7. arithmetic & number theoretic recreations a. Fibonacci numbers

136. (10, 30, 50) yielding 10 each. He gives the solution with prices 3 and 1/7. There are (25, 16) solutions.

137. (11, 33, 55) yielding 11 each. He gives the solution with prices 2 and 1/6. There are (30, 20) solutions.

138. (16, 48, 80) yielding 16. He gives the solution with prices 3 and 1/11. There are (64, 49) solutions.

139. (10, 20, ..., 90), yielding 100. He gives the unique positive solution. There are (3, 1) solutions.

Bachet. Problemes. 1612. Prob. XXI, 1612: 106-115. Prob. XXIV, 1624: 178-186; 1884: 122 126. He gives a new "general and infallible rule", which is fairly general -- Glaisher says it produces a selection of the solutions. He applies the idea to (20, 30, 40), exhibiting 4 solutions. It has (100, 81) solutions. Cf. van Etten. 1612 also does (20, 25, 32).

In the 5th ed., the general material is dropped and replaced by some vague algebra. Labosne gives two solutions for (18, 40), but one of them uses fractions. It has (171, 36) solutions. He then considers (18, 40, 50) and gives one fractional solution -- there are (3, 1) solutions. He then makes some discussion of (10, 12, 15) (which has (7, 4) solutions) and (31, 32, 37) (which has (70, 60) solutions).

van Etten. 1624. Prob. 69 (62), pp. 64 65 (90 91). (20, 30, 40). Gives one solution with prices 3 and 1. There are (100, 81) solutions. Cf. Bachet. Henrion's 1630 Notte, p. 22, states that Bachet found many other solutions and gives a solution with prices 2  & 7.

Hunt. 1651. Pp 282-283: Of three women that sold apples. (20, 30, 40). Gives one solution with prices 1 and 3.

Ozanam. 1694. Prob. 24, 1696: 77-80; 1708: 68 70. Prob. 28, 1725: 201 210. Prob. 12, 1778: 199-204; 1803: 196-201; 1814: 170-174; 1840: 88-90. (10, 25, 30). Glaisher describes the material in the 1696 ed. and says that Ozanam first considers the same general form that Bachet considered and then applies it to the example. Glaisher indicates that Ozanam's and Bachet's methods are essentially the same, but Ozanam certainly gets all solutions, while I am not sure that Bachet can do so. 1696 gives two solutions at prices 7 and 2 and at prices 6 and 1. 1725 et seq. gives a general method and finds all 10 solutions of the specific problem. (Glaisher notes that the Remarques on pp. 203 210 are new to the 1723 ed. They give an algebraic form of the solution.) 1725 refers to the second part of Arithmétique Universelle, p. 456 (more specifically identified as by M. de Lagny [1660-1734] in 1778 et seq.), where 6 solutions are found. 1725 says there are 10 solutions and 1778 says de Lagny is mistaken -- but in fact, there are (10, 6) solutions and de Lagny probably meant just the positive ones. 1778 drops the preliminary general general form.

Amusement for Winter Evenings. A New and Improved Hocus Pocus; or Art of Legerdemain: Explaining in a Clear and Comprehensive Manner Those Apparently Wonderful and Surprising Tricks That are performed by Slight of Hand and Manual Dexterity: Including Several Curious Philosophical Experiments. M. C. Springsguth, London, nd [c1800 -- HPL], 36pp. Pp. 22-23: Of three sisters. (10, 16, 22) sold at 7 a penny and then a penny apiece, i.e. prices 1/7 and 1, each earning 4.

Bestelmeier. 1801. Item 718: Das Eyerverkauf. Three women sell different numbers of eggs and make the same. Further details not given.

Jackson. Rational Amusement. 1821. Curious Arithmetical Questions. No. 14, pp. 17 & 74. (10, 30, 50). Gives the solution with prices 3 and 1/7.

John Badcock. Domestic Amusements, or Philosophical Recreations. Op. cit. in 6.BH. [1823]. P. 155, no. 192: Trick in reasoning. (10, 30, 50) -- gives the solution with prices 3 and 1/7.

Rational Recreations. 1824. Exer. 21, p. 98. (10, 30, 50) eggs.

Boy's Own Book. 1843 (Paris): 341. "Three country-women and eggs." (10, 30, 50) -- gives the solution with prices 3 and 1/7. = Boy's Treasury, 1844, p. 299. = de Savigny, 1846, p. 289: Les trois paysannes et les œufs.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 564-21, pp. 254 & 395-396: Die Blumenmädchen. (27, 29, 33). This has (117, 100) solutions -- she gives one with prices ⅓ and 1, each earning 13.

Hugh Rowley. More Puniana; or, Thoughts Wise and Other-Why's. Chatto & Windus, London, 1875. P. 179. Man with three daughters and lots of apples -- (10, 30, 50). Solution with prices 3 and 1/7.

Mittenzwey. 1880. Prob. 122, pp. 25 & 76; 1895?: 140, pp. 29 & 79; 1917: 140, pp. 27 & 77. Three sisters selling bunches of violets. (27, 29, 33). This has (117, 100) solutions -- he gives one with prices ⅓ and 1, each earning 13. = Leske.

Hoffmann. 1893. Chap. IV, pp. 160 & 215-216 = Hoffmann-Hordern, p. 140.

No. 65: The three market women. (27, 29, 33). c= Leske.

No. 66: The farmer and his three daughters. (10, 30, 50). He gives the solution with prices 3 and 1/7.

Dudeney?? Breakfast Table Problems No. 328: "How were the oranges sold". Daily Mail (27  & 28 Jan 1905) both p. 7. (20, 30, 40, 50, 60). Gives the solution with prices 1  and 1/11. There are (45, 36) solutions.

