Sources page probability recreations

Knot III, pp. 13 18, 90 95. Circular railway with trains going at different frequencies in the two directions. How many are met on a round trip in each direction?

Knot V, pp. 27 28. Comment on Knot III problem.

Knot VIII, pp. 52 53, 55 57, 132 134. Buses in both directions pass depot every 15 minutes. Walker starts from depot at same time as a bus and meets a bus in 12½ minutes. When is he overtaken by a bus?

Laisant. Op. cit. in 6.P.1. 1906. Chap. 49: Du Havre à New-York, pp. 123-125. Definitely asserts that this problem was posed by Lucas at a meeting 'longtemps déjà'. Boats leave every noon each way between le Havre and New York and take exactly seven days to make the trip -- how many do they pass?

Pearson. 1907. Part II, no. 106, pp. 136 & 212 213. Tube trains run every 2 minutes. How many are met in a 30 min journey? Answer: 30.

Peano. Giochi. 1924. Prob. 11, p. 4. Buses take 7 minutes from the centre to the terminus. Buses leave from the centre and from the terminus every minute. How many buses are met in going one way? Observes that it takes 14 buses to run such a service and so one bus meets the other 13.

McKay. At Home Tonight. 1940. Prob. 14: Passing the trains, pp. 65 & 79. 12 hour journey and trains start from the other end every hour.

Sullivan. Unusual. 1943. Prob. 24: Back in the days of gasoline, tires, and motorists. 4½ hours from Chicago to Indianapolis, buses leaving every hour [on the hour].

Anonymous. The problems drive, 1956. Eureka 19 (Mar 1957) 12-14 & 19. No. 10. Circular line with trains departing each way every 15 minutes, but the east bound trains take 2 hours for the circuit while the west bound ones take 3 hours for a circuit. Starting after one leaves the station, how many trains does one see?

Doubleday - 1. 1969. Prob. 36: Traveler's tale, pp. 48 & 162. = Doubleday - 5, pp. 55-56. Trains going between Moscow and Paris, taking seven days.

[Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. 1973. Op. cit. in 5.E. How many buses, pp. 7-8.

10.A.3. TIMES FROM MEETING TO FINISH GIVEN
Simpson. Algebra. 1745. Section XI (misprinted IX in 1790), prob. XLVI, pp. 110-111 (1790: prob. LIX, pp. 111 112). Travellers set out from each of two cities toward the other, at the same time. After meeting, they take 4 and 9 hours to finish their journeys. How long did they take? He gives a general solution -- if x is the time before meeting and a, b are the times from meeting to finishing, then x² = ab. [I have seen a 20C version where only the ratio of velocities is asked for -- indeed I used it in one of my puzzle columns, before I knew that the times could be found.]

Dodson. Math. Repository. 1775.

P. 67, Quest. CXXV. Travellers set out from London and York at the same time. When they meet, they observe that A had travelled 30 miles more than B and that A expected to reach York in 4 days and B expected to reach London in 9 days. What is the distance between London and York?

P. 68, Quest. CXXVII. Travellers set out from London and Lincoln at the same time. When they meet, they observe that A had travelled 20 miles more than B and that A travelled in 6⅔ days as much as B had gone and B expected to get to London in 15 days. What is the distance from London to Lincoln?

Ozanam Montucla. 1778. Supplement, prob. 42, 1778: 432; 1803: 425; 1814: 360; 1840: 186. Couriers set out toward one another from cities 60 apart. After meeting, they take 4 and 6 hours to reach their destinations. What are their velocities?

Thomas Grainger Hall. The Elements of Algebra: Chiefly Intended for Schools, and the Junior Classes in Colleges. Second Edition: Altered and Enlarged. John W. Parker, London, 1846. P. 136, ex. 38. Simpson's problem with cities of London and York and times of 9 and 16.

T. Tate. Algebra Made Easy. Op. cit. in 6.BF.3. 1848. P. 89, no. 4. Same as Simpson.

Todhunter. Algebra, 5th ed. 1870. Examples XXIV, nos. 20 & 22, pp. 211-212 & 586.

No. 20. Simpson's problem between London and York, with times 16 & 36.

No. 22. = Dodson, p. 67.

William J. Milne. The Inductive Algebra .... 1881. Op. cit. in 7.E. No. 132, pp. 306 & 347. Travellers between London and York reach their destinations 25 & 36 hours after meeting. How long did each take?

