7. arithmetic & number theoretic recreations a. Fibonacci numbers

Pp. 304-306 (S: 429-431). Several simple divinations, e.g. x*2+5*5+10+y*10+z  =  350 + 100x + 10y + z.

Pp. 307 308 (S: 433-434). If a₁ + a₂ + a₃ = a, he uses M = (2, a 1, a) and subtracts from a² to obtain (a 2) a₁ + a₂. He also uses M = (2, k, k+1) and subtracts from a(k+1) to get (k 1) a₁ + a₂. These work whether the a_i are a permutation or not. He uses a = 6, k = 9, hence M = (2, 9, 10). For permutations (?) of 2, 3, 4, he uses a = 9, k = 9. No mnemonics.

Abbott Albert. c1240. Prob. 2, pp. 332 333. M = (2, 9, 10). He then subtracts from 60 which produces 8a₁ + a₂. This is identical to Fibonacci's first example, but Albert gives a complete table of all the partitions of 6 into 3 non negative summands and computes 2x + 9y + 10z for each, showing that P determines the a_i if a₁ + a₂ + a₃  = 6, whether it is a permutation of 1, 2, 3 or not. No mnemonics. This can also be viewed as a form of 7.P.1.

BR. c1305.

No. 38, pp. 56 59. Determine values of three dice from a + b + c and 2a + 8b + 9c, so this is essentially Fibonacci's second case with k = 8.

No. 39, pp. 58 59. Same using a + b + c and 3a + 9b + 10c.

No. 100, pp. 118 119. Determine values of two dice from a + b and 2a + 10b.

Munich 14684. 14C. No. IX. This is obscure, but is repeated more clearly as No. XIX. M = (2, 9, 10). IX is followed by a two line verse. Curtze could make no sense of it, but I wonder if it might be a mnemonic??

Folkerts. Aufgabensammlungen. 13-15C.

Discusses various simple divinations, citing Fibonacci.

He cites 18 sources using a = 6, k = 9 in Fibonacci's first form. For a permutation of three things, one can take M = (m-n, m-1, m). Subtracting from am leaves na₁ + a₂. This is equivalent to using (0, 1, n). 10 sources for divining a permutation of 2, 3, 4 using m = 10, n = 8. Cites several older versions.

AR. c1450. Prob. 269, p. 122, 180 181, 227 228. M = (2, 9, 10). Then 60   P = 8a₁ + a₂. Vogel cites several earlier appearances, but mentions no mnemonics. ??NYS.

Chuquet. 1484. Prob. 159. M = (1, 2, 4). Starts with 24, but 6 are used to label the people, so this is really subtracting from 18. Table, but no mnemonic. FHM 232.

Pacioli. De Viribus. c1500.

Ff. 76v - 77v. XXXV effecto de saper trovare .3. varie cose divise fra .3. persone et .4. divise fra .4. et de qua(n)te vorrai etc. (35th effect to know how to find 3 different things distributed among 3 persons and 4 among 4 and as many as you want etc.) = Peirani 112-114. Gives 12, 24, 36 counters to three people and says the person with the first object is to discard 1/2 of his pile, the person with the second is to discard 2/3 of his pile and the person with the third object is to discard 3/4 of his pile. What is left takes on the values 23, 24, 25, 27, 28, 29. The method is taking M = (6, 4, 3), which is equivalent to (0, 1, 3). There is no mention of doing it with four persons.

Ff. 78v - 79r. XXXVII. commo el mode precedente se po far con fave et quartaruoli etc. (37th. How the preceding method can be done with beans or farthings etc.) = Peirani 115. Same as the above.

Tagliente. Libro de Abaco. (1515). 1541. Prob. 142, f. 63v. x*2+5*5+10+y*10+z  =  350 + 100x + 10y + z. Cf Fibonacci.

Cardan. Practica Arithmetice. 1539. Chap. 61, section 18, f. T.iv.v (p. 113). Mentions divination of a permutation of three things by use of 18 counters, so this is probably M = (1, 2, 4), subtracting from 18, as in Chuquet, Baker, etc.

Baker. Well Spring of Sciences. 1562? Prob. 4, 1580?: ff. 197r 198r; 1646: pp. 310-312; 1670: pp. 354-355. M = (1, 2, 4), subtracting from 18. Vowel mnemonic: Angeli, Beati, Taliter, Messias, Israel, Pietas with explanatory table.

Recorde-Mellis. Third Part. 1582. Ff. Yy.v.r - Yy.v.v (1668:. 477-478: Another [divination] of things hidden. M = (1, 2, 4). No mnemonics.

John Wecker. Op. cit. in 7.L.3. (1582), 1660. Book XVI -- Of the Secrets of Sciences: Chap. 20 -- Of Secrets in Arithmetick: To discover to one a thing that is hid, pp. 289 290. M = (1, 2, 4). No mnemonics. Cites Gemma Frisius, ??NYS.

Prévost. Clever and Pleasant Inventions. (1584), 1998.

Pp. 177-180. M = (1, 2, 4), subtracting from 18. No mnemonics.

Pp. 180-182. M = (1, 2, 4), subtracting from 23. No mnemonics.

Bachet. Problemes. 1612. Prob. XXII: De trois choses et de trois personnes proposées deviner quelle chose aura été prise par chaque personne, 1612: 115-126. Prob. XXV, 1624: 187-198; 1884: 127 134. M = (1, 2, 4). Vowel mnemonic: Par fer César jadis devint si grand prince. Gives a four person version, S = 78, M = (1, 4, 16, 0), referring to Forcadel as giving an erroneous method. Labosne says Diego Palomino (1599) has studied the four person case. [This must be Jacobo Palomino; Liber de mutatione aeris in quo assidua et mirabilis mutationis temporum historia cum suis caussis enarratur. -- Fragmentum quodam ex libro de inventionibus scientiarum; Madrid, 1597 or 1599, ??NYR.] Labosne adds some explanation, another mnemonic: Avec éclat L'Aï brillant devint libre, and expands on Bachet's work on the case of four objects, but eliminates reference to Forcadel.

van Etten. 1624. Prob. 8 (8), pp. 9 11 (19 22). M = (1, 2, 4). Vowel mnemonics: Salve certa anima semita vita quies; Par fer Cesar Iadis Devint si grand Prince. Henrion's 1630 Notte, pp. 10 11, says that Bachet has extended it to 4 objects.

Hunt. 1631 (1651). Pp. 255-261 (247-253). Usual form with mnemonic: Angeli Beati Pariter Elias Israel Pietas. Then does another version.

Schott. 1674. Art. V, p. 58. M = (1, 2, 4). Vowel mnemonics: Salve Certa Animæ Semita Vita Quies; Pallētis Evandri Sanguine Feritas Imane (the m should have an overbar) Vigebat.

Ozanam. 1694. Prob. 28, 1696: 83; 1708: 74. Prob. 32: 1725: 217-218. Prob. 10, 1778: 154-155; 1803: 154-155; 1814: 136-137. Prob. 9, 1840: 70. M = (6, 4, 3). No mnemonics.

Ozanam. 1725. Prob. 46, 1725: 250-253. Prob. 12, 1778: 158-161; 1803: 159-161; 1814: 140-142. Prob. 11, 1840: 72. M = (1, 2, 4). Par fer Cesar jadis devint si grand Prince and Salve certa anima semita vita quies. 1778 et seq. has César and animæ.

Minguet. 1733. Pp. 176-180 (1755: 127-129; not noticed in 1822; 1864: 164-166). M = (1, 2, 4) as in Chuquet. Vowel mnemonic: Aperi, Premati, Magister, Nihil, Femina, Vispane, Vispena.

Alberti. 1747. Part 2, p. ?? (69). M = (6, 4, 3) translated from Ozanam, 1725, prob. 32.

Alberti. 1747. Part 2, pp. ?? (76-77). M = (1, 2, 4) as in Ozanam, 1725, prob. 46, but he first gives an Italian mnemonic: Aperì Prelati Magister Camille Perina Quid habes Ribera. He explains the usage using 4 = Camille as example, but later notes that 4 never occurs! Then gives Salva certa anima semita vita quies; Perfer Cesar Jadis devint sigrand Prince; Pare ella ai segni; Vita, Piè.

Les Amusemens. 1749. Prob. 13, pp. 134-135: Les trois Bijoux. M = (1, 2, 4). Par fer, César jadis devint si grand Prince.

Hooper. Rational Recreations. Op. cit. in 4.A.1. 1774. Recreation IX: The confederate counters, pp. 34-36. M = (1, 2, 4). Par fer Cesar jadis devint si grand prince.

Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 85-86, no. 131: Three persons having each chosen privately one out of three things, to tell them which they have chosen. Salve certa anima semita vita quies.

Manuel des Sorciers. 1825. ??NX