7. arithmetic & number theoretic recreations a. Fibonacci numbers



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Use the same odd figure five times to make 14.

Four 9s to make 100.

Four 5s to make 6½.


Depew. Cokesbury Game Book. 1939.

Three eights, p. 216. Use three 8s to make 7.

Twenty-four, p. 227. Use a digit three times to make 24. Answers: 33 - 3, 22 + 2.


McKay. Party Night. 1940. No. 9, p. 177.

(a) Use three 9s to make 10. Answer: 9 9/9.

(b) Use four 9s to make 20. Answer: 9 99/9.

(c) Use three 9s to make 100. Answer: 99.9 (or 99.9 for clarity).

(d) Use two 9s to make 10. Answer: 9/.9 or 9.9 (or 9.9).


Meyer. Big Fun Book. 1940. A half dozen equals 12, pp. 119 & 738. Use six 1s to make 12. Answer: 11 + 11/11.

George S. Terry. The Dozen System. Longmans, Green & Co., NY, 1941. ??NYS -- quoted in: Underwood Dudley; Mathematical Cranks; MAA, 1992, p. 25. What numbers can be expressed with four 4s duodecimally? About 5 dozen. How many numbers can be expressed using each of the digits 1, 2, 3, 4 once only, again duodecimally? About 9 dozen and nine. Terry (or Dudley) also gives the results for decimal working as 22 and 88.

Sullivan. Unusual. 1943. Prob. 17: Five of a kind. Write 100 with the same figure five times "and the usual mathematical symbols". Says it can be done with 1s, 2s, 5s (two ways), 9s and perhaps others.

J. A. Tierney, proposer; Manhattan High School of Aviation Trades, D. H. Browne, H. W. Eves, solvers. Problem E631 -- Two fours. AMM 51 (1944) 403 & 52 (1945) 219. Express 64 using two 4s.

Vern Hoggatt & Leo Moser, proposers and solvers. Problem E861 -- A curious representation of integers. AMM 56 (1945) 262 & 57 (1946) 35. Represent any integer with p a's, for any p  3 and any a  1. Solution for ±n uses log to base ...a, with n radicals.

S. Krutman. Curiosa 138: The problem of the four n's. SM 13 (1947) 47.

Sullivan. Unusual. 1947. Prob. 28: A problem in arithmetic. What is the smallest number of eights which make 1000?

G. C. S[hephard, ed.] The problems drive. Eureka 11 (Jan 1949) 10-11 & 30.


No. 4. Use 1, 2, 3, once each to make 19. Answer: (2/.1) - [3]. Ibid. 12 (Oct 1949) 17 gives a simpler answer: (1 + 3!!/2).

No. 7/ Use four 1s to express 7, 37, 71, 99. Answers: (1+1+1)! + 1; 111 x .1 [.1 is .111..., but may not show up clearly]; .1 x ((1/.1)!! - 1 [same comment on .1]; 1/(.1 x .1) - 1.


Anonymous. The problems drive. Eureka 13 (Oct 1950) 11 & 20-21.

No. 4: Start with 2 and use cubing and integral part of square root to form any positive integer. m cubings, followed by n roots gives 2^(3m/2n) = 2^(2ma-n), where a = log2 3. Since a is irrational, we can choose ma - n so that 2^(2ma-n) is arbitrarily close to N + ½, so the integer part of it is N.

No. 6: Use four 4s to approximate π. They get 3.14159862196..., using a nine-fold root.


Anonymous. The problems drive. Eureka 17 (Oct 1954) 8-9 & 16-17. No. 5. Use four 4s to express 37; 57; 77; 97; 123.

D. G. King Hele. Note 2509: The four 4's problem. MG 39 (No. 328) (May 1955) 135. n  =  log[{log 4}/{log n 4}]/log 4 expresses any positive integer n in terms of three 4s. A slight variation expresses n in terms of four x's, for any real x  0, 1. 1  can be used by taking x = .1. He also expresses n in terms of m x's for real x  0, 1 with m > 5 and also with m = 5.

Anonymous. Problems drive. Eureka 18 (Oct 1955) 15-17 & 21. No. 6. Use four 4s to make 7; 17; 37; 3,628,800.

Anonymous. Problems drive, 1958. Eureka 21 (Oct 1958) 14-16 & 30. No. 10. Use 1, 2, 3, 4, 5, in order to form 100; 3 1/7; 32769.

