7. arithmetic & number theoretic recreations a. Fibonacci numbers

Kaye I 41; III 171, f. 3r says there is a fragment of another problem of this type with values 4, 5, 6

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Kaye I 41; III 171, f. 3r says there is a fragment of another problem of this type with values 4, 5, 6.

Kaye I 40-42, III 168-169, ff. 1r-2r, sutra 11. Gupta says the MS is poor and Kaye has misinterpreted it. Gupta interprets it as leading to x₁/2 + x₂ + x₃ + x₄ + x₅ = h, etc., where h is the price of a jewel. The i-th equation then reduces to T = h + x_i (i 1)/i, where T is the total wealth. Kaye considers it as T x_i = h x_i/i. Gupta converts this to a present of gems problem, but since the multipliers are non-integral, it takes a little more work. Answer: 120, 90, 80, 75, 72; 377. I can't see that Kaye treats this any differently.

Kaye III 170, f. 2v is another example, with three values and diagonal coefficients 7/12, 3/4, -5/6 and answer: 924, 836, 798; 1095.

Mahavira. 850. Chap. VI, v. 162 166, pp. 137 138. Gupta (op. cit. under Bakhshali MS) says the rule given is similar to the Bakhshali rule.

164. 6, 7, 8 giving one each.

165. 16, 10, 8 giving two each.

Sridhara. c900. Gupta (op. cit. under Bakhshali MS) says that Sridhara gives the same rule as the Bakhshali MS, but allowing n people. This rule is quoted in the Kriyākramakari, a 1534 commentary on the Lilavati and Gupta quotes and translates it. The Kriyākramakari also quotes Mahavira, without attribution. Gupta cannot locate this rule in Sridhara's extant works.

Bhaskara II. Lilavati. 1150. Chap. IV, sect. IV, v. 100. In Colebrooke, p. 45. Also in his Bijaganita, chap. IV, v. 111, pp. 195 196. 8, 10, 100, 5 giving one each to others and all are equal.

7.P.5 . SELLING DIFFERENT AMOUNTS 'AT SAME PRICES' YIELDING THE SAME
NOTATION: (a, b, c, ...) means the sellers initially have a, b, c, .... They all sell certain amounts at one price, then sell their remnants at a second price so that each receives the same amount. Western versions give a, b, c, ... and sometimes the amount each receives. The Indian versions give the proportion a : b : c : ... (by stating each person's capital, but not the cost price of the items; they invest their capital in the items and then sell them) and the larger price for selling the remnant of the items. Further, the price for selling the first part of the items is the reciprocal of an integer. (However the remnant price is sometimes a fraction.) In both versions, the problem is indeterminate, with a 3 parameter solution set, but scaling or similarity or fixing the yield reduces this to 2. There are also non negativity and integrality conditions. The Indian version has infinitely many solutions, while the Western version gives a finite number of solutions. I have recently found a relatively simple way to generate and count the solutions in the Western version, which is basically a generalization of Ozanam's example -- see my paper below. The article by Glaisher discusses many of these problems. As in 7.P.1, (a, b) solutions means a non-negative solutions of which b are positive solutions.

Versions where the earnings are different: Ghaligai.

See Tropfke 651.
Index of western versions.
(10, 20) Abraham

(10, 30) Fibonacci

(12, 32) Fibonacci

(12, 33) Fibonacci

(18, 40) Labosne

( 7, 18, 29) McKay

( 8, 17, 26) Blasius

(10, 12, 15) Labosne

(10, 16, 22) Amusement

(10, 20, 30) Pacioli

(10, 25, 30) Ozanam

(10, 30, 50) Munich 14684, Folkerts, Marliani, Provençale Arithmétique, Pseudo dell'Abbaco, Chuquet, HB.XI.22?, Widman, Demaundes Joyous, Tagliente, Ghaligai, Tartaglia, Jackson, Badcock, Rational Recreations, Boy's Own Book, Rowley, Hoffmann

(11, 33, 55) Tartaglia

(16, 48, 80) Tartaglia

(18, 40, 50) Labosne

(19, 25, 27) Williams & Savage

(20, 25, 32) Bachet

(20, 30, 40) Bachet, van Etten, Hunt

(20, 40, 60) Tagliente

(27, 29, 33) Leske, Mittenzwey, Hoffmann, Pearson

(30, 56, 82) Widman

(31, 32, 37) Labosne

(60, 63, 66) Bath

(17, 68, 119, 170) Widman

(20, 30, 40, 50, 60) Dudeney

(305, 454, 603, 752, 901) Widman

(20, 40, ..., 140) Glaisher, Gould

(10, 20, ..., 90) Tartaglia

Mahavira. 850. Chap. VI, v. 102 110, pp. 113 116. He gives a rule which gives one special solution of Sridhara's set of solutions.

V. 103. Capitals: 2, 8, 36; remnant price 6.

V. 104. Capitals: 1½, ½, 2½; remnant price 6.

V. 105. Each receives 41; remnant price 6. What is the largest of the capitals? (The other capitals are not determined.)

V. 106. Each receives 35; remnant price 4. (Cf. v. 105.)

V. 108. Capitals: ½, ⅓, ¼; remnant price 6/5.

V. 110. Capitals: ½, ⅔, ¾; remnant price 5/4.

Sridhara. c900. V. 60 62, ex. 76 77, pp. 44 49 & 94. The verses are brief rules, which are expanded by editorial algebra, giving a one parameter family of solutions.

Ex. 76. Capitals: 1, 3, 5 or ⅓, ¼, ½; remnant price 3.

Ex. 77. Capitals: 3/2, 2, 3, 5; remnant price ½.

Bhaskara II. Bijaganita. 1150. Chap. 6, v. 170. In Colebrooke, pp. 242 244. Capitals 6, 8, 100; remnant price 5. Solution given is 3294, 4392, 54900, which is one solution from Sridhara's set of solutions, but not by the same method as Mahavira. The method is not clearly described. Bhaskara says: "Example instanced by ancient authors .... This, which is instanced by ancient writers as an example of a solution resting on unconfirmed ground, has been by some means reduced to equation; and such a supposition introduced, as has brought out a result in an unrestricted case as in a restricted one. In the like suppositions, when the operation, owing to restriction, disappoints; the answer must by the intelligent be elicited by the exercise of ingenuity."

Fibonacci. 1202. Pp. 298 302 (S: 421-423): De duobus hominibus, qui habuerunt poma [On two men who had apples]. He clearly states that there are two forums where the same prices are different.

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