(10, 30) -- he gives 5 solutions, there are (55, 36)

(10, 30) -- he gives 5 solutions, there are (55, 36).

(12, 32) -- he gives 6 solutions, there are (78, 55).

(12, 33) -- he gives 1 solution, there are (78, 55).

Glaisher, pp. 79-107, analyses this text in detail and finds that Fibonacci gives a reasonably general method which would give the 24 positive solutions with smaller price 1 in the first example. He then considers the amounts received as being fixed. He then permits the amounts received to differ, one receiving a multiple of what the other receives. He also considers whether a solution exists for given amounts and prices and how to find solutions with one price given.

Munich 14684. 14C. Prob. XIII, pp. 79 80. (10, 30, 50). Gives the solution with prices 1/7  and 3. There are (25, 16) solutions.

Folkerts. Aufgabensammlungen. 13-15C. 21 sources for (10, 30, 50). Says the problem goes back to Fibonacci, but Fibonacci only has examples with two vendors.

Giovanni Marliani. Arte giamata arismeticha. In codex A. II. 39, Biblioteca Universitaria de Genova. Van Egmond's Catalogue 139 dates it c1417. Described and partly transcribed by Gino Arrighi; Giuochi aritmetici in un "Abaco" del Quattrocento Il matematico milanese Giovanni Marliani. Rendiconti dell'Istituto Lombardo. Classe di Scienze (A) 99 (1965) 252 258. Prob. V: (10, 30, 50).

Provençale Arithmétique. c1430. Op. cit. in 7.E. F. 116v, pp. 62-63. (10, 30, 50). Gives the solution with prices 3 and 1/7 and then gives a solution with three prices!

Pseudo-dell'Abbaco. c1440. Prob. 101, pp. 85 87. (10, 30, 50). Gives the solution with prices 3  and 1/7.

Chuquet. 1484. English in FHM 227-228. Prob. 145. (10, 30, 50). Same two solutions as in Provençale Arithmétique. FHM says it appears in one of Dudeney's books, where he expresses "grave dissatisfaction with the answer".

HB.XI.22. 1488. P. 54 (Rath 247). Rath doesn't give the numbers and says it is similar to Munich 14684 and p. 73v of Cod. Vindob. 3029. Glaisher dates Cod. Vindob. 3029 as c1480.

Johann Widman. Op. cit. in 7.G.1. 1489. F. 134v+. ??NYS -- discussed by Glaisher, pp. 1 18. (10, 30, 50) -- one solution with prices 3 and 1/7. He then generalises this example to construct single solutions for other examples: (30, 56, 82), (17, 68, 119, 170), (305, 454, 603, 752, 901), which have (225, 196), (45, 35), (11552, 11400) solutions respectively. Glaisher then describes several examples that Widman might have constructed.

Pacioli. De Viribus. c1500. Ff. 119r - 119v.

LXV. C(apitolo). D dun mercante ch' a .3. factori et atutti ma'da auno mercato con p(er)le. (10, 20, 30). Gives the solution with prices 1 and 1/6 and result 5, selling 4, 2, 0 at the higher price. There are (25, 16) solutions.

Ff. IIIv - IVr. = Peirani 7. The Index lists the above as Problem 69 and then gives the following.

Problem 70: De unaltro mercante ch' pur a .3. factori et mandali a una fiera con varia quantita de perle' et vendano a medesimo pregio et portano acasa tanti denari al patrone uno quanto laltro (Of another merchant who sends three agents to a fair with varying numbers of pearls and they sell them at the same price and they each carry as many pence as the others to the master at home).

Problem 71: De unaltro vario dali precedenti ch' pur a .3. factori con vari quantita de perle' pregi pari et medesimamente portano al patrone d(enari) pari (Of another variant of the preceding with three agents having various quantities of pearls at equal prices and likewise take as many pence to the master).

Problem 72: De unaltro mercante ch' ha 4. factori ali quali da quantita varie di perli ch' amedisimi pregi le vendino et denari equalmente portino (Of another merchant who has 4 agents to whom he gives various numbers of pearls which they sell at the same prices and receive equal money).

Problem 73: De un altro ch' pur a .4. factori con quanti(ta) varie di perle apari pregi et pari danari reportano a casa vario dali precedenti (Of another who sends 4 agents with varying numbers of pearls and they report back to the house the same prices and the same money, variation of the preceding).

Anon. Demandes joyeuses en manière de quodlibets. End of 15C. ??NYS. Selected and translated as: The Demaundes Joyous. Wynken de Worde, London, 1511. [The French had 87 demandes, but the English has 54. This is the oldest riddle collection printed in England, surviving in a single example in Cambridge Univ. Library. Often attributed to de Worde. Santi 9 uses Yoyous and Wynkyn and list de Worde as author.] Facsimile with transcription and commentary by John Wardroper, Gordon Fraser Gallery, London, 1971, reprinted 1976. Prob. 50, pp. 6 of the facsimile, 26-27 of the transcription. (10, 30, 50) apples. One solution with prices 3 and 1/7.

Blasius. 1513. F. F.iii.r: Decimaquarta regula. Selling eggs -- (8, 17, 26). There are (16, 9) solutions. He give one with prices 2 and 1/5 and each sold as many batches of 5 as possible. Discussed by Glaisher.

Tagliente. Libro de Abaco. (1515). 1541. Prob. 115, ff. 57r-57v. Women selling eggs -- (10, 30, 50). One solution with prices 1/7 and 3. Glaisher, below and in op. cit. in 7.G.1, cites Hieronymus Tagliente and says the 1515 & 1527 editions give (10, 30, 50) with solution at prices 1/7 & 3, and the 1525 ed. has (20, 40, 60) with solution at prices 3 and 1/7. The latter has (100, 81) solutions.

Ghaligai. Practica D'Arithmetica. 1521.

Prob. 21, ff. 65r-65v. (10, 30, 50). One solution, with prices 1/7 and 3. Glaisher says it is p. 66 (misprinted 64) in the 1548 ed. (H&S 53 gives Italian from 1552 ed., but no solution.) Ghaligai says the problem was known to Benedetto (da Firenze, flourished 1470s) and Ghaligai's teacher Giovanni del Sodo, as a problem outside of any rule, and Ghaligai labels it "Ragione apostata" (Exceptional problem).

Prob. 22, f. 65v. (10, 50) with first making twice the second. Solution with prices 13  and 1/7.

Tartaglia. General Trattato, 1556, art. 136 139, pp. 256r 256v.

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