7. arithmetic & number theoretic recreations a. Fibonacci numbers

Prob. 84, pp. 294-296. Relates 2ⁱ and 5ⁱ for i = 1, ..., 7. He determines 2^3.5 as 128 which he estimates as 11⅓. Similarly he estimates 5^3.5 as 279.

Jacques Chauvet Champenois. Les Institutions de L'Arithmetique. Hierosme de Marnef, Paris, 1578, p. 70. ??NYS. Problem of tailor and robe involving 4888 divided by 2 twenty times. (French quoted in H&S 14 15.)

van Etten. 1624. Prob. 87 (84): Des Progressions & de la prodigieuse multiplication des animaux, des Plantes, des fruicts de l'or & de l'argent quand on va tousjours augmentant par certaine proportion, pp. 111 118 (177 183). Numerous examples including horseshoe problem and chessboard problem, with ratios 1000, 4, 50. Henrion's Notte, p. 38, observes that there are many arithmetical errors which the reader can easily correct. In part X: Multiplication des Hommes, he considers one of the children of Noah, says a generation takes 30 years and that, when augmented to the seventh, one family can easily produce 800,000 souls. The 1674 English ed. has: "... if we take but one of the Children of Noah, and suppose that a new Generation of People begin at every 30 years, and that it be continued to the Seventh Generation, which is 200 years; ... then of one only Family there would be produced 111000 Souls, 305 to begin the World: ... which number springing onely from a simple production of one yearly ...."

W. Leybourn. Pleasure with Profit. 1694. Chap. VI, pp. 24-28: Of the Increase of Swine, Corn, Sheep, &c. Examples with ratios 4, 40, 2, 1000, 2, mostly taken from van Etten. Then art. VI: Of Men, discusses the repopulation of the world from Noah's children: "... if we take but one of the Children of Noah, and suppose that a New Generation of People begin at every 30 years, and that it be continued to the seventh Generation, which is 210 years; ... then, of one only family there would be produced 111305, that is, One hundred and eleven thousand, three hundred and five Souls to begin the World .... ... such a number arising only from a simple production of only One yearly ...." I cannot work out how 111305 arises -- the fact that he spells it out makes it unlikely to be a misprint.

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, .... 1778 et seq. also mention 1, 3, 9, ....

Ozanam. 1725. Prob. 11, questions 6 & 7, 1725: 79 82. Prob. 3, parts 1-3, 1778: 80-82; 1803: 82-84; 1814: 72-75; 1840: 38-39. Examples of population growth in Biblical and biological contexts. In 1725, he has ratios of 2, 50, 3, 4, 1000, The examples vary a bit between 1725 and 1778.

Walkingame. Tutor's Assistant. 1751. The section Geometrical Progression gives several problems with powers of 2 and the following less common types.

Prob. 5, 1777: p. 95; 1835: p. 103; 1860: p. 123. Find 1 + 4 + 16 + ... + 4¹¹ farthings.

Prob. 8, 1777: p. 96; 1835: p. 104; 1860: p. 123. Find 2 + 6 + 18 + ... + 2 x 3²¹. If these are pins, worth 100 to the farthing, what is the value?

Vyse. Tutor's Guide. 1771? The section Geometrical Progression, 1793: 35, pp. 138-143; 1799: XXXV, pp. 146-151 & Key pp. 190-192, gives several examples with doublings and triplings as well as examples with ratios of 3/2 and 10. There is a major error in the solution of prob. 7, to find 2 + 6 + 18 + ... + 2 x 3¹⁹.

Pike. Arithmetic. 1788. Pp. 237-239. Numerous fairly standard examples, mostly doubling, but with examples of powers of 3 and of 10 and the following. D. Adams, 1835, copies two examples, but not the following.

Pp. 239-240, no. 8. One farthing placed at 6% compound interest in year 0 is worth what after 1784 years? And supposing a cubic inch of gold is worth £53 2s 8d, how much gold does this make? This is very close to 2¹⁵⁰ farthings and makes about 4 x 10¹⁴ solid gold spheres the size of the earth!

Eadon. Repository. 1794. P. 241, ex. 3. Doubling 20 times from a farthing.

John King, ed. John King 1795 Arithmetical Book. Published by the editor, who is the great-great-grandson of the 1795 writer, Twickenham, 1995. P. 100. 10 + 10² + ... + 10¹¹ grains of wheat, converted to bushels and value at 4s per bushel.

(Beeton's) Boy's Own Magazine 3:6 (Jun 1889) 255 & 3:8 (Aug 1889) 351. (This is undoubtedly reprinted from Boy's Own Magazine 1 (1863).) Mathematical question 59. Seller of 12 acres asks 1 farthing for the first acre, 4 for the second acre, 16 for the third acre, .... Buyer offers £100 for the first acre, £150 for the second acre, £200 for the third acre, .... What is the difference in the prices asked and offered? Also entered in 7.AF.

Lewis Carroll. Sylvie and Bruno Concluded. Macmillan, London, 1893. Chap. 10, pp. 131 132. Discusses repeated doubling of a debt each year as a way of avoiding paying the debt -- "You see it's always worth while waiting another year, to get twice as much money!" = Carroll-Wakeling II, prob. 5: A new way to pay old debts, pp. 9 & 66, where Wakeling adds some problems based on repeated doubling and gives the chessboard problem.
7.L.1. 1 + 7 + 49 + ... & ST. IVES
See Tropfke 629.
Papyrus Rhind, c 1650, loc. cit. in 7.C. Problem 79, p. 112 of vol. 1 (1927) (= p. 59 (1978)). 7 + 49 + 343 + 2401 + 16807. (Sanford 210 and H&S 55 give Peet's English.) Houses, cats, mice, ears of spelt, hekats.

L. Rodet. Les prétendus problèmes d'algèbre du manuel du calculateur Égyptien (Papyrus Rhind). J. Asiatique 18 (1881) 390 459. Appendice, pp. 450 454. Discusses this problem and its appearance in Fibonacci (below).

F. Cajori. History of Mathematics. 2nd ed., Macmillan, 1919; Chelsea, 1980. P. 90 gives the legend that Buddha was once asked to compute 7¹⁷.

Shakuntala Devi. The Book of Numbers. Orient Paperbacks (Vision Books), Delhi, 1984. This gives more details of the Buddha story, saying it occurs in the Lalitavistara and Buddha finds the number of atoms (of which there are seven to a grain of dust) in a mile, obtaining a number of 'about 50 digits'. Note: 7⁵⁸ = 1.04 x 10⁴⁹.

Alcuin. 9C. Prob. 41: Propositio de sode et scrofa. This has sows which produce 7 piglets, but this results in a GP of ratio 8.

Fibonacci. 1202.