7. arithmetic & number theoretic recreations a. Fibonacci numbers

Part II, no. 5: Fill in the gaps, pp. 114 & 191. Division layout with some numbers given.

Part II, no. 11: Find the multiplier, pp. 115 116 & 193. Multiplication layout with some numbers given.

Anon. Sol. 38. Hobbies 30 (No. 756) (9 Apr 1910) 37 & 46. (I don't have the problem proposal -- ??NYS.) ...1 * 7. = 6.... + .6.2._ = ....5. with solution 8061 * 78  =  628758.

??? J. Indian Math Club, 1910. ??NYS -- cited by Archibald who says it has skeleton divisions with 4 digits given.

Smith. Number Stories. 1919. Pp. 111 112 & 139 140. Two cryptarithmic multiplications and two cryptarithmic divisions, but with full layouts.

W. E. H. Berwick. MG (Mar 1920) 43. ??NYS -- cited by Archibald. Four fours.

F. Schuh. Een tweetal rekenkundige ardigheden. Nieuw Tijdschrift voor Wiskunde 8 (1920 1921) 64. Skeleton division with no digits given, but the quotient has a repeating decimal and the divisor and dividend are relatively prime. The problem is reproduced as Note 16, AMM 28 (1921) 278, signed ARC [= R. C. Archibald], with solution by D. R. Curtiss and comment by A. A. Bennett in AMM 29 (1922) 210 213. Bennett shows that relatively primality is not essential. The problem and solution are given as Section 258: Repeating division puzzle, pp. 320 322, in Schuh's The Master Book of Mathematical Recreations, Dover, 1968. (Originally Wonderlijke Problemen; Leerzaam Tijdverdrijf Door Puzzle en Spel, Thieme, Zutphen, 1943.) 7752341 / 667334  =  11.6168830001168830001....

R. C. Archibald. AMM 28 (1921) 37 -- sketches the history: J. Indian Math Club; Berwick's 'seven sevens' and 'four fours' cited to MG (Mar 1920) 43. See The President, 1941.

W. E. H. Berwick. Problem 555: The four fours (in Dudeney's column). Strand Mag. (1921 or 1922?) ??NYS. Four fours given in the skeleton of 1200474 / 846 = 1419, but there are other fours present and there are three other solutions.

Egbert F. Odling. Problem 627: Solitary seven (in Dudeney's column). Strand Mag. (Nov & Dec 1922), ??NYS. Skeleton of 12128316 / 124  =  97809, with only the 7 given. Unique solution and the 7 only occurs once. Repeated as Problem 1105: Skeleton sum; Strand Mag. (Jul 1932) 104 & (Aug 1932) 216.

Anon. Note 671. MG 11 (1922 23) 338. Gives Odling's problem and solution and some comments.

Ackermann. 1925. Pp. 109 115. Discusses Berwick's problems, referring to MG of Mar 1920, Dec 1921 and Jan 1922 (??NYS) for four fours, five fives, three threes and six sixes. Gives one of Smith's problems, Berwick's seven sevens and five fives, Schuh's problem (attributed to Ball) and Odling's problem.

Dudeney. MP. 1926. Prob. 70: The solitary seven, pp. 26 27 & 118. (= 536, prob. 144, pp. 43 & 259.) Cites EFO [= Odling]. "It is the first example I have seen ... in which only one figure is given."

A. A. Bennett, proposer; H. Langman, solver. Problem 3212. AMM 33 (1926) 429 & 34 (1927) 538-540. Skeleton division xxxxxxxxxxxxxccfx / xxxxabxxxx = xcxxxxx, with numerous further positions given by definite letters. Solution notes that some of the information is not needed, e.g. f can be replaced by x. Answer is 70900515872010075 / 68253968253 = 1038775.

Collins. Fun with Figures. 1928. Through a knot-hole, p. 189. *7*9* / 215 = 1** with some further figures given.

H. E. Slaught, proposer; C. A. Rupp, solver. Problem E1. AMM 39 (1932) 489 & 40 (1933) 111-112. Odling's 'Solitary Seven' problem. Editor's note says it has appeared in Le Sphinx.

