Sources page biographical material

5.C. FALSE COINS WITH A BALANCE
See 5.D.3 for use of a weighing scale.

There are several related forms of this problem. Almost all of the items below deal with 12 coins with one false, either heavy or light, and its generalizations, but some other forms occur, including the following.

8 coins, 1 light: Schell, Dresner

26 coins, 1 light: Schell

8 coins, 1 light: Bath (1959)

9 coins, 1 light: Karapetoff, Meyer (1946), Meyer (1948), M. Adams, Rice

I have been sent an article by Jack Sieburg; Problem Solving by Computer Logic; Data Processing Magazine, but the date is cut off -- ??
E. D. Schell, proposer; M. Dernham, solver. Problem E651 -- Weighed and found wanting. AMM 52:1 (Jan 1945) 42 & 7 (Aug/Sep 1945) 397. 8 coins, at most one light -- determine the light one in two weighings.

Benjamin L. Schwartz. Letter: Truth about false coins. MM 51 (1978) 254. States that Schell told Michael Goldberg in 1945 that he had originated the problem.

Emil D. Schell. Letter of 17 Jul 1978 to Paul J. Campbell. Says he did NOT originate the problem, nor did he submit the version published. He first heard of it from Walter W. Jacobs about Thanksgiving 1944 in the form of finding at most one light coin among 26 good coins in three weighings. He submitted this to the AMM, with a note disclaiming originality. The AMM problem editor published the simpler version described above, under Schell's name. Schell says he has heard Eilenberg describe the puzzle as being earlier than Sep 1939. Campbell wrote Eilenberg, but had no response.

Schell's letter is making it appear that the problem derives from the use of 1, 3, 9, ... as weights. This usage leads one to discover that a light coin can be found in 3ⁿ coins using n weighings. This is the problem mentioned by Karapetoff. If there is at most one light coin, then n weighings will determine it among 3ⁿ   1 coins, which is the form described by Schell. The problem seems to have been almost immediately converted into the case with one false coin, either heavy or light.

Walter W. Jacobs. Letter of 15 Aug 1978 to Paul J. Campbell. Says he heard of the problem in 1943 (not 1944) and will try to contact the two people who might have told it to him. However, Campbell has had no further word.
V. Karapetoff. The nine coin problem and the mathematics of sorting. SM 11 (1945) 186 187. Discusses 9 coins, one light, and asks for a mathematical approach to the general problem. (?? -- Cites AMM 52, p. 314, but I cannot find anything relevant in the whole volume, except the Schell problem. Try again??)

Dwight A. Stewart, proposer; D. B. Parkinson & Lester H. Green, solvers. The counterfeit coin. In: L. A. Graham, ed.; Ingenious Mathematical Problems and Methods; Dover, 1959; pp. 37 38 & 196 198. 12 coins. First appeared in Oct 1945. Original only asks for the counterfeit, but second solver shows how to tell if it is heavy or light.

R. L. Goodstein. Note 1845: Find the penny. MG 29 (No. 287) (Dec 1945) 227 229. Non optimal solution of general problem.

Editorial Note. Note 1930: Addenda to Note 1845. Ibid. 30 (No. 291) (Oct 1946) 231. Comments on how to extend to optimal solution.

Howard D. Grossman. The twelve coin problem. SM 11:3/4 (Sep/Dec 1945) 360 361. Finds counterfeit and extends to 36 coins.

Lothrop Withington, Jr. Another solution of the 12 coin problem. Ibid., 361 362. Finds also whether heavy or light.

Donald Eves, proposer; E. D. Schell & Joseph Rosenbaum, solvers. Problem E712 -- The extended coin problem. AMM 53:3 (Mar 1946) 156 & 54:1 (Jan 1947) 46 48. 12 coins.

Jerome S. Meyer. Puzzle Paradise. Crown, NY, 1946. Prob. 132: The nine pearls, pp. 94 & 132. Nine pearls, one light, in two weighings.

N. J. Fine, proposer & solver. Problem 4203 -- The generalized coin problem. AMM 53:5 (May 1946) 278 & 54:8 (Oct 1947) 489 491. General problem.

H. D. Grossman. Generalization of the twelve coin problem. SM 12 (1946) 291 292. Discusses Goodstein's results.

F. J. Dyson. Note 1931: The Problem of the Pennies. MG 30 (No. 291) (Oct 1946) 231 234. General solution.

C. A. B. Smith. The Counterfeit Coin Problem. MG 31 (No. 293) (Feb 1947) 31 39.

C. W. Raine. Another approach to the twelve coin problem. SM 14 (1948) 66 67. 12 coins only.

K. Itkin. A generalization of the twelve coin problem. SM 14 (1948) 67 68. General solution.

Howard D. Grossman. Ternary epitaph on coin problems. SM 14 (1948) 69 71. Ternary solution of Dyson & Smith.

Jerome S. Meyer. Fun-to-do. A Book of Home Entertainment. Dutton, NY, 1948. Prob. 40: Nine pearls, pp. 41 & 188. Nine pearls, one light, in two weighings.

Blanche Descartes [pseud. of Cedric A. B. Smith]. The twelve coin problem. Eureka 13 (Oct 1950) 7 & 20. Proposal and solution in verse.

