We distinguish between a level on a sequence of charts (e.g. top level or 5th level) and a place on a single chart (e.g. 5th place on the 22nd week). A song occupies a place by halting there prior to a continued rise or fall of its path, or by stopping there if that place is at the highest level attained. A song which passes to a level higher (or lower) than level p without halting there bye-passes level p. The upward path of a song terminates at the first stopping place and the downward path commences at the last stopping place. Note : Level No. 1 is the lowest level and level P is the uppermost one. This is contrary to popular usage but makes the mathematics more conventional.

No field of endeavour, trivial or non-trivial, escapes man’s quest for the superlatives—always the smallest or the largest, the fewest or the most is his goal. Whether this drive stems from a thirst for knowledge or a desire for leisure in which to pursue other work is a good question. In this massive and perpetual search for the ultimate a trivial game is no exception. No doubt a fantastic amount of time has been spent seeking to determine how many little counters could be arranged in a certain pattern, then this pattern re-arranged by jumping one unit over the other, removing the counter jumped, until the greatest number is lined up to be taken in a single move, or the pattern is reduced to a single unit with the fewest moves. Erom available published accounts, the single move limit has been 8 and the fewest moves have been 16 since this particular pattern called a 33-Solitaire board was conceived, the 33 referring to the number of spaces in this pattern.

The first reference to the Problem Bureau in The Mathematical Gazette was in July 1928:

Ladies and Gentlemen,

There are many different forms of Presidential Address. To glance back at those delivered to this Association is at once an exhilarating and a humbling procedure. Indeed, at one stage during it I toyed with the notion of filling this session with an amalgam of abstracts. You might have been better served had I done so; as it is I shall be quoting from three only and the reasons for that particular selection should become apparent before long.

Few could cavil at the general views expressed about the place of mathematics in schools by the writers of the Newsom Report, as most of the ideas have been advocated in many Training Colleges for at least ten years. It is strange, however, to see that the section devoted to mathematics is the second part of Chapter 18 on “Science and Mathematics.” Does not this lend colour to the heresy promulgated by Hogben in “Mathematics for the Million” that the subject is to be valued principally, if not entirely, as the “Handmaid of Science”? Everyone will agree that “Mathematics has much to offer to boys and girls of average ability at the Secondary stage. It has a self-evident usefulness which can be a great advantage in arousing the interest of pupils.” Is it true that the only appeal to such pupils is through experience of the practical utility of mathematics in the world around them—the modern world of technology and science and commerce? Doubtless we need more and better mathematicians in all such activities, but they will not be drawn in any large numbers from these pupils. Numbers, shapes and patterns have an intrinsic fascination of their own that is independent of their practical value. For instance, it might be argued that the fact that “all numbers divisible exactly by 9 have a digit sum that is a multiple of nine” may be of some practical value—it was the basis of the process of “casting out the nines” that was formerly advocated as a quick check on the result of a calculation. Yet the pattern of the “nine-times table” is of aesthetic value as well as an aid to memory. Is the test to discover whether a number is a multiple of 11 of much practical value? Lewis Carroll used it to devise a method of dividing by 11 that depends solely on addition and subtraction, but its real value is the insight into number relations that is gained from demonstrating its validity: it certainly has an appeal to most people as a kind of “mathematical magic.” Do you know the test for discovering whether an amount of money expressed in shillings and pence is divisible exactly by 11?—just add the number of shillings to the number of pence, and if their sum is a multiple of 11, the amount of money is divisible by 11. There is an equally simple test for an amount of money to be exactly divisible by 13—can you find it? Experience shows that facts such as these stimulate the interest of those who dislike arithmetical calculations, whether young people or adults: the film-strip “Alice in Numberland” was designed to undermine the prejudice that many women students have against mathematics, by showing Lewis Carroll’s delight in puzzles and paradoxes, and his humorous approach to Symbolic Logic. Even in Secondary Modern Schools at all levels it should not be forgotten that mathematics can have an “entertainment value” comparable to that of music or any other form of art, if presented in the right way.

The problem of arranging dominoes with like numbers in groups of four was proposed by Édouard Lucas. Although Lucas published only symmetrical solutions for the double-six set, generalizations of the problem have led to some interesting results. For the double-six set, it has been shown there are hundreds of shapes meeting the requirement that like numbers appear in groups of four.

A sequence of athletic records forms by definition a monotonic sequence. Since it is reasonable to assume this to be bounded, it follows that a limit must exist to future performance. It has been proposed [1, 2] that this can be estimated by use of a curve-fitting procedure on existing data, such as may be found in many compendia of athletic records (e.g. [3]).