Pearson. 1907. Part II, no. 37, pp. 121 & 199. (27, 29, 33) c= Leske.

J. W. L. Glaisher. On certain puzzle-questions occurring in early arithmetical writings and the general partition problems with which they are connected. Messenger of Mathematics 53 (1923-24) 1-131. Discusses the versions in Blasius, Widman, Tagliente and attempts to explain the methods used. Unfortunately he is rather prolix and I often get lost in the many examples and special cases, but he seems to have general solutions. On p. 12, he mentions that Tagliente's problem could be extended to (20, 40, ..., 140) -- cf. Gould below. On p. 77, he says more results will appear in a later paper -- check index of Messenger??

A. A. Krishnaswami Ayyangar. A classical Indian puzzle-problem. J. Indian Math. Soc. 15 (1923-24) 214-223. Responding to Glaisher, he analyses the Indian version, obtaining a simple complete solution with two parameters having an infinite range and a third parameter being bounded. I found this a bit confusing since he sometimes uses price to mean the number of items per unit cost. He says many of the solutions are not in Glaisher's system given on p. 19, but I can't tell if Glaisher intends this to be a complete solution.

Rupert T. Gould. The Stargazer Talks. Geoffrey Bles, London, 1944. A Few Puzzles -- write up of a BBC talk on 10 Jan 1939, pp. 106-113. Seven applewomen with 20, 40, 60, 80, 100, 120, 140. One solution with prices 3 and 1/7 -- cf. Glaisher. There are (27, 21) solutions.

McKay. At Home Tonight. 1940. Prob. 16: Extraordinary sales, pp. 65 & 81. (7, 18, 29) eggs. This has (12, 6) solutions. He asks for a solution where all make 10d. He gives one solution with prices 1/4 and 3 and selling as many eggs in batches of 4 as possible.

W. T. Williams & G. H. Savage. The Penguin Problems Book. Penguin, 1940. No. 75: A falling market, pp. 43 & 128. Cauliflowers (19, 25, 27) with each making 85d, both prices being integral and each sells some at the lower price, not to be less than 2d. Actually there is only one solution with integral prices and each making 85d.

Philip E. Bath. Fun with Figures. Op. cit. in 5.C. 1959. No. 45: Potatoes for sale, pp. 19 & 47. (60, 63, 66). This has (900, 841) solutions. He wants solutions where all make 9/6 = 114d. There are two solutions with integral prices, but he gives the solution with prices 2d and 12/7 d, i.e. 7 for 1s, but he doesn't require maximum numbers of batches of 7 to be sold.

David Singmaster. Some diophantine recreations. In: Papers Presented to Martin Gardner on the occasion of the opening of the exhibition: Puzzles: Beyond the Borders of the Mind at the Atlanta International Museum of Art and Design; ed. by Scott Kim, 16 Jan 1993, pp. 343-356 AND The Mathemagician and Pied Piper A Collection in Tribute to Martin Gardner; ed. by Elwyn R. Berlekamp & Tom Rodgers;. A. K. Peters, Natick, Massachusetts, 1999, HB, pp. 219-235. Sketches some history, gives complete solutions for the Western and Indian cases (filling a gap in Ayyangar) and finds a new simple formula for the number of solutions in the Western case.
7.P.6. CONJUNCTION OF PLANETS, ETC.
See Tropfke 642.

Some overtaking problems in 10.A take place on a circular track and are related to or even identical to these problems. In particular, if two persons start around an island of circumference D, from the same point and in the same direction at rates a, b, this is the same as O-(a, b; D) of Section 10.A. This is easily adapted to dealing with different starting points and going in opposite directions (which gives a meeting problem). Clock problems, 10.R, are also related to these.

Sun Zi. Sun Zi Suan Ching. Op. cit. in 7.P.2. 4C. ??NYS. Sisters come home every 5, 4, 3 days, when do they all come together?. (Mikami 33 gives English.)

Zhang Qiujian. Zhang Qiujian Suan Jing. Op. cit. in 7.E. 468, ??NYS -- translated on p. 139 of: Shen Kangsheng; Mutual subtraction algorithm and its applications in ancient China; HM 15 (1988) 135 147. Circular route of length 325. Three persons start with speeds 150, 120, 90. When do they all meet at the start?

Brahmagupta. Brahma sphuta siddhanta. 628. Several of the sections of Chap. XVIII discuss astronomical versions of this problem, but with complicated values and unclear exposition.

Gherardi. Libro di ragioni. 1328. P. 47. Two men start going in a circuit. One can do it in 4 days, the other in 5½ days. When do they meet again? Same as O-(1/4, 1/5½), D = 1, in the notation of Section 10.A.

AR. c1450. Prob. 148, pp. 72, 164 165, 214. Though titled 'De planetis' and described as conjunction by Vogel, this is really just an overtaking problem -- see 10.A.

H&S 74 75 says sun and moon problems are in van der Hoecke and Trenchant (1556).

Cardan. Practica Arithmetice. 1539. Chap. 66, sections 20-24, ff. CC.vii.v - DD.i.r (pp. 142 143). Several versions concerning conjunctions of planets, including irrational ratios and three planets. Examples with periods: 7, 5; 18, 30; 8, 20; 1000, 999; and 5, 4, 3. In section 23, he gives periods of Saturn and Jupiter as 30 & 12 years and periods of Jupiter and Mars as 144 & 23 months. (H&S 75 gives English and some of the Latin.)

Cardan. Opus Novum de Proportionibus Numerorum. Henricpetrina, Basil, 1570, ??NYS. = Opera Omnia, vol. IV, pp. 482-486. General discussion and examples, e.g. with periods 2, 3, 7.

Wells. 1698.