Apart from Fermat's famous proof,
mathematics has more obvious ...

genuine fakes

Barry Pilton

Throughout history there has been one major problem which has faced scientist and mathematician alike. I am not referring to the problem of Squaring the Circle, but to the more practical one of Cooking the Books.

Faking statistical results is a hazardous enterprise; there are times when the observed results simply do not fit the theory one is trying to prove, and then the only possible recourse is to fiddle the results (or some similar viol practice). The great thing about statistical analyses is that they can be carried out years after the figures are produced (this is the customary procedure in government circles); and advances in statistical methods can sometimes reveal faking where no faking was previously observed.

The 'fit' of experimental theory with actual figures, as well an being too bad, can also be too good. If you throw 1000 coins and claim to have got exactly 500 heads and 500 tails, don't claim it too often; for the probability of getting a fit this good is only 1 in a hundred or therabouts. Recent analysis of certain experiments bearing this in mind has come up with a number of cases where fakery of some form is apparent.

One remarkable instance is that of Mendel, whose experiments with plants (the good man was a monk and so confined his genetic experiments to botany) form the foundation of modern theories of heredity. Taken as a whole Mendel's results are too good to be true. No-one is suggesting that he knowingly faked his results, but it does seem that when he was deciding whether a particular plant was 'dwarf' or 'tall' his judgement was not quite impartial enough. Another concerns a man by the name of Moewus who counted different types of algae (he claimed), getting results agreeing so well with the expected figures that, had the experiment been repeated by the whole human race every day for 10000000000 years, as good a fit might have happened once. Maybe this was just luck, but it's not the sort of luck to put in a Ph.D. thesis.

OK, so the scientists fake their figures; we mathematicians have always suspected as much anyway. But what of mathematical fakery? Not much chance there... but indeed there is, and an instance of it is a very famous piece of experimental work by Lazzerini to find the value of p by dropping needles onto ruled lines. This has been reported by various eminent authorities with a little too much respect. That Lazzerini was a successful hoaxer has been shown independently by O'Beirne (4) and Gridgeman(2); this article is taken largely from (4).

First a piece of theory, due to Count Buffon. Suppose we have a needle of length d, and drop it on to a grid of parallel lines distance a apart, where d<a. What is the probability that the needle lands across a line? The answer, which we shall not calculate here, turns out to be p = 2d/pa so by doing an experiment taking ratio of successes to total trials as p, we get an experimental value for p. In 1901 Lazzerini made 3408 tosses and obtained p = 3.1415929 in error by about 0.0000003. Now this is a remarkably close agreement but until recently only one authority cast any doubt on it. In (1) p.176, Coolidge says: "... it seems quite likely that Lazzerini 'watched his step' and stopped his experiment at the moment he got a good result."

In the wake of these doubts come many more. If the experiment was accurate to one part in a million, presumably the measurements of d and a were also that accurate; if not then the result is presented to too great degree of accuracy. And in only 3408 trials.

Suppose in n trials there are s successes. Then we estimate p = 2dn/as. Now the ration d/a is most likely to be chosen as a fairly simple fraction, so this expression gives an approximation to p in rational numbers. A good approximation is 355/113; the next better is 52163/16604. And 355/113 happens to equal 3.1415929 - exactly what Lazzerini obtained. Furthermore the unusual prime 71 divides not only 355, but also 3408. Hence iof d/a is a fraction with a numerator of 5, the most likely denominators are 6 or 8, and Lazzerini's results would be obtained if the number of successes was respectively 1808 or 1356.

Having come to these conclusions O'Beirne looked up Lazzeririi's original paper (3). The first discovery was that the rnan's name was actually Lazzarini, which meant that every-one else had been copying each other. The next was that d/a was 5/6; and the number of successes indeed 1808. It now became obvious what happened. Arrange that d/a is nearly 5/6, and assume it is exactly this. Count your successes for each multiple of 213 trials; continue until the number of successes is the same multiple of 113, then stop. You ought equality somewhere before 5000 trials. But Lazzarini's fakery does not stop here. He also lists results after various numbers of trials; again his agreement with theory is far too good.

Incidentally, for those who still think this method is worth trying, to have confidence in one's first n figures would necessitate making 102n+2 trials or so.

Bibliography
(1) J. L. Coolidge: The Mathematics of Great Amateurs
(2) N.T. Gridgeman: Geometric Probability and the Number; scripta mathematica 25 (1960)
(3) M.lazzarini: Un' applicazione del calcolo della probabilita all ricerca sperimentale di un valore approssimato di p. Periodico di Matematica (1902).
(4) T.H. O'Beirne puzzles and Paradoxes.