To Find Fake Coin

December 22, 2006

Ian Stewart

Mathematicians who work with the media need to be versatile, quick on their feet, and endowed with a capacious memory, an extensive library, or both. What interests the media, at any given moment, is usually unpredictable and occasionally bizarre. One of my weirdest media interludes occurred in February 2003, when my long-term collaborator (and outgoing SIAM president) Marty Golubitsky was visiting, and it took both of us to deal with it. We lost a day we had set aside for some research, but it was time well spent.

It all began with a short letter to the Daily Telegraph newspaper, from Harold Hop-wood of Gravesend. Mr. Hopwood, who had solved the newspaper's crossword every day since 1937, was writing about a conundrum that had been nagging at the back of his mind ever since he was alerted to it, during a boat trip on the Danube. He and his family had devoted many hours to the puzzle, without success, and at the age of 82 he had finally decided to enlist some help. The puzzle described in his letter concerns 12 balls, 11 of which have the same weight and one that is either lighter or heavier than the others. The problem is to find out which ball is different, and whether it is light or heavy, using at most three weighings on a pair of scales.

A few days later the paper reported that "by teatime yesterday, the Daily Telegraph had received its biggest mailbag in living memory, and our telephones were still ringing off the hook." The letters desk handled 362 letters and calls, nearly all asking for the answer. A few readers offered solutions, but the paper felt unable to judge their correctness, and they phoned me.

I recognized the problem as one of the classic puzzles, typical of the "weights and scales" genre, but I'd forgotten the answer. Marty, who was in the room when I answered the phone, also recognized the problem, and had some welcome news. The selfsame puzzle had inspired him as a teenager, and his success in solving it had led him to become a mathematician. Less welcome was the news that he had forgotten how the solution went, but after some heated discussion we came up with an algorithm, weighing various sets of coins against various others, which did the job. We then had to explain the solution to the newspaper's editor, so that he could get an artist to draw suitable diagrams.

This was challenging. The entire discussion was carried out at breakneck speed by phone, fax, and e-mail, because the paper wanted an answer for the next day. The process took several hours, and we just scraped through inside the deadline.
There are, of course, many answers, including a very clever and elegant one, which I finally remembered on the day the paper printed our kludge. The clever solution is, naturally, far superior, except that it comes as a deus ex machina without the slightest motivation. I later realized that I had read it as a schoolboy, in New Scientist, and that it was sitting on my bookshelf as chapter two of Thomas O'Beirne's Puzzles and Paradoxes, a collection of puzzle columns from the magazine.

Hoary old puzzles like this seem to come round every twenty years or so, presumably when a new generation is exposed to them, a bit like an epidemic that gets a new lease of life when the population loses all immunity. O'Beirne traced it back to Howard Grossman in 1945, but it is almost certainly much older, going back at least to the 17th century. It wouldn't surprise me if one day we found it on a Babylonian cuneiform tablet.

O'Beirne offered a "decision tree" solution, along the lines that Marty and I had concocted. He also recalled the elegant 1950 solution published by "Blanche Descartes" in Eureka, the journal of the Archimedeans, Cambridge University's undergraduate mathematics society. I edited Eureka in 1965, as it happens, and therefore knew the editorial folklore---in particular, that Ms. Descartes was in actuality one Cedric A.B. Smith. I even own a copy of the 1950 issue, now a collector's piece.

Smith's solution is a masterpiece of ingenuity. It is presented as a poem about a certain Professor Felix Fiddlesticks, and the key part (edited for conciseness) goes like this:

F set the coins out in a row
And chalked on each a letter, so,
To form the words "F AM NOT LICKED"
(An idea in his brain had clicked).
And now his mother he'll enjoin:

This cryptic list of three weighings, one set of four against another, solves the problem, and this can be proved by exhaustive enumeration---as the solution in Eureka explains, still in verse.

I have a feeling that the entire incident typifies a particularly European attitude, perhaps even a British one. I find it hard to imagine anyone writing a letter to The New York Times about such a question, and even harder to imagine the paper printing it. But perhaps I am wrong---consider Marilyn and the goats, a.k.a. the Monty Hall puzzle, in Parade magazine.

Publication of a valid solution did not end the matter. It never does. Readers wrote in to object to our answer, on spurious grounds. They wrote to improve it, not always by valid methods. They e-mailed to point out Ms. Descartes' solution or similar ones. They told us about other weighing puzzles. They thanked us for setting their minds at rest. They cursed us for reopening an old wound. It was as if some vast, secret reservoir of folk wisdom had suddenly been breached, spilling its contents all over the collective consciousness.

There were critics. One reader objected to the paper's publication of a solution by two math professors, on the grounds that it suggested that you had to be a math professor to solve the puzzle. Quite so---and for the same reason, politicians should never be asked to comment on terrorism, which would suggest that you have to be a politician to have an opinion about terrorists. The real reason that the paper had consulted math professors was slightly different: The professors had the credentials to convince the newspaper, and its readers, that their answer was right. Whether it was or not.

Another correspondent remembered the puzzle featuring on BBC television in the 1960s, presented by anchorman Cliff Michelmore in an evening program, with the solution given the following night. Ominously, the letter continued, "I do not recall why it was raised in the first place, or whether that was my first acquaintance with it; I have a feeling that it was not." I am absolutely certain that the Daily Telegraph's disinterment of this ancient chestnut will not be our last acquaintance with the 12-ball puzzle.

Er--please don't write. . . .

Ian Stewart is a professor of mathematics at Warwick University, engaged in research and efforts to enhance public awareness of mathematics. He is also a science writer and a science fiction writer.

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