7. arithmetic & number theoretic recreations a. Fibonacci numbers

No. 20, pp. 19 & 76. Problem B-9 (= Euler).

Rational Recreations. 1824. Exer. 11, p. 56. A-7 (= Bhaskara I). Answer: 301.

Unger. Arithmetische Unterhaltungen. 1838. Pp. 145-172 & 259-262, nos. 541-645. He treats the topic of 7.P.1 and 7.P.2 at great and exhausting length, starting with solving ax = by, then x + y = c, then ax + y = c, then ax = by + 1 and ax = by + c, also phrased as ax  c (mod b). On pp. 156-158 & 260, nos. 588-597, are Chinese Remainder problems, but with just two moduli, so I will not describe them.

The New Sphinx. c1840. No. 45, pp. 24 & 121. Problem A-7 in verse, with bunch of walnuts.

Magician's Own Book. 1857. The basket of nuts, pp. 245-246. Problem A-7 (= Bhaskara I), general solution. = Book of 500 Puzzles, 1859, pp. 59-60. = Boy's Own Conjuring Book, 1860, p. 217.

Illustrated Boy's Own Treasury. 1860. Prob. 14, pp. 428 & 431. Problem D-5, one solution (= Baker).

Todhunter. Algebra, 5th ed. 1870. Examples XLVI, nos. 22-24, pp. 394 & 596.

No. 22. n  21 (28), 17 (19).

No. 23. n  -1 (3, 5, 7).

No. 24. n  13 (28), 2 (19), 7 (15). This is an example of calculating the Julian year, but he does not state this.

Wehman. New Book of 200 Puzzles. 1908. The basket of nuts, p. 55. A-7 (= Bhaskara I) = Magician's Own Book, except that a line of the statement has been dropped by the typesetter, making the problem unintelligible.

Ripley's Puzzles and Games. 1966.

P. 15, repeated on p. 66. Problem C-10, with solution 14,622,047,999. Though a correct answer, I cannot fathom how this was obtained.

P. 16. Problem B-10, asking for smallest example. Answer is 2520, which is the smallest positive example.

P. 68. n  1 (2, ..., 10). Answer: 2521.

7.P.3. ARCHIMEDES' CATTLE PROBLEM
Archimedes? Letter to Eratosthenes, c 250.

Greek text, with commentaries, first published by Gotthold Ephraim Lessing in Beiträge zur Geschichte und Literatur 1 (1773) 421 ???.

Archimedes. Opera Omnia. Ed. by. J. L. Heiberg. 2nd ed., vol. II, Teubner, 1913. Problema Bovinum, pp. 527 534. Heiberg gives the same classical references as Dijksterhuis (below), cites Lessing as the first editor of the problem (from Gud. Graec. 77, f. 415v), gives the later commentators and editors and says the problem also appears in Cod. Paris Gr. 2448, f. 57. He then gives the Greek and a Latin translation.

The first edition of Heiberg's edition is the basis of:

T. L. Heath. The Works of Archimedes (CUP, 1897) + The Method of Archimedes (CUP, 1912); reprinted in one volume by Dover, 1953. The Cattle Problem, pp. 319 326, discusses the problem and the attempts at solving it.

Dijksterhuis, p. 43, says Lessing's article occurs in the Zweiter Beitrag (1773), 2nd ed., Braunschweig, 1773 (??). = Sämtliche Schriften; ed. by K. Lachmann, vol. IX, p. 285+; 3rd ed., much corrected by F. Muncker, Leipzig, 1897, vol. XIII (or 12??), pp. 99-115.

English verse version in H. Dorrie; 100 Great Problems of Elementary Mathematics; Dover, 1965, pp. 5 6. Dorrie also cites the 19C historians on the question of authenticity.

Greek and English in SIHGM II 202-207. Thomas notes that the epigram is unlikely to have been actually written by Archimedes. SIHGM I 16-17, is a Scholium to Plato's Charmides 165 E which states "logistic ... treats on the one hand the problem called by Archimedes the cattle-problem" and Thomas gives some of the standard references in a note.

English translation in D. H. Fowler; Archimedes Cattle Problem and the Pocket Calculating Machine; Preprint, 1980, plus addenda. (Based on SIHGM.)

Dijksterhuis. Archimedes. Op. cit. in 6.S.1. P. 398 gives some classical references to the problem: a scholium to Plato's Charmides; Heron; two references in Cicero -- all ??NYS.

T. L. Heath. Diophantos of Alexandria. Op. cit. as Diophantos. 1910. Pp. 121 124 discusses the problem.
J. F. Wurm. Review of J. G. Hermann's pamphlet: De archimedis problemate bovino; Leipzig, 1828. In: Jahn's Jahrbücher für Philologie und Pädagogik 14 (1830) 195-??. ??NYS -- cited by Archibald. Solves the easier interpretation, getting 5,916,837,175,686 cattle in all.

B. Krumbiegel. Das problema bovinum des Archimedes. Zeitschrift für Mathematik und Physik -- hist.-litterar. Abt. 25 (1880) 121-136. ??NYS -- cited by Archibald. Survey of earlier historical work.

A. Amthor. ???. Ibid., pp. 153-171. ??NYS -- cited by Archibald. Survey of the mathematics.

H. E. Licks. Op. cit. in 5.A. 1917. Art. 54: The cattle problem of Archimedes, pp. 33 39. discusses the work of Amthor & A. H. Bell (AMM (May 1895), ??NYS), who started the calculation of the answers. This is an abridgement of an article by Mansfield Merriman in Popular Science Monthly (Nov 1905) ??NYS, o/o, but omitting the author's name, which leads us to believe that H. E. Licks is a pseudonym of Mansfield Merriman.

[R. C. Archibald.] Topics for club programs -- 14: The cattle problem of Archimedes. AMM 25 (1918) 411-414. He gives a detailed history, but some details vary from Dijksterhuis. He cites Krumbiegel and Amthor as the basic works, Wurm as the first solver with the simpler interpretation and numerous other works.

H. C. Williams, R. A. German & C. R. Zarnke. Solution of the cattle problem of Archimedes. Math. Comp. 19 (1965) 671. Plus comments by D. Shanks, ibid., 686 687. Describes full solution, but doesn't print it.

Harry L. Nelson. A solution to Archimedes' cattle problem & Note. JRM 13 (1980 81) 162 176 & 14 (1981 82) 126. First printed version of the solution -- 206,545 digits. The note clarifies the formulation of the problem.
7.P.4. PRESENT OF GEMS
Bakhshali MS. c7C.

Kaye I 40-42 interprets all of the following as examples of the form described for the third problem, which has some connection with 7.R.1.

Kaye I 40-42; III 170-171, f. 3v. Hoernle, 1886, p. 129; 1888, pp. 33 34, and Gupta describe it fully. Three men have 7 horses, 9 ponies, 10 camels. Each puts in three animals which are then divided equally, which is the same as each giving one to both others. Then they are equally wealthy. Solution gives 42, 28, 24 as values of the animals, though 21, 14, 12 is the smallest integral solution. The original wealths ("capitals") of each merchant are 294, 252, 240 and the final wealth is 262. (Kaye III 170 has 242 for 240.)