the hands can meet, be opposite, be interchangeable, etc. There are also questions with fast and slow clocks. Many examples are in 19C arithmetic and algebra books and in Loyd, Dudeney, etc. I am somewhat surprised that my earliest examples are 1678? and 1725, as clocks with minute hands appeared in the late 16C. The problems here are somewhat related to conjunction problems -- see 7.P.6.

See also 5.AC for digital clocks, which are a combinatorial problem.

These are related to Section 7.P.6.
Wingate/Kersey. 1678?. Quest. 20, p. 490. Clock with an hour hand and a day hand, which goes round once every 30 days. They are together. When are they together again? In 30 days, the faster hand must pass the slower 59 times, so the time between coincidences is 30/59 of a day.

Ozanam. 1725. Prob. 11, question 3, 1725: 76 77. Prob. 2, 1778: 75-76; 1803: 77-78; 1814: 69; 1840: 37. When are the hands together? 1725 does it as a geometric progression, like Achilles and the tortoise. 1778 adds the idea that there are 11 overtakings in 12 hours, but this does not appear in the later eds.

Les Amusemens. 1749. Prob. 122, p. 264. When are clock hands together?

Vyse. Tutor's Guide. 1771? Prob. 7, 1793: p. 304; 1799: p. 317 & Key pp. 357-358. When are the hands together between 5 and 6 o'clock?

Dodson. Math. Repository. 1775.