Philip E. Bath. Fun with Figures. Op. cit. in 5.C. 1959.

No. 12: One hundred every time, pp. 10 & 41. Make an arrangement of x's which gives the result 100 for x = 1, 2, ..., 9. Answer: xxx/x - xx/x.

No. 20: Form fours, pp. 12 & 42. Eight 4s to make 500.

No. 74: Signs wanted, pp. 28 & 54. Insert signs (+, -, x, /) into a row of four x's to make 10 - x, for x = 2, ..., 9.


M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 & 22-23. No. 3. Use three 1s to make the integers from one to twelve, using only arithmetic symbols. (No trigonometric functions or integer parts allowed.)

Young World. c1960. P. 54. Use five 9s to make 1000.

B. D. Josephson & J. M. Boardman. Problems drive 1961. Eureka 24 (Oct 1961) 20-22 & 24. Prob. K. Use three 7s to express 1, ..., 11, using only arithmetic symbols.

J. H. Conway & M. J. T. Guy. π in four 4's. Eureka 25 (Oct 1962) 18 19. Cite Eureka 13 (1950). Note that π  =  [( 4/4)!]4, if non integral factorials are allowed. Show that any real number can be arbitrarily well approximated using four 4s.

R. L. Hutchings & J. D. Blake. Problems drive 1962. Eureka 25 (Oct 1962) 20-21 & 34-35. Prob. H. Use four identical digits to represent 100 in as many ways as possible, but not using representations which are independent of the digit used, like (5x5)/(.5x.5). The give eight examples, using 9, 5, 5, 4, 4, 3, 3, 9, 1, and say there are more.

D. E. Knuth. Representing numbers using only one 4. MM 37 (1964) 308 310.

Gardner. SA (Jan 1964) adapted as Magic Numbers, chap. 5. Cites Knowledge as the origin. Magic Numbers gives numerous other citations.

Marjorie Bicknell & Verner E. Hoggatt. 64 ways to write 64 using four 4's. RMM 14 (Jan Feb 1964) 13 15.

Jerome S. Meyer. Arithmetricks. Scholastic Book Services, NY, 1965. Juggling numbers, no. 3, pp. 83 & 88.

Make 100 with four 7s. 77/.77.

Make 20 with two 3s. 3!/.3.

Make 7 with four 2s. (2/.2)/2 + 2.

Make 37 with six 6s. 6*6 + 66/66.


Ripley's Puzzles and Games. 1966. Pp. 16-17, item 2. Use 13 3s to make 100.

Steven Everett. Meanwhile back in the labyrinth. Manifold 10 (Autumn 1971) 14 16. (= Seven Years of Manifold 1968 1980; ed. by I. Stewart & J. Jaworski; Shiva Publishing, Cheshire, 1981, pp. 64 65.) n =  4 * log4 log4n4, where log4 means log to the base 4 and n means n fold iterated square root. This is a variant of King Hele's form. This article is written in a casual style and seems to indicate that this formula was devised by Niels Bohr. He gets a form for e, but it uses infinitely many factorial signs!!!!...

Editor's note on p. 2 (not in the collection) gives an improvement due to Michael Gerzon, but it is unclear what is intended. The note gives another method due to Professor Burgess using sec tan-1 m = (m+1) and 1 = nn which expresses n by one 1.

[Henry] Joseph and Lenore Scott. Master Mind Pencil Puzzles. 1973. Op. cit. in 5.R.4. Numbers-numbers, part 3, pp. 109-110. Use 13 3s to make 100. The give 33 + 33 + 33 + (3/3)3 + 3x3 + 3x3. I found 33 + 33 + 33 + 33/3 - 3x3 - 3/3, which seems simpler.

Ball. MRE, 12th ed., 1974. Pp. 15-17. Under "Four fours problem", the material of the 9th ed. and the footnotes mentioned at 7th ed are repeated, but the bound for four 9s is increased.

Bronnie Cunningham. Funny Business. An Amazing Collection of Odd and Curious Facts with Some Jokes and Puzzles Too. Puffin, 1978. Pp. 38 & 142. Arrange three 9s to make 20. Answer: (9 + 9)/.9.

Putnam. Puzzle Fun. 1978.



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