7. arithmetic & number theoretic recreations a. Fibonacci numbers

Chap. 31, quest. 2. 1678: p. 325 (misprinted 305 in 1678); 1715: p. 210; 1787: p. 182. B says he is 3/2 A. C  says he is twice B. A says the sum of their ages is 165.

Chap. 32, quest. 2. 1678: pp. 332-333; 1715: p. 214; 1787: p. 186. A says he is 18. B says he is A + C/2. C says he is A + B.

Anonymous proposer and solver. Ladies' Diary, 1708-09 = T. Leybourn, I: 3, quest. 5. [I have a reference to this as Question V and Leybourn also gives this number.] III (3, 15, 2). "A person remarked that upon his wedding day the proportion of his own age to that of his wife was as three to 1; but 15 years afterwards the proportion of their ages was as 2 to 1. What were their ages upon the day of marriage."

Also reproduced in The Diarian Repository, 1774, Wallis 341.5 DIA, ??NX, but I copied it as follows. [This seems to indicate that Leybourn has condensed the material -- ??]

When first the marriage knot was tied

Betwixt my wife and me,

My age to her's we found agreed

As nine doth unto three;

But after ten and half ten years,

We man and wife had been,

Her age came up as near to mine,

As eight is to sixteen.

Now tell me if you can, I pray,

What was our age o' th' wedding day?

Dilworth. Schoolmaster's Assistant. 1743.

P. 92, no. 4. A is 20; B is A + C/2; C = A + B.

P. 167, no. 104. III-(2, 30, 7/5). After another 30 years, we have III-(2, 60, 5/4), when they died. "I demand ... also the reason why the lady's age, which was continually gaining upon her husband's, should, notwithstanding, be never able to overtake it."

Simpson. Algebra. 1745. Section XI (misprinted IX in 1790), prob. XVI, p. 87 (1790: prob. XXII, pp. 85-86). Essentially III (4, 14, 2), though the present time is halfway between the times involved in the problem -- i.e. 7 years ago, the father was 4 times the son's age; in 7 years, he will be 2 times.

Les Amusemens. 1749. Prob. 97, p. 239. Father is three times the age of his son, when will he be only twice as old? -- i.e. III-(3, b, 2) -- with answer b = Y, the son's present age.

Walkingame. Tutor's Assistant. 1751.

1777, p. 82, prob. 5; 1835: p. 91, prob. 4; 1860: p. 11, prob. 5. H is 30. K = H + L/4. L = H + K.

1777, p. 177, prob. 120; 1860: p. 185, prob. 113. (The 1835 edition differs -- see under 1835 below.) III-(3, 10½, 2).

When first the marriage knot was ty'd,

Between my wife and me,

Her age did mine as far exceed,

As three times three does three;

But when seven years, and half seven years,

We man and wife had been,

My age came then as near to her's,

As eight is to sixteen.

Quest. What was each of our ages when we married?

(I find it extraordinary that the man is younger in this version and in no other that I have seen. The 1860 makes a few minor changes and puts it on four lines. Wehman, New Book of 200 Puzzles, 1908, p. 51 is an incomplete copy of this, comprising the first two lines of a four-line version, with the answer.)

Mair. 1765?

P. 454, ex. 5. Same as Cocker, chap. 31, quest. 2.

P. 458, ex. 5. Same as Cocker, chap. 32, quest. 2.

P. 143, no. 11. III (3, 15, 2).

When first the marriage-knot was tied

Betwixt my wife and me,

My age did hers as far exceed,

As three times three doth three;

But after ten and half ten years,

We man and wife had been,

Her age came up as near to mine,

As eight is to sixteen.

Now, Tyro, skill'd in numbers, say,

What were our ages on the wedding-day?

Answer.

Sir, Forty-five years you had been,

Your Bride no more than just fifteen.

(Text copied from 2nd ed, ??NX, and differs very slightly from the 3rd ed.)

Vyse. Tutor's Guide. 1771?

Prob. 3, 1793: p. 128; 1799: pp. 135-136 & Key p. 177. A = C + 4, B = A + C + 9, D = 45 = A + B + C.

Prob. 5, 1793: p. 128; 1799: p. 137 & Key pp. 180-181. B = 5A = 7(A-4).

Prob. 8, 1793: p. 130; 1799: p. 138 & Key pp. 181-182. III-(3, 15, 2).

When first the Marriage-Knot was ty'd

Betwixt my Wife and me,

My Age did her's as far exceed

As three Times three does three;

But when ten Years, and Half ten Years,

We Man and Wife had been,

Her Age came up as near to mine

As eight is to sixteen.

Now, tell me, I pray,

What were our Ages on the Wedding Day?

Dodson. Math. Repository. 1775.

P. 6, Quest. XIV. A + B + 25 = 2A; A - B - 8 = B.

P. 15, Quest. XL. III-(4, 14, 2) stated at the middle of the 14-year period.

P. 31, Quest. LXXVI. Find x/y such that (x+4)/(y+4) = 4/3, (x-4)/(y-4) = 3/2. [In general, (x-a)/(y-a) = r, (x+b)/(y+b) = s has solution x = {br(s 1) + as(r 1)}/(r-s), y = {b(s-1) + a(r-1)}/(r-s). Cf Vinot, below.]

Charles Hutton. A Complete Treatise on Practical Arithmetic and Book-keeping. Op. cit. in 7.G.2. [c1780?] 1804: prob. 2, p. 132. III-(5, 5, 4).

Bonnycastle. Algebra. 1782. Prob. 11, p. 84 (1815: prob. 10, p. 105). A = 2B; B = 3C; A + B + C = 140.

Pike. Arithmetic. 1788. P. 335, no. 7. B = 3A/2; C = 21 (A + B)/10; A + B + C = 93.

Anon. The American Tutor's Assistant. (1st ed. is unknown; 2nd ed., Philadelphia, 1791); 1810 ed., Joseph Crukshank, Philadelphia. ??NYS -- quoted in: Lucas N. H. Bunt et al.; The Historical Roots of Elementary Mathematics; Prentice-Hall, 1976; p. 33. III (3, 15, 2).

When first the marriage knot was ty'd

Between my wife and me,

My age was to that of my bride

As three times three to three

But now when ten and half ten years,

We man and wife have been,

Her age to mine exactly bears,

As eight is to sixteen;

Now tell, I pray, from what I've said,

What were our ages when we wed?

Answer:

Thy age when marry'd must have been

Just forty-five; thy wife's fifteen.

Eadon. Repository. 1794. P. 297, no. 16. III-(3, 15, 2) in verse.

When first the marriage knot was ty'd

Betwixt my wife and me,

My age did her's as far exceed,

As three times three doth three;

But after ten, and half ten years,

We man and wife had been,

Her age came up as near to mine,

As eight is to sixteen.

Now tell me (you who can) I pray,

What were our ages on the wedding day?

Hutton. A Course of Mathematics. 1798? Prob. 1, 1833 & 1857: 80. III-(5, 5, 4). On 1833: 224-231; 1857: 228-235, he has an extensive discussion Remarks upon Equations of the First Degree concerning possible negative roots and considers IV (42, 12, 4) whose solution is -2.

Bonnycastle. Algebra. 10th ed., 1815. P. 104, no. 7. III-(3, 15, 2).

L. Murray. The Young Man's Best Companion, and Book of General Knowledge; .... Thomas Kelly, London, (Preface dated 1814; BL has 1821), 1824. P. 177, example 2. III-(3, 14, 2) stated at the middle of the 14 years.

Manuel des Sorciers. 1825. P. 81, art. 41. ??NX IV-(54, 18, 2).

Augustus De Morgan. Arithmetic and Algebra. (1831 ?). Reprinted as the second, separately paged, part of: Library of Useful Knowledge -- Mathematics I, Baldwin & Craddock, London, 1836. Art. 113, p. 29. IV-(40, 8, 3) -- "in how many years hence will the father's age be just three times the son's?" Then gives general solution of IV-(a, b, 3). Observes that a = 40, b = 18 leads to -7 and that a = 3b leads to 0 and discusses the meaning of both situations. Then solves the general case IV-(a, b, n).

Child. Girl's Own Book. Arithmetical Puzzles. 1832: No. 8, pp. 171 & 179; 1833: No. 8, pp. 185 & 193 (answer numbered 6); 1839: No. 8, pp. 165 & 173; 1842: No. 8, pp. 283 & 291; 1876: No. 6, pp. 231 & 244. III (3, 15, 2) given in verse which is the same in the 1832, 1839 and 1876 eds. (Except 1839 & 1842 have her's in line three.)

When first the marriage knot was tied

Between my wife and me,

My age exceeded hers as much,

As three times three does three.

But when ten years and half ten years

We man and wife had been,

Her age approached as near to mine

As eight is to sixteen.

The 1833 ed has the first lines of the second verse garbled as: But when the man and wife had been, / For ten and half ten years.

= Fireside Amusements; 1850: Prob. 7, pp. 132 &184.

Bourdon. Algèbre. 7th ed., 1834. Art. 58, pp. 87-89. Discusses the problem IV-(a, b, 4) and the significance of negative solutions, using IV-(45, 15, 4) as an example.

Walkingame. Tutor's Assistant. 1835. P. 180, prob. 59. III-(3, 15, 2).

When first the marriage knot was ty'd

Between my wife and me,

My age did hers as far exceed,

As three times three does three;

But when ten years, and half ten years,

We man and wife had been,

Her age came then as near to mine,

As eight is to sixteen.

= Depew; Cokesbury Game Book; 1939; Marriage problem, pp. 207-208.

Augustus De Morgan. On The Study and Difficulties of Mathematics. First, separately paged, part of: Library of Useful Knowledge -- Mathematics I, Baldwin & Craddock, London, 1836. P. 53. IV-(56, 29, 2) -- "when will the father be twice as old as the son?" Answer is -2 and he discusses the meaning of this.

Unger. Arithmetische Unterhaltungen. 1838.