7. arithmetic & number theoretic recreations a. Fibonacci numbers



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F. 77v (p.83). Lion, bear and wolf eating a sheep, (4, 2, 1) hours.


Johann Faulhaber. Arithmetischer Wegweiser .... Ulm, 1614. ??NYS. A 1708 ed. is quoted in Hugo Grosse; Historische Rechenbücher des 16. und 17. Jahrhunderts; (1901); reprinted by Sändig, Wiesbaden, 1965, p. 120.

No. 91, p. 228: wolf, sheepdog and dog eating a sheep, (1, 3, 6).

No. 92, p. 229: ship with three sails, (2, 3, 4).


van Etten. 1624. Prob. 83 (76): Du Lyon de Bronze posé sur une fontaine avec cette epigraphe, pp. 94 95 (140). (2, 3, 4, 1/2) = Metrodorus, art. 7.

Georg Meichsner. Arithmetica Historica. Hieronymus Körnlein, Rotenburg/Tauber, 1625. No. 68, p. 209. ??NYS. Quoted in Hugo Grosse, op. cit. under Faulhaber, above, p. 77. Three men with devices to pump out flooded lands in Holland, (60, 30, 20).

Schott. 1674. Ex. 1, pp. 570-571. Cistern: (2, 3, 4, 1/4) done several ways. Cites Clavius for the lion fountain (Metrodorus 7).

Wingate/Kersey. 1678?.


Quest. 4, pp. 476-477. Workmen: (20, 30).

Quest. 5, pp. 477-478. "I am a brazen lion ..." in Latin. (2, 3, 4, 1/2), where 6 hours is counted as 1/2 day, i.e. a day has 12 hours. = Metrodorus, Art. 7.

Quest. 6, pp. 478-479. (1/2, -10/7, -7/3).

Quest. 7, pp. 479-480. Dog, wolf, lion eating a sheep: (1, 3/4, 1/2), but the lion has a head-start of 1/8 hour.

Quest. 12, p. 484. (20, x) in 12. Man and wife drinking beer.

Quest. 13, pp. 484-485. (30, 40, x) in 15. Carpenters building a house.


W. Leybourn. Pleasure with Profit. 1694. Prob. 14, p. 39. Cistern emptying: (6, 4, 3, 2).

Wells. 1698.


No. 105, p. 206. (20, 30) and (a, b).

No. 106, p. 206. (20, x) in 12 and (a, x) in c.


Isaac Newton. Arithmetica Universalis, 1707. ??NYS. English version: Universal Arithmetic, translated by Mr. Ralphson, with revisions and additions by Mr. Cunn, Colin Maclaurin, James Maguire and Theaker Wilder; W. Johnston, London, 1769. (De Morgan, in Rara, 652 653, says there were Latin editions of 1722, 1732, 1761 and Raphson's English editions of 1720 and 1728, ??NYS.) Resolution of Arithmetical Questions, Problem VII, pp. 184 185. "The Forces of several Agents being given, to determine x the Time, wherein they will jointly perform a given Effect d." Gives general approach for three workers. Example is (3, 8/3, 12/5), where the Force of the second is expressed as saying he can do the work "thrice in 8 weeks".

Ozanam. 1725.


Prob. 24, question 9, 1725: 180 181. Prob. 5, 1778: 188-189; 1803: 185-186; 1814: 161-162; 1840: 84. Same as Metrodorus 7, except that a day is considered as 24 hours, so the problem is done as (2, 3, 4, ¼).

Prob. 24, question 10, 1725: 181. (6, 9, 12) months to print a book.


Simpson. Algebra. 1745. Section XI (misprinted IX in 1790).

Prob. XI, pp. 83-84 (1790: prob. XXX, pp. 91 92). General problems, (a, b) and (a, b, c). Example: (10, 12, 16).

Prob. XXXI, pp. 90-92 (1790: prob. XXXVI, pp. 96 98). General problem: given (x, y) in a, (x, z) in b, (y, z) in c, determine x, y, z. Example with a, b, c   =  8, 9, 10.


Alexis-Claude Clairaut. Elémens d'Algèbre. 1746. Vol. I, art. LVI (my source quotes from the 6th ed. of 1801). ??NX. He uses the context of this type of problem to study the meaning of negative solutions. A cistern of size a is filled by a source running for time b together with another source running for time c. Another reservoir of size d is filled by the sources in times e and f. Determine the rate of each source.

Les Amusemens. 1749.


Prob. 173, p. 321. Cistern: (9, -12).

Prob. 174, pp. 322-323. Reservoir with three nymphs: (3, 4, 5).


"By his Holiness the Pope". The Gentleman and Lady's Palladium (1750) 22. Qn. 11. (??NYS, cited by E. H. Neville; Gleaning 1259: On Gleaning 1146; MG 23 (No. 254) (May 1939) 149. "If a Cardinal can pray a soul out of purgatory ...." See Welch, 1833, below.

Dictator Roffensis, proposer; Steph. Hodges & Will. Smith, solvers. Ladies' Diary, 1750-51 = T. Leybourn, II: 45-46, quest. 334. [??NX of p. 46.] Three drinkers: (10, 12, 15) for 12 hour days, how long together for 10 hour days.

Arthur Young. Rural Oeconomy: or, Essays on the Practical Parts of Husbandry. Dublin, 1770, p. 32. ??NYS - described in: Keith Thomas; Children in early modern England; IN: Gillian Avery & Julia Briggs; Children and Their Books A Celebration of the Work of Iona and Peter Opie; OUP, (1989), PB ed, 1990, pp. 45-77, esp. pp. 66 & 76. Proverb: one boy, one day's work; two boys, half a day's work; three boys, no work at all.

Euler. Algebra. 1770. I.IV.III: Questions for practice.


No. 14, pp. 204 205. Cistern, (x, 20) in 12.

No. 22, p. 205. Same as the example in Simpson's prob. XXXI


Vyse. Tutor's Guide. 1771?

Prob. 61, 1793: p. 69; 1799: p. 74 & Key p. 100. Two workers, (10, 13).

Prob. 62, 1793: p. 69; 1799: pp. 74-75 & Key p. 100. Boatbuilders, (18, x) in 11.

Prob. 6, 1793: p. 128; 1799: p. 136 & Key p. 178. Cistern emptying, (1, 2, 3,). Gives volume of cistern but never uses it. = Metrodorus 133.

Prob. 15, 1793: p. 156; 1799: p. 167 & Key p. 208. Builders, (34, x, 24) in 12.

Prob. 5, 1793: p. 189; 1799: p. 201 & Key pp. 244-245. Trenching a field. A, B, C can do in 12; B, C, D can do in 14; C, D, A can do in 15; D, A, B can do in 18. How long for all four together and for each one singly? Solution in decimals.


Dodson. Math. Repository. 1775.

P. 22, Quest. LVIII. Man and wife drinking beer. (30, x) in 12.

P. 23, Quest LIX. Cistern: (20, x) in 12.

Pp. 52-53, Quest. CV. Workers. A & B in 8; A & C in 9; B & C in 10.

P. 56, Quest 56. (3, 8/3, 12/5), phrased as in Newton. Does the problem in general, then applies to the data.


Ozanam-Montucla. 1778.

Question 5, 1778: 193-194; 1803: 190-191; 1814: 165-166; 1840: omitted. Same as Metrodorus 135.



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