W. W. Rouse Ball. Elementary Algebra. CUP, 1890 [the 2nd ed. of 1897 is apparently identical except for minor changes at the end of the Preface]. Ex. 5, p. 231. A and B are 168 miles apart. Trains leave each end for the other starting at the same time. They meet after 1 hour 52 minutes. The train from A reaches B half an hour before the other reaches A.

Haldeman-Julius. 1937. No. 133: Racers' problem, pp. 15 & 28. Racers on a circular track start in opposite directions. After meeting, they take 4 and 9 minutes to pass the starting point, but they continue. When do they meet each other at the starting point?

Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. Prob. b, pp. 188-190. Trains start toward each other at 7 am; one takes 8 hours, the other takes 12. When do they meet?

Shakuntala Devi. Puzzles to Puzzle You. Op. cit. in 5.D.1. 1976. Prob. 37: Walking all the way, pp. 29 & 106-107. Similar to Simpson, but they start at given times and the time of meeting is given and they get to their destination at the same time. This is thus the same as Simpson if it is considered with time reversed. The elapsed times to meeting are 25/12 and 49/12 hours.

Birtwistle. Calculator Puzzle Book. 1978.

Prob. 60: Meeting point, pp. 42 & 102. Travellers start at the same time from opposite ends of a journey. When they meet, they find that the first has 11 1/5 hours to go, while the other has 17 1/2. Also the first has travelled 7 miles further than the second. How long is the journey?

Prob. 71: Scheduled flight, pp. 50-51 & 108. Planes A and B start toward each other at 550 mph. Five minutes later, plane C starts from the same place as A at 600 mph. It overtakes A and then meets B 36 minutes later. When does A land?

10.A.4. THE EARLY COMMUTER
A man is usually met by a car at his local train station, but he arrives A minutes early and begins walking home. The car meets him and picks him up and they arrive home B minutes early. How long was he walking? The car's trip is B/2 minutes shorter each way, so the commuter is met B/2 minutes before his usual time and he has been walking A - B/2 minutes. New section -- there must be older examples.
M. Adams. Puzzles That Everyone Can Do. 1931. Prob. 155, pp. 61 & 151: Catching the postman. Man driving 10 mph usually overtakes the postman walking 4 mph at the same point every morning. One morning, the man is four minutes late and he overtakes the postman half a mile beyond the usual point. Was the postman early or late, and by how much? (Note that the man is four minutes late starting, not in overtaking.)

Meyer. Big Fun Book. 1940. No. 10, pp. 162-163 & 752. A = 60, B = 10.

Harold Hart. The World's Best Puzzles. Op. cit. in 7.AS. 1943. The problem of the commuter, pp. 8 & 50. A = 60, B = 20.

The Little Puzzle Book. Op. cit. in 5.D.5. 1955. Pp. 59-60: An easy problem. A = 60, B = 20, he walks at 4 mph. How fast does the chauffeur drive?

Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. Prob. f, pp. 189 & 191. A = 60, B = 16. The time of arrival is also given, but is not needed.

Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. The Commuter's tale, pp. 87-88 & 136. A = 60, B = 10.

10.A.5. HEAD START PROBLEMS
Doubleday - 2 gives a typical example. New section -- I have seen other examples but didn't record them.

Gardner, in an article: My ten favorite brainteasers in Games (collected in Games Big Book of Games, 1984, pp. 130-131) says this is one of his favorite problems. ??locate

Todhunter. Algebra, 5th ed. 1870. Examples XIII, no. 23, pp. 103 & 578. In racing a mile (= 1760 yds), A gives B a headstart of 44 yd and wins by 51 sec; but if A gives B a headstart of 75 sec, A loses by 88 yd. Find the times each can run a mile.

Charles Pendlebury. Arithmetic. Bell, London, (1886), 6th ed, 1893. Section XXXV (b): Races and games of skill, pp. 267-268, examples XXXV (b) & answers, p. viii. Does an example: A can give B 10 yards in 100 yards and A can give C 15 yards in 100 yards. How much should B give C in 150 yards? He gives 11 similar problems. On pp. 363-364 & answers, part II, p. xix, he gives some further problems, 45-48.

W. W. Rouse Ball. Elementary Algebra. CUP, 1890 [the 2nd ed. of 1897 is apparently identical except for minor changes at the end of the Preface].