The pitch of a musical note is defined by its frequency, measured in cycles per second. The frequencies of pure musical tones form an infinite set of real numbers lying between the lower and upper limits of audibility. The notes of the pianoforte form a finite subset of this infinite ‘spectrum’ containing usually 88 members. Other instruments provide different subsets of available notes, these subsets being infinite in the case of the strings and the trombone, whereas in the case of most other instruments, we have finite subsets of discrete musical tones.

In an article [1] in The Mathematical Gazette, S. N. Collings discusses and proves certain properties of projective polygons. Some of these results are contained in a previous article [3], published in the American Mathematical Monthly, or can be proved from the results obtained in that article. However, the language of [3] is different from that of [1], and the purpose of this note is to show how some of the results of [1] could be translated into the language of [3].

If we wish to plot the results of the election on a graph, constituency by constituency, we face the problem that a sheet of paper has two degrees of freedom, but there are three political parties. We may overcome this by using only the ratio of the votes obtained by each party. In any constituency,

One approach to investigating the structure of a group is to examine the family of subgroups of the group. In particular, one often looks for normal subgroups and considers things such as direct products and composition series. However, fruitful investigation usually involves establishing that certain subgroups do, in fact, exist. Hence it is important to find conditions which will guarantee the existence of subgroups. This note indicates the usefulness of one such set of conditions—the Sylow theorems for finite groups.

Everyone connected with the teaching or practice of mathematics is aware that in recent years there has been a great increase in the number and variety of opportunities for the employment of mathematicians. In particular, industry, and especially engineering industry, has found a need to recruit mathematicians in large numbers with a view to employing them to do mathematics as a full-time occupation. This is quite a new development; although the value of the mathematically trained mind has long been recognised in many places, it is only quite recently that mathematics, as opposed to mathematicians, has been extensively employed by industry This change has, of course, largely coincided with the advent of the electronic digital computer, and the computer has certainly had a great influence on it; new mathematical disciplines have grown up as a result of the existence of computers, and it is now practicable to solve problems whose solution would be inconceivable without the use of a computer. As a result of this it has become fashionable to attribute the spread of mathematics to the computer. I think it more likely that the computer has accelerated rather than created the demand for mathematicians, and that the development of new industries and the introduction of new techniques to old ones would have brought about a demand for mathematicians even in the absence of computers. However there is no doubt that this demand exists, and it has certainly been stimulated by the existence of the computer.

This September the transition from Grammar to Senior High School will be virtually complete. The transfer age is 14, and is, at the moment, by parental choice. We still have all boys, but receive our first intake of girls this year. Our first non-selective entry was in 1964, and all members of the mathematics staff, only one of whom had met them previously, have now taught the lower ability groups. We have had our problems, none insurmountable, and more of administration than of adaptation. Anyone embarking on a transition to Comprehensive education and needing advice is recommended to read Canon Eperson’s admirable article in the recent Gazette. Much of what he says we have learned by experience, and if I requote from Newsom, it is because the quotations are particularly apt.

In The Gazette for February 1966 (Vol. L, No. 371) Mr. C. A. Parrack introduced cliques to serve as counter-examples for groups. The purpose of this note is to show how cliques are related to groups. This leads to the classification of cliques in terms of commutative groups, and to considerable simplifications in clique calculations.

Given points P1 P2, …, Pn on a circle centre 0, they have a centroid G. We shall show that they have also a generalised orthocentre H, a generalised nine point circle with centre N, and that OGNH lie on a line forming a harmonic range.

The older sixth form masters have seen a fundamental change from a unified programme guided by the form master, to an à la carte curriculum where the form master may see some members of his form only once or twice a week. The old liberal character of form work must now be transformed into certain elusive elements in the subjects themselves and in the way they are studied. In this respect sixth form teachers are in the same vulnerable position as the editors of newspapers: they must be resolute in resisting the monopolists, whether inside or outside the school.

An excellent S-level question asks how far a tender capable of 12 knots has to travel to intercept a ship travelling at 20 knots along a course passing within 1 nautical mile of port, the tender leaving port at the latest possible moment. The problem leads to interesting ramifications.

Among the greatest advances in knowledge this century one would include relativity, quantum mechanics and the explanation of hereditary replication in living things by the structure of the DNA molecule. But it would not be absurd to include in this category a discovery which is not at all well-known and goes back to 1930, namely, that logic and mathematics as we know them have undecidable propositions.