Cyclic pairs

When you transpose the digits of a two digit number and obtain a multiple of the original, the original may be called a “cyclic pair”. I have before me, a computer printout obtained by Mr F. B. Satterthwaite of 439 East 51st Street, New York 10022, exhibiting all the cyclic pairs for which the multiple of the original obtained by transposing the digits is ⩽ 16. This computer printout exhibits patterns of many kinds. The most striking pattern that Satterthwaite identified was that cyclic pairs do not exist when the base is one less than a prime number, but do exist for every other base ⩾ 5. His claim may be substantiated as follows.

In his interesting article on Newton in the July Gazette Dr. Rouse Ball had space only for a slight reference to Newton’s views on what we should now call the Foundations of the Differential Calculus. The subject “ involves,” as Dr. Rouse Ball has remarked, “ some awkward questions of philosophy which before Weierstrass’s researches were usually slurred over.” Some of the logical difficulties incident to Newton’s method of approach were clearly seen and pointed out by a contemporary, the celebrated Bishop Berkeley, in a tract which drew much attention at the time, and was followed by a considerable controversy.

The most surprising thing about this formula is its use of the Bernoulli numbers, and it is natural to ask why they appear. The answer is that it is not the B’s which insist on entry, but the numbers Ar = Br/r!. We use the notation where all other B with odd subscripts vanish, and those with even subscripts are alternately positive and negative. We shall often ignore the fact that A3, A5, .. vanish, thereby refusing to complicate formulae unnecessarily.

The investigations of Professor Karl Pearson and his coadjutors on the applications of the theory of errors to statistics have greatly increased the importance and extent of the subject.

It is hoped that the following version of one of the methods of arriving at the so-called normal law of error may be of interest from an elementary point of view: affording, as it does, a simple application of the binomial theorem and some interesting graphical work.