Perelman. FFF. 1934. Mysterious division & Another division. 1957: probs. 104 & 105, pp. 138 & 145; 1979: probs. 107 & 108, pp. 167 & 176. = MCBF, probs. 107 & 108, pp. 168 & 178-179. Berwick's 'four fours' and 'seven sevens', with all four solutions of the latter, rather poorly attributed to "the American publications School World (1906) and Mathematical Magazine (1920)".

Perelman. FMP. c1935? Mysterious division, pp. 256 & 268 269. Skeleton of 11268996 / 124  =  90879, with only the 7 given. The quotient is unique, but there are 11 possible divisors: 114, 115, ..., 124, of which 115, 116 and 120 give no other 7s in the layout.

A. G. Sillito. Note 1424: Division without figures. MG 23 (No. 257) (Dec 1939) 467 468. Two divisions like Schuh's: 16 / 41 and 81 / 91.

W. T. Williams & G. H. Savage. The Penguin Problems Book. Penguin, 1940.

No. 67: A skeleton square root, pp. 40 & 125. A dotted diagram with two letters marked in, one 8 times, the other 4 times.

No. 87: A skeleton division, pp. 49 & 132. = Odling, 1922.

The President. Some missing-figure divisions. Eureka 6 (May 1941) 21-24. Studies skeleton divisions. Cites 'the solitary seven' and 'the numberless decimal division' 'in one of Dudeney's Puzzle Books' -- presumably MP, 1926. Then gives six problems, four given by Berwick in MG -- ??NYS. I is Four-threes, with complete solution. II is Three-threes, proposed by Berwick, the same as Four-threes with one three not given, which is a harder version for which he does not have a satisfactory solution, though the answer is the same as for case I. III is Berwick's Four-fours, for which Berwick gives the complete solution so here the four answers are only stated. IV is Berwick's Five-fives, again completely solved by Berwick with just the answer here. V is Four-sixes, apparently novel, with complete solution. VI is Seven-sevens, proposed by Berwick, with complete solution here.

Sullivan. Unusual. 1947. Prob. 37: Lost and found. Skeleton division of 1089708 / 12  =  90809 with only the 8 of the quotient given.

P. L. Chessin, proposer; ???, solver. Problem E1111. AMM 61 (Apr 1954) 712 & ???. ??NYS -- given in the Otto Dunkel Memorial Problem Book, ed. by Howard Eves and E. P. Starke, AMM 64:7 part II (Aug-Sep 1957) 6, where it is described as the most popular problem ever published in the AMM, with 70 solvers. Also given by Gardner, SA (May 1959) = 2nd Book, chap. 14, prob. 5: The lonesome 8, pp. 154-155 & 160 161. Skeleton division of 10020316 / 124 = 80809 with only the middle 8 of the quotient given. Answer is unique.

William R. Ransom. Op. cit. in 6.M. 1955. Only one digit known, p. 134. Skeleton of 11260316 / 124  =  90809 with only the 8 given. Answer is unique and 8 only appears once.

Anonymous. Problems drive, 1957. Eureka 20 (Oct 1957) 14-17 & 29-30. No. 9. Gives complete layout of 1345 x 32 and asks to find five digits which would determine the rest.

G. A. Guillotte. Note 2865: Missing digits. MG 43 (No. 345) (Oct 1959) 200. Long division with 17 0s specified.

B. D. Josephson & J. M. Boardman. Problems drive 1961. Eureka 24 (Oct 1961) 20-22 & 24. Prob. E. Full skeleton of 5980482 / 498 with a single 8 given in the last line.

Anonymous postcard to The Science Correspondent, "The Glasgow Herald", 8 May 1963, found in Prof. Lenihan's copy of Gardner's More Mathematical Puzzles and Diversions and given in Jay Books sale catalogue 129 (Feb? 1992) and 130 (Jun 1992). Full skeleton of 1062 / 16 = 66.375 with no digits specified. The solution is unique.

L. S. Harris & J. M. Northover. Problems drive 1963. Eureka 26 (Oct 1963) 10-12 & 32. Prob. J. Complete skeleton of 32943 / 139 with all six 3s given.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 69, pp. 70 & 104. Same as Sullivan, 1947.

Birtwistle. Math. Puzzles & Perplexities. 1971.