7. arithmetic & number theoretic recreations a. Fibonacci numbers

Part 3, F. 281v, no. 133: Dimme come farrai a partir vinti in 5 parti despare (Tell me how to divide 'vinti' into five odd parts) = Peirani 407. Divides as v.i.n.t.i, and mentions dividing 20 into 7 pens.

W. Leybourn. Pleasure with Profit. 1694. ??NYS -- described in Cunnington, op. cit. in 7.G.2, 1904, p. 151 and in De Morgan, Rara, p. 633. "How can you put five odd numbers to make twenty?" "Write three nines upside down and two ones." De Morgan says he does not recall ever seeing this problem, that Leybourn considers the answer a fallacy, but that he thinks "the question more than answered, viz. in very odd numbers."

Les Amusemens. 1749.

P. xxv.

Quatre fois trois font quinze, il n'en faut rien rabattre;

Neuf cing et un font douze, et rien de surplus;

Deux sept et six font treize, et sur ce je conclus

Par ce juste calcul que tout ne fait que quatre.

Solution is to take the number of letters in the words -- but 'et' is printed like '&' which makes it a bit hard to recognize it as a two-letter word.

P. 52. 1^o: Exprimer un nombre pair par 3 impairs. General solution: a a/a = a+1. Examples: 7 7/7, 21 21/21. Pp. 53-54 give other problems but they are not relevant, e.g. 33 3/3 -- see entry under 7.I.

Philip Breslaw (attrib.). Breslaw's Last Legacy. 1784? Op. cit. in 6.AF. 1795: 78-81. 'How to rub out Twenty Chalks at five Times rubbing out, every time an odd one.' Set out marks numbered 1 to 20. Then erase the last four, which are those starting at 17, which is odd, etc.

Henri Decremps. Codicile de Jérôme Sharp, .... Op. cit. in 4.A.1. 1788. Avant Propos, pp. 20-21. Statement is the same as in Breslaw, but solution is not given.

Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Parlour magic, no. xx, p. 202 (1868: 203): How to rub out twenty chalks in five rubs, each time erasing an odd number. "Begin at the bottom and rub out upwards, four at a time." See Breslaw for clarification.

Magician's Own Book. 1857. How to rub out twenty chalks at five times, rubbing out every time an odd one, p. 239. = Boy's Own Conjuring Book, 1860, How to rub twenty chalks at five times rubbing out, every time an odd one, pp. 205 206. Cf Breslaw.

Magician's Own Book (UK version). 1871. The Arabian trick, p. 313. "To take up twenty cards, at five times, and each time an odd-numbered one", he lays out twenty cards, 1   10, 1 - 10 (considered as 1 - 20) and proceeds as in Breslaw.

Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 136, no. 3. "Place 15 sheep in 4 pens, so there will be the same number of sheep in each pen." Though not of the same type as others in this section, and no solution is given, I think the solution is similar to other solutions here, namely to put the pens concentrically around the inner pen and put all the sheep inside the inner pen.

Pearson. 1907. Part II.

No. 111, pp. 137 & 213. Five odd figures to make 14. Gives 1 + 1 + 1 + 11 = 14 and 1 + 1 + 1 + 1 = 4 with another 1 makes 14.

No. 115: What are the odds?, pp. 137 & 214. Place 20 horses in three stalls with an odd number in each. Puts 1, 3, 16 and says 16 is "an odd number to put into any stall".

Part III, no. 61: The shepherd's puzzle, p. 61. Put 21 sheep in 4 pens, an odd number in each. Uses concentric pens.

Mr. X [cf 4.A.1]. His Pages. The Royal Magazine 27:1 (Nov 1911) 89: The Ass-tute farmer. Farmer has 17 asses and a friend bets he can't put an odd number in each of four stalls. Farmer puts 7, 5, 3, 2. After inspecting the first three stalls, the friend says the farmer has lost, but the farmer says to go into the last stall to check, whereupon he shuts and locks the door, announcing there are three asses in the stall!

Loyd. Cyclopedia. 1914. The pig sty problem, pp. 37 & 343. = MPSL2, prob. 7, pp. 6 7 & 123. = SLAHP: Pigs in pairs, pp. 51 & 104. = Pearson's shepherd's puzzle.

Loyd. Cyclopedia. 1914. A tricky problem, p. 38. = SLAHP: Torturing Dad, pp. 75 & 115. Five odd figures to make 14. Gives Pearson's first solution.

Smith. Number Stories. 1919. Pp. 126 & 146. Put 10 pieces of sugar in three cups so each cup has an odd number. Put 7 & 3 and put one cup inside another cup.

Blyth. Match-Stick Magic. 1921. The twenty game, p. 79. As in Breslaw, etc., but more clearly expressed: "The matchsticks have now to be removed in five lots, .... Each time ... the last of the group must be an odd number."

Hummerston. Fun, Mirth & Mystery. 1924. Puzzle no. 64, pp. 149 & 182. "How would you arrange twenty horses in three stalls so as to have an odd number of horses in each stall?" Arranges as 1, 3, 16 -- "sixteen is a very odd number of horses to put into any stall."

Wood. Oddities. 1927. Prob. 50: The lumps of sugar, pp. 42-43. Ten lumps of sugar into three cups so each cup contains an odd number of lumps. Confusing solution, but lets one cup be inside another and lists all 15 possible solutions.

Evelyn August. The Black-Out Book. Op. cit. in 5.X.1. 1939. Number, Please!, pp. 20 & 210. Use the same odd figure five times to make 14. 1 + 1 + 1 + 11.

McKay. Party Night. 1940. Pill-taking extraordinary, p. 152. "A man had a box holding 100 pills. He took an odd number of pills on each of the seven days of the week, and at the end of the week all the pills were gone. How could he manage that?" 1 on each of the first 6 days and 94 on the last -- "you must admit that 94 is a very odd number of pills to take on any day."

Jerome S. Meyer. Fun-to-do. Op. cit. in 5.C. 1948. Prob. 17: Pigs and pens, pp. 27 & 184. Put nine pigs in four pens, an odd number in each pen. Three pens of three with a big pen around them all.

The Little Puzzle Book. Op. cit. in 5.D.5. 1955. P. 19: Pigs in pens. Same as Meyer.

John Paul Adams. We Dare You to Solve This!. Op. cit. in 5.C. 1957?

Prob. 40: A pen pincher, pp. 24 & 40. 9 pigs in four pens, an odd number in each. Three pens of three, contained in one large pen.