J. S. Robertson. Those twelve coins again. SM 16 (1950) 111 115. Article indicates there will be a continuation, but Schaaf I 32 doesn't cite it and I haven't found it yet.

E. V. Newberry. Note 2342: The penny problem. MG 37 (No. 320) (May 1953) 130. Says he has made a rug showing the 120 coins problems and makes comments similar to Littlewood's, below.

J. E. Littlewood. A Mathematician's Miscellany. Methuen, London, 1953; reprinted with minor corrections, 1957 (& 1960). [All the material cited is also in the later version: Littlewood's Miscellany, ed. by B. Bollobás, CUP, 1986, but on different pages. Since the 1953 ed. is scarce, I will also cite the 1986 pages in (  ).] Pp. 9 & 135 (31 & 114). "It was said that the 'weighing pennies' problem wasted 10,000 scientist hours of war work, and that there was a proposal to drop it over Germany."

John Paul Adams. We Dare You to Solve This! Berkley Publishing, NY, nd [1957?]. [This is apparently a collection of problems used in newspapers. The copyright is given as 1955, 1956, 1957.] Prob. 18: Weighty problem, pp. 13 & 46. 9 equal diamonds but one is light, to be found in 2 weighings.

Hubert Phillips. Something to Think About. Revised ed., Max Parrish, London, 1958. Foreword, p. 6 & prob. 115: Twelve coins, pp. 81 & 127 128. Foreword says prob. 115 has been added to this edition and "was in oral circulation during the war. So far as I know, it has only appeared in print in the Law Journal, where I published both the problem and its solution." This may be an early appearance, so I should try and track this down. ??NYS

Dan Pedoe. The Gentle Art of Mathematics. (English Universities Press, 1958); Pelican (Penguin), 1963. P. 30: "We now come to a problem which is said to have been planted over here during the war by enemy agents, since Operational Research spent so many man hours on its solution."

Philip E. Bath. Fun with Figures. The Epworth Press, London, 1959. No. 7: No weights -- no guessing, pp. 8 & 40. 8 balls, including one light, to be determined in two weighings. Method actually works for  1 light.

M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15-17 & 22-23. No. 9. Five boxes of sugar, but some has been taken from one box and put in another. Determine which in least number of weighings. Does by weighing each division of A, B, C, D into two pairs.

Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. The "False Coin" problem, pp. 178-182. Sketches history and solution.

Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 46: Dud reckoning, pp. 21 & 94. Find one light among eight in two weighings.

Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden-Bartell Books, 1965. Prob. 55, pp. 57 & 98. Six identical appearing coins, three of which are identically heavy. In two weighings, identify two of the heavy coins.

Charlie Rice. Challenge! Hallmark Editions, Kansas City, Missouri, 1968. Prob. 7, pp. 22 & 54-55. 9 pearls, one light.

Jonathan Always. Puzzling You Again. Tandem, London, 1969. Prob. 86: Light weight contest, pp. 51 52 & 106 107. 27 weights of sizes 1, 2, ..., 27, except one is light. Find it in 3 weighings. He divides into 9 sets of three having equal weights. Using two weighings, one locates the light weight in a set of three and then weighing two of these with good weights reveals the light one. [3 weights 1, 2, 3 cannot be done in one weighing, but 9 weights 1, 2, ..., 9 can be done in two weighings.]

Robert H. Thouless. The 12 balls problem as an illustration of the application of information theory. MG 54 (No. 389) (Oct 1970) 246 249. Uses information theory to show that the solution process is essentially determined.

Ron Denyer. Letter. G&P, No. 37 (Jun 1975) 23. Asks for a mnemonic for the 12 coins puzzles. He notes that one can use three predetermined weighings and find the coin from the three answers.

Basil Mager & E. Asher. Letters: Coining a mnemonic. G&P, No. 40 (Sep 1975) 26. One mnemonic for a variable method, another for a predetermined method.

N. J. Maclean. Letter: The twelve coins. G&P, No. 45 (Feb 1976) 28-29. Exposits a ternary method for predetermined weighings for (3ⁿ-3)/2 in n weighings. Each weighing determines one ternary digit and the resulting ternary number gives both the coin and whether it is heavy or light.

Tim Sole. The Ticket to Heaven and Other Superior Puzzles. Penguin, 1988. Weighty problems -- (iii), pp. 124 & 147. Nine equal pies, except someone has removed some filling from one and inserted it in a pie, possibly the same one. Determine which, if any, are the heavy and light ones in 4 balancings.

Calvin T. Long. Magic in base 3. MG 76 (No. 477) (Nov 1992) 371-376. Good exposition of the base 3 method for 12 coins.

Ed Barbeau. After Math. Wall & Emerson, Toronto, 1995. Problems for an equal-arm balance, pp. 137-141.

1. Six balls, two of each of three colours. One of each colour is lighter than normal and all light weights are equal. Determine the light balls in three weighings.

2. Five balls, three normal, one heavy, one light, with the differences being equal, i.e. the heavy and the light weigh as much as two normals. Determine the heavy and light in three weighings.

3. Same problem with nine balls and seven normals, done in four weighings.