7. arithmetic & number theoretic recreations a. Fibonacci numbers

274: 32 horseshoe nails.

317: 16 cow nails.

318: 32 horseshoe nails.

353: 28 nails -- text is obscure.

Riese. Rechenung nach der lenge .... 1525. (Loc. cit. under Riese, Die Coss.) Prob. 32, p. 20. 32 horseshoe nails.

Christoff Rudolff. Künstliche rechnung mit der ziffern und mit den zal pfenninge. Vienna, 1526; Nürnberg, 1532, 1534, et seq. F. N.viii.v. ??NYS. 32 horseshoe nails. (H&S 56 gives German.)

Apianus. Kauffmanss Rechnung. 1527. Ff. D.vi.r - D.vi.v. 32 horseshoe nails.

Anon. Trattato d'Aritmetica, e del Misure. MS, c1535, in Plimpton Collection, Columbia Univ. ??NYS. Horseshoe problem: 1 + 2 + 4 + ... + 2²³. (Rara, 482 484, with reproduction on p. 484.)

Recorde. First Part. 1543. Ff. L.ii.r - L.ii.v (1668: 141-142: A question of an Horse). 24  horseshoe nails.

Buteo. Logistica. 1559. Prob. 34, pp. 237-238. 24 horseshoe nails. (H&S 56.)

van Etten. 1624. Prob. 87, pp. 111 118 (not in English editions). Includes 24 horseshoe nails problem as part VII on p. 115. Henrion's Notte, p. 38, observes that there are many arithmetical errors in prob. 87 which the reader can easily correct.

Wells. 1698. No. 102, p. 205. 24 nails.

Ozanam. 1725. Prob. 11, question 4, 1725: 77 78 & 80. Part of prob. 3, 1778: 79-80; 1803: 81; 1814: 72; 1840: 38. 24 nails -- first asks for the price of the 24th, then the total.

Dilworth. Schoolmaster's Assistant. 1743. P. 96, no. 1. 32 nails, starting with a farthing.

Walkingame. Tutor's Assistant. 1751. Geometrical Progression, prob. 6, 1777: p. 95; 1835: p. 103; 1860: p. 123. 32 nails, one farthing for the first, wants total, which he gives in £ s/d.

Mair. 1765? P. 493, ex. III. "What will a horse cost by tripling the 32 nails in his shoes with a farthing?" I.e., 32 horseshoe nails, but with tripling!

Euler. Algebra. 1770. I.III.XI.511, p. 166. Horse to be sold for the value of 32 nails, 1  penny for the first, ....

Vyse. Tutor's Guide. 1771? Prob. 2, 1793: p. 140; 1799: p. 148 & Key p. 190. 36 horseshoe nails. Want value of last one, starting ¼, ½, 1, ....

Bullen. Op. cit. in 7.G.1. 1789. Chap. 31, prob. 1, p. 215. Same as Walkingame.

Bonnycastle. Algebra. 10th ed., 1815. P. 80, no. 6. Same as Walkingame.

Manuel des Sorciers. 1825. P. 84. ??NX 24 horseshoe nails.

The Boy's Own Book. The horsedealer's bargain. 1828: 182; 1828-2: 238; 1829 (US): 106; 1843 (Paris): 346; 1855: 393 394; 1868: 431-432. Wants value of 24th nail, starting with a farthing. = Boy's Treasury, 1844, p. 304. = de Savigny, 1846, p. 292: Le marché aux chevaux.

Nuts to Crack XIV (1845), no. 74. The horsedealer's bargain. Almost identical to Boy's Own Book.

Walter Taylor. The Indian Juvenile Arithmetic .... Op. cit. in 5.B. 1849. P. 199. 32 nails.

[Chambers]. Arithmetic. Op. cit. in 7.H. 1866? P. 268, quest. 64. 24 nails, starting with a farthing. Finds total.

Mittenzwey. 1880. Prob. 93, pp. 19 & 68; 1895?: 108, pp. 23-24 & 71; 1917: 109, pp. 21-22 & 68. 32 horseshoe nails, starting at 1 pf.

Cassell's. 1881. P. 101: The horse dealer's bargain. 24 nails, unclear, but uses 2²³ farthings as the answer.
7.L.2.c. USE OF 1, 2, 4, ... AS WEIGHTS, ETC.
See Tropfke 633.

A special case of this is the use of such amounts to make regular unit payments, e.g. rent of one per day. See: Knobloch; Fibonacci; BR; Widman; Tartaglia; Gori; Les Amusemens.

Eberhard Knobloch. Zur Überlieferungsgeschichte des Bachetschen Gewichtsproblems. Sudhoffs Archiv 57 (1973) 142-151. This describes the history of this topic and 7.L.3 from Fibonacci to Ozanam (1694). He gives a table showing occurrences of: powers of two, powers of three, weight problem, payment problem. I am not entirely clear what he means in the first three cases -- I would have two kinds of weight problem corresponding to the first two cases and perhaps some of his references in the first case are listed under 7.L.2. However, the last case clearly corresponds to the problem of making a payment of one unit per day as in Fibonacci. He lists this as occurring in Fibonacci, BR, Widmann and Tartaglia and notes that Sanford, H&S 91, only noticed Fibonacci. Knobloch notes that Ball's citations are not very good and that Ahrens' note about them does not go much deeper. I have a number of references listed below which were not available to Knobloch.
Fibonacci. 1202. P. 298 (S: 421). Uses 5 ciphi of value 1, 2, 4, 8, 15 to pay a man at rate of 1 per day for 30 days.

BR. c1305. No. 93, pp. 112 113. Use of 1, 2, 4 as payments at rate of one per year for 7 years.

Widman. Op. cit. in 7.G.1. 1489. Ff. 138v-139r. ??NYS -- Knobloch says he uses values of 1, 2, 4, 8, 16 to pay for 31 days.

Pacioli. Summa. 1494. Ff. 97v-98r, no. 35. Use five cups to pay daily rent for 30 days. Uses cups of weight 1, 2, 4, 8, 15. In De Viribus, c1500, F. XIIIv, item 86 in the Indice for the third part is: De 5 tazze, diversi pesi ogni di paga l'oste (Of 5 cups of diverse weights to pay the landlord every day) = Peirani 20, but at the end Pacioli says this problem is in 'libro nostro', i.e. the Summa. Cf Agostini, p. 6.

Tartaglia. General Trattato, 1556, part 2, book 1, chap. 16, art. 32: Di una particolar proprieta della progression doppia geometrica, p. 17v. Weights: 1, 2, 4, 8, ... (See MUS I 89.). Also does payments with 1, 2, 4, 8, 16, 29. Knobloch also refers to art. 33-35 -- ??NYS -- and notes that the folios are misnumbered, but miscites 'doppia' as 'treppia' here. This covers the powers of 3 also.

Buteo. Logistica. 1559. Prob. 91, pp. 309-312. Use of 1, 2, 4, 8, 16, ... as weights. (Cited by Knobloch.)

Knobloch also cites Ian Trenchant (1566), Daniel Schwenter (1636), Franz van Schooten (1657).

Gori. Libro di arimetricha. 1571. Ff. 71r 71v (p. 76). Use of cups weighing 1, 2, 4 to make all weights through 7, to pay for days at one per day.

Bachet. Problemes. 1612. Addl. prob. V & V(bis), 1612: 143-146; as one prob. V, 1624: 215-219; 1884: 154-156. Mentions 1, 2, 4, 8, 16 and cites Tartaglia, art. 32 only. This was omitted in the 1874 ed. Knobloch cites 1612, pp. 127 & 143-146, but but p. 127 is Addl. prob. I, which is a Chinese Remainder problem?

van Etten/Henrion. 1630. Notte to prob. 53, pp. 20 21. Refers to Bachet and compares with ternary weights.

Ozanam. 1694.