Sources page probability recreations

P. 147, Quest. CCXXXIV. When are hour and minute hands together?

P. 147, Quest. CCXXXV. When are hour, minute and second hands together?

Charles Hutton. A Complete Treatise on Practical Arithmetic and Book-keeping. Op. cit. in 7.G.2. [c1780?] 1804: prob. 35, p. 136. When are watch hands together between 4 and 5?

Pike. Arithmetic. 1788. P. 352, no. 27. When are the clock hands together next after noon?

Bonnycastle. Algebra. 1782.

P. 86, no. 22. When are the hands next together after 12:00?

P. 201, no. 1 (1815: p. 226, no. 1). When are the hands together between 8:00 and 9:00?

Eadon. Repository. 1794. P. 195, no. 13. When do the hands meet after noon?

Hutton. A Course of Mathematics. 1798? Prob. 36, 1833: 223; 1857: 227. When do the hands next meet after noon?

Robert Goodacre. Op. cit. in 7.Y. 1804. Miscellaneous Questions, no. 128, p. 205 & Key p. 270. When are clock hands next together after 12:00?

Silvestre François Lacroix. Élémens d'Algèbre, a l'Usage de l'École Centrale des Quatre-Nations. 14th ed., Bachelier, Paris, 1825. Section 82, ex. 6, p. 122. When are clock hands together after noon? Notes that this is related to overtaking problems.

Bourdon. Algèbre. 7th ed., 1834. Art. 57, prob. 10, p. 85. When are clock hands together?

D. Adams. New Arithmetic. 1835. P. 244, no. 83. When are hands next together after 12:00? Observes that the minute hand gains 11 spaces (i.e. hour marks) per hour on the hour hand.

Augustus De Morgan. Examples of the Processes of Arithmetic and Algebra. Third, separately paged, part of: Library of Useful Knowledge -- Mathematics I, Baldwin & Craddock, London, 1836. P. 90. How much time elapses between times when the hands are together? Suppose one hand revolves in a hours, the other in b hours -- how long between conjunctions? In the latter case, how long does it take the faster hand to gain p/q of a whole revolution on the other?

Hutton-Rutherford. A Course of Mathematics. 1841?

Prob. 11, 1857: 81. When are clock hands next together after 12:00?

Prob. 19, 1857: 82. When are clock hands together between 5 and 6 o'clock?

T. Tate. Algebra Made Easy. Op. cit. in 6.BF.3. 1848.

Pp. 46-47, no. 19. When are the hands together between 2:00 and 3:00?

P. 46, no. 20. Same between 3:00 and 4:00?

Anonymous. A Treatise on Arithmetic in Theory and Practice: for the use of The Irish National Schools. 3rd ed., 1850. Op. cit. in 7.H.

P. 356, no. 4. When are hands together between 5 and 6?

P. 360, no. 46. When are hands together between 1 and 2?

Dana P. Colburn. Arithmetic and Its Applications. H, Cowperthwait & Co., Philadelphia, 1856. Miscellaneous Examples.

No. 12, p. 359. When are hands together after 12:00?

No. 13, p. 359. When are hands opposite after 12:00?

No. 32, p. 361. When is the hour hand as much after 12 as the minute hand is before 1?

Todhunter. Algebra, 5th ed. 1870.

Examples X, no. 35, p. 87 & 577. Between 1:00 and 2:00, when is the minute hand one minute in front of the hour hand?

Miscellaneous Examples, no. 17, pp. 546 & 604. When are clock hands together between 9:00 and 10:00?

Colenso. Op. cit. in 7.H. These are from the (1864), 1871 material.

No. 3, p. 186. When are hour and minute hands at right angles, or coincident, between 5:00 and 6:00?

No. 21, pp. 190 & 215. When are hour and minute hands 162^o apart between 11:00 and 12:00?

Problem 203. Zeitschrift für mathematische und naturwissenschaftlichen Unterricht 15 (1874) 197-198. Interchanging clock hands. Taken from Journ. élém., which must be a French journal, but I don't recognise it. Shows that interchanging hands gives a legal appearance when the hour hand is at 60k/143 minutes, k = 0, ..., 143. Copy sent by Heinrich Hemme.

William J. Milne. The Inductive Algebra .... 1881. Op. cit. in 7.E.

No. 95, p. 140. When are hands together after 2:00?

No. 96, pp. 140 & 332. When are hands together after 5:00?

No. 97, pp. 141 & 332. When are hands together after 8:00?

No. 98, pp. 141 & 332. When are hands at 180^o after 4:00?

No. 99, pp. 141 & 332. When are hands at 180^o after 5:00?

No. 100, pp. 141 & 332. When are hands at 90^o after 6:00?

No. 101, pp. 141 & 332. When are hands at 90^o after 8:30?

No. 104, pp. 303 & 347. When are hands at 180^o after 11:00?

Carroll-Wakeling. 1888 to 1898. Prob. 14: Looking-glass time, pp. 17 & 66-67. This was on one of the undated typed sheets Carroll sent to Bartholomew Price. "A clock face has all the hours indicated by the same mark, and both hands the same in length and form. It is opposite to a looking-glass. Find the time between 6 and 7 when the time as read direct and in the looking-glass shall be the same." This seems to be the first example of looking at the mirror image of a clock. This can also occur by reading a transparent clock from the wrong side. Mentioned in Carroll-Gardner, p. 53. Wakeling has sent me a copy of the typescript.

Anon. Prob. 80. Hobbies 31 (No. 795) (7 Jan 1911) 350 & (No. 798) (28 Jan 1911) 412. Three pendulum clocks, supposed to have one second swings. One gains 5 minutes per day, the second loses ten minutes per day and the third is correct. Assuming we start all three pendula at the same time at the same end of their swing, when will they all be together at this point again? Answer: 9 min 36 sec = 1/150 of a day.

Ernest K. Chapin. Loc. cit. in 5.D.1. 1927. Prob. 5, p. 89 & Answers p. 8. Can the three hands make equal angles? [Last line of the answer has slipped into the next column.]

W. B. Campbell, proposer; W. E. Buker, solver. Problem ??. AMM 41 (Sep 1934) 447 & 42 (Feb 1935) 110-111. Interchanging clock hands. ??NYS - information sent by Heinrich Hemme.

Perelman. 1937. MCBF. Interchanging the hands of a clock, prob. 141, pp. 235-238. Says the following was posed to Einstein by his friend A. Moszkowski: at what times can one exchange the hour and minute hands and get a valid position of the hands? There are 143 solutions, of which 11 are the positions where the hands coincide, which Perelman discusses as prob. 142, pp. 238-239.

Sullivan. Unusual. 1943. Prob. 1: Watch and see. How many places do hands meet?

Anonymous. Problems drive, 1957. 20 (Oct 1957) 14-17 & 29-30. No. 10. B reckons A's clock gains one minute per day. C reckons B's clock gains one minute per day. They synchronise watches. Assuming rates are constant, when do all three agree again?

Pierre Berloquin. The Garden of the Sphinx, op. cit. in 5.N. 1981.