7. arithmetic & number theoretic recreations a. Fibonacci numbers

Prob. 8, 1696: 33-35; 1708: 29-32. Prob. 11, 1725: 68-75. Section II, 1778: 68-74; 1803: 70-76; 1814: ??NYS; 1840: 34-36. A discussion of geometric progression and a mention of 1, 2, 4, ..., without any application to weighing. 1778 et seq. also mentions 1, 3, 9, ....

Prob. 12, vol. II, 1694: 18-19 (??NYS). Prob. 12, 1696: 284 & fig. 131, plate 46, p. 275; 1708: 360 & fig. 26, plate 14, opp. p. 351. Prob. 8, vol. II, 1725: 345 348 & fig. 131, plate 46 (42). Prob. 14, vol. I, 1778: 206 207; 1803: 201-202. Prob. 13, vol. II, 1814: 174-175; 1840: 90 91. Gives double and triple progressions. Knobloch gives the 1694 citation. The figure is just a picture of a balance and is not informative -- the same figure is also cited for other sets of weights.

Les Amusemens. 1749. Prob. 8, p. 128. Coins of value 1, 2, 4, 8, 15 to pay for a room at a rate of 1 per day for 30 days.

The Bile Beans Puzzle Book. 1933. No. 42: Money juggling. Place £1000 in 10 bags so any amount can be paid without opening a bag. Solution has bags of:

1, 2, 4, 8, 16, 32, 63, 127, 254, 493. I cannot see why the solution isn't:

1, 2, 4, 8, 16, 32, 64, 128, 256, 489.

7.L.3. 1 + 3 + 9 + ... AND OTHER SYSTEMS OF WEIGHTS
See MUS I 88-98; Tropfke 633.
Tabari. Miftāh al-mu‘āmalāt. c1075. P. 125ff., no. 43. ??NYS -- Hermelink, op. cit. in 3.A, and Tropfke 634-635 say this gives 1, 3, 9, ..., 19683 = 3⁹ to weigh up to 10,000.

Fibonacci. 1202.

P. 297 (S: 420-421). Weights 1, 3, 9, 27 and 1, 3, 9, 27, 81 'et sic eodem ordine possunt addi pesones in infinitum' [and thus in the same order weights can be added without end].

Pp. 310 311 (S: 437). Finds 1 + 2 + 6 + 18 + ... + 2*3⁶² = 3⁶³ by repeated squaring to get 3⁶⁴ and then divides by 3.

Gherardi. Libro di ragioni. 1328. P. 53. Weights 1, 3, 9, 27, 80 to weigh up through 120.

Columbia Algorism. c1350. Prob. 71, pp. 92 93. Weights 1, 3, 9, 27.

AR. c1450. Prob. 127, pp. 67 & 182. 1, 3, 9, 27.

Chuquet. 1484. Prob. 142.

1, 2, 7 to weigh up to 10 (English in FHM 225);

1, 2, 4, 15 up to 22;

1, 3, 9 up to 13;

1, 3, 9, 27 up to 40; (original of this and the next case reproduced on FHM 226)

1, 3, 9, 27, 81 up to 121. Knobloch says Chuquet gives a general solution, but I don't see that Chuquet is general.

Pacioli. Summa. 1494. Ff. 97r-97v, no. 34. General discussion of 1, 3, 9, 27, 81, 243, .... In De Viribus, c1500, F. XIIIv, item 85 in the Indice for the third part is: De far 4 pesi che pesi fin 40 (To make four weights which weigh to 40) = Peirani 20, but at the end Pacioli says this problem is in 'libro nostro', i.e. the Summa. Cf Agostini, p. 6.

Cardan. Practica Arithmetice. 1539. Chap. 65, section 12, ff. BB.vii.r - BB.vii.v (p. 136). Weights 1, 3, 9, 27, ....

Knobloch also cites: Giel vanden Hoecke (1537); Gemma Frisius (1540); Michael Stifel (1553); Simon Jacob (1565); Ian Trenchant (1566); Daniel Schwenter (1636); Kaspar Ens (1628); Claude Mydorge (1639); Frans van Schooten (1657).

