7. arithmetic & number theoretic recreations a. Fibonacci numbers

The cancelled figure guessed. 1828: 177; 1828-2: 237; 1829 (US) & 1881 (NY): 105; 1843 (Paris): 346: A person striking a figure out of the sum of two given numbers, to tell what that figure was; 1855 & 1859: 392-393; 1868: 430-431; 1880: 460 461. Give a person several multiples of nine, then ask him to add two of them and strike out one digit from the total and tell you the sum of the other digits in the total. No consideration of the case when the cancelled digit could be 0 or 9. = Boy's Treasury, 1844, pp. 303-304. = de Savigny, 1846, p. 346.

Boy's Own Book. 1843 (Paris): 342. "To make any number divisible by nine, by adding a figure to it." Only appends or inserts the necessary digit. = Boy's Treasury, 1844, p. 299.

Lewis Carroll. Diary entry for 8 Feb 1856, In Carroll-Gardner, pp. 43-44. Observes that a number minus its reverse is divisible by nine, so you can ask someone to delete a digit and show you the rest and you announce the deleted digit. Gardner points out that one can subtract any permutation of the original digits.

Mittenzwey. 1880.

Prob. 40-41, pp. 9 & 60-61; 1895?: 46-47, pp. 14 & 64; 1917: 46-47, pp. 13 & 58-59. Deduce the figure deleted in 9x; in x minus the sum of its digits.

1895?: prob. 118, pp. 74-75; 1917: 118, pp. 23 & 71-72. Casting out 9s as a method of checking multiplication, claiming that the verification shows the calculation is correct.

Parlour Games for Everybody. John Leng, Dundee & London, nd [1903 -- BLC], p. 42: The expunged figure. Have someone write a number, form the sum of its digits and subtract that from the given number. Get him to strike out any figure and tell you the sum of the remaining figures. Says that if the result is a multiple of nine, then a nine was struck out.

Peano. Giochi. 1924. Prob. 50, p. 13. Take x, form 10x and subtract x. Cancel a non-zero figure from the result and tell me the other figures. I will tell what number you cancelled.

7.L. GEOMETRIC PROGRESSIONS
See Tropfke 628.

These also occur in 7.M, 7.S.1 and 10.A.

I am starting to include early problems which involve interpolation in a geometric series here -- these were normally solved by linear interpolation. From about 1400, such problems arise in compound interest but I will omit most such problems. See Chuquet here and Chiu Chang Suan Ching & Cardan in 10.A.
H. V. Hilprecht. Mathematical, Metrological and Chronological Tablets from the Temple Library of Nippur. Univ. of Pennsylvania, Philadelphia, 1906. Pp. 13, 28 34, 62, 69, pl. 15, PL. IX, are about a tablet which has a geometric progression from c 2300. The progression is double: a_n = 125 * 2ⁿ and 60⁴/a_n for n = 0, 1, ..., 7. There is no summation.

Tablet K 90 of the British Museum. A moon tablet deciphered by Hincks containing 5, 10, 20, 40, 80 followed by 80, 96, 112, 128, ..., 240. Described in The Literary Gazette (5 Aug 1854) -- ??NYS. Cited and described in: Nicomachus of Gerasa: Introduction to Arithmetic; Translated by Martin Luther D'Ooge, with notes by Frank Egleston Robbins and Louis Charles Karpinski; Macmillan, London, 1926; p. 12.

Euclid. IX: 35, 36. This gives the general rule for the sum of a geometric progression.

The Friday Night Book (A Jewish Miscellany). Soncino Press, London, 1933. Mathematical Problems in the Talmud, pp. 132-133. The Talmud says that any one visiting a sick person takes away a sixtieth of his illness. This led to the question of what happened if sixty people visited the person. This was answered by saying that the visitor took away a sixtieth of the illness that the person had, i.e. the patient was left with 59/60 of his illness, so that 60 visits left him still with (59/60)⁶⁰ of his illness. The text quoted in the source says this 'is still approximately one-quarter of the original illness', but it is .36479. The modern compiler adds that 'The Talmud does not indicate the method of working out the remainder after each visitor, and it is to be noted that although the summation of series was known to the Greeks, there is no mention of it anywhere in the Talmud.' To me, this shows some confusion as I don't see that summation of series is needed! [The Talmud was compiled in the period -300 to 500, but nothing in the source gives any more precise dating for this problem.]

Zhang Qiujian . Zhang Qiujian Suan Jing. Op. cit. in 7.E. 468, ??NYS. Mikami 42 gives: "A horse, halving its speed every day, runs 700 miles in 7 days. What are his daily journeys?" -- i.e. x*(1 + 1/2 + ... + 1/64) = 700. Solved by adding up.

Mahavira. 850.

Chap. IV, v. 28, p. 74: x   x/2   x/4   ...   x/256 = 32.

Chap. VI, v. 314, pp. 175 176: Let a_i+1 = r*a_i + c. He sums such terms.

Fibonacci. 1202. Pp. 313 316 (S: 439-443). Man has 100 and gives away 1/10 of his wealth 12 times. This has been described under 7.E.

Lucca 1754. c1330. F. 10v, pp. 36 37. Computes 2⁴⁰ & 2¹⁰⁰ by repeated squaring.

Columbia Algorism. c1350. Prob. 63, pp. 84 85. Same as the Fibonacci, but he converts to pounds, shillings and pence!

Folkerts. Aufgabensammlungen. 13-15C. Four sources with progressions with ratio 7 and seven sources with ratio 12.

Chuquet. 1484. He gives a number of such problems -- see also 7.E.

Prob. 96, English in FHM 219. Cask of 9½ drains so first barrel takes 1 hour, second barrel takes 2 hours, third barrel takes 4 hours, .... How long to empty? I.e. he wants the sum of 9½ terms of a geometric progression. He gets the correct answer of 2^9.5 - 1 hours.

Prob. 97, English in FHM 219. Man travels 1, 3, 9, ... leagues per day. How far has he travelled in 5½ days? He gets the correct answer of (3^5.5-1)/2 days.

Ghaligai. Practica D'Arithmetica. 1521. Prob. 29, f. 66v. Same as Fibonacci. (H&S 59-60.)

Buteo. Logistica. 1559.

Prob. 69, pp. 276-278. 1 + .9 + (.9)² + ... + (.9)^x = 7.5. The phrasing of the problem is unclear, but this is what he considers. He interpolates linearly between 12 and 13, getting 12.164705107, while the exact answer is (log .25/log .9)   1  =  12.15762696.