There seems to be doubt about the meaning of the word “function.” Roughly speaking applied mathematicians tend to use it in its old meaning as a synonym for “dependent variable”, while some pure mathematicians use it to mean a mapping in which no element of the domain can have more than one image.

Most readers probably had to solve at school quite enough quadratic equations of the type x2–x–6=0. Look then at the following problem. It is typical of the work Professor Zoltan P. Dienes is getting 10-year-olds to do.

“ As many tanks go into battle as the number of each tank crew. A direct hit wipes out one tank, and everyone in it. Six more tank c:ew are lost. At battle's end the survivors pick up some abandoned enemy tanks.

Generalization of the ellipse to

is credited to Gabriel Lamé, the 19th-century French physicist and the eponym of this family of curves. Lamé curves have recently been taken out of mathematical limbo due to their appeal to certain designers and architects. Particularly to be mentioned is Piet Hein, the contemporary Danish poet-designer-scientist (and inventor of mathematical games) who rediscovered the curves and has been using “superellipses” (Lamé curves with n > 2, and therefore oval) in objets d’art, furniture, pottery, fabric patterns, and so on. His major achievement to date is a sunken oval shopping plaza, promenade, and pool in the centre of Stockholm; its shape is a superellipse with and . Derivatively, Gerald Robinson, a Toronto architect, has incorporated a superelliptic parking garage into a shopping complex (called a Superblock) in downtown Peterborough, Ontario; the parameters are and n = 2.71828 … (the exponential coefficient). In such urban contexts the two-parameter super-ellipse is efficiently tailorable to the dimensions of the site. It is said to be very elegant.

Most people agree that some sort of revolution has taken place in this country during the past decade in the teaching of school mathematics. But it has been a very British revolution, and therefore confusing. No Minister or Inspector General has issued a new syllabus to be adopted by all schools with effect from midnight 31st December 1969/1st January 1970; even the British equivalent of the foreign Ministries, the examination boards, have come out in a rash of syllabuses A, B, C, D,…, modes I, II, III and given their blessing to special papers by the S.M.P., M.M.E., M.E.I., C.S.M., …, The bewildered foreigner equates U.K. with Nuffield, Biggs, Southampton, Exeter, Leicestershire, Brianthwaites, Skempproject, Schools Council, Blackpaper, according to which he has heard of first or whom he has met last.

It has been my pleasure lately to be involved in the organizing and teaching of elementary computer techniques in a college of further education. Whilst the courses were not intended to give tuition in the electronic side of computing it was felt, nevertheless, that some simple explanation of logic circuits and their adaptability to binary adders would not only be an advantage to the students but, if supported by practical classes, would create some considerable interest.

Glancing through George Gamow’s popular excursion into modern biology Mr. Tompkins Inside Himself, written with Martynas Yeas, my eye fell on an amusing account of parity. In it a matador faces a bull in a Möbius ring. He would find, should he run right round the Möbius ring by-passing the animal, that he would face the bull again, but the wrong way up, doing, as it were, a hand-stand. The matador, the authors point out, changes his parity. And standing on one’s head is no way to fight a bull.

Pantactic squares were invented and studied by Astle [1]. From a review by Holzer [2] we learned that the number of distinct pantactic squares remained unknown; Astle himself only mentions that the number of distinguishable pantactic squares runs into hundreds. Therefore we turned the problem over to the computer. By a simple programme in FORTRAN, run on our CDC 3600/3200 computer, we found altogether 800 distinguishable pantactic squares.

In a recent paper (1) P. Holgate obtains the following equations for pk(t) [k = 1,2,…, n], the probability that, at time t, the n animals in a given area are grouped into k herds.

The Fifteen puzzle is well known. In its original form it consists of a shallow square box that can just hold sixteen small squares. Squares, numbered 1 to 15, are placed in any order in the box, leaving the sixteenth position empty (e.g. Fig. 1). The problem is to slide the squares, without lifting them out, until the numbers are in their natural order (Fig. 2).

1. We determine first an infinity of solutions of the equation xn-yn=zm when m, n are relatively prime. Let p be an odd prime; then q= pm has a primitive root g, say, and so, if ϕ is Euler's function, x= gϕ(q) satisfies xn= gnϕ(q)) = l (mod q).

A selection process used by some compilers of big-prize competitions is to present a triangle (usually equilateral) divided into (say) nine congruent sub triangles by the lines joining corresponding trisection points of the sides, and to ask how many triangles there are in the figure. It is amusing to obtain the general result when each side of the original triangle A is divided into n equal parts and pairs of dividing points are joined by parallels to the sides in all possible ways.

In Classroom Notes No. 178 (Math. Gazette, 1968, pp. 376-380), a problem is set and formally solved on a body falling through a tube in the interior of the Earth. One can condone taking the ideal conditions of a frictionless tube and even overlook the fact that no such tube could withstand the stresses present in the Earth below some 30 kilometres depth. But when, as in this Note, it is further assumed (without even reference to any assumption) that g is proportional to r, the calculations, whatever mathematical interest they might have, cannot be regarded as having any relation whatever to the planet Earth.

“It is clear that desk machines will increasingly be used in schools at all levels”

Some Lessons in Mathematics, 1964

“Machines can illuminate the idea of place value”

Nuffield Project Desk