Tartaglia, 1556 -- see in 7.L.2.c.

Buteo. Logistica. 1559. Prob. 91, pp. 309-312. Use of 1, 3, 9, 27, ... as weights. (Cited by Knobloch.)

John [Johann (or Hanss) Jacob] Wecker. Eighteen Books of the Secrets of Art & Nature Being the Summe and Substance of Naturall Philosophy, Methodically Digested .... (As: De Secretis Libri XVII; P. Perna, Basel, 1582 -- ??NYS) Now much Augmented and Inlarged by Dr. R. Read. Simon Miller, London, 1660, 1661 [Toole Stott 1195, 1196]; reproduced by Robert Stockwell, London, nd [c1988]. Book XVI -- Of the Secrets of Sciences: chap. 19 -- Of Geometricall Secrets: To poyse all things by four Weights, p. 289. 1, 3, 9, 27; 1, 3, 9, 27, 81; 1, 3, 9, 27, 81, 243. Cites Gemma Frisius.

Bachet. Problemes. 1612. Addl. prob. V & V(bis), 1612: 143-146; as one prob. V, 1624: 215-219; 1884: 154 156. Weights: 1, 3, 9, 27, ..., and the general case via the sum of a GP. In the 1612 ed., Bachet only does the cases 40 and 121, then does the general case. Knobloch cites 1612, pp. 127 & 143-146, but p. 127 is Addl. prob. I, which is a Chinese Remainder problem. He also says this is the first proof of the problem, excepting Chuquet, though I don't see such in Chuquet.

van Etten. 1624. Prob. 53 (48), pp. 48 49 (72). 1, 3, 9, 27; 1, 3, 9, 27, 81; 1, 3, 9, 27, 81, 243. Henrion's Notte, pp. 20 21, refers to Bachet and compares this with binary weights.

Ozanam. 1694.

Prob. 8, 1696: 33-35; 1708: 29-32. Prob. 11, 1725: 68-75. Section II, 1778: 68-74; 1803: 70-76; 1814: ??NYS; 1840: 34-36. A discussion of geometric progression and a mention of 1, 2, 4, ..., without any application to weighing. 1778 et seq. also mentions 1, 3, 9, ....

Prob. 12, vol. II, 1694: 18-19 (??NYS). Prob. 12, 1696: 284 & fig. 131, plate 46, p. 275; 1708: 360 & fig. 26, plate 14, opp. p. 351. Prob. 8, vol. II, 1725: 345 348 & fig. 131, plate 46 (42). Prob. 14, vol. I, 1778: 206 207; 1803: 201-202. Prob. 13, vol. II, 1814: 174-175; 1840: 90 91. Gives double and triple progressions. Knobloch gives the 1694 citation. The figure is just a picture of a balance and is not informative -- the same figure is also cited for other sets of weights.

Les Amusemens. 1749. Prob. 18, p. 140: Les Poids. Weights 1, 3, 9, 27, 81, 243.

Vyse. Tutor's Guide. 1771? Prob. 2, 1793: p. 303; 1799: p. 316 & Key pp. 356-357. Weights 1, 3, 9, 27.

Bonnycastle. Algebra. 1782. P. 202, no. 13. 1, 3, 9, 27, 81, 243, 729, 2187 to weigh to 29 hundred weight -- an English hundred weight is 112 pounds. c= 1815: p. 230, no. 33. 1, 3, 9, 27, 81 to weigh to a hundred weight.

Eadon. Repository. 1794. Pp. 297-298, no. 1. 1, 3, 9, ..., 3¹³.

Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. P. 50, no. 77: To find the least Number of Weights that will weigh from One Pound to Forty. 1, 3, 9, 27.

Jackson. Rational Amusement. 1821. Curious Arithmetical Questions. No. 24, pp. 20 & 78 79. 1, 2, 4, 8, 16, ... and 1, 3, 9, 27, 81, ....

Rational Recreations. 1824. Exer. 22, p. 131. 1, 3, 9, 27.

Endless Amusement II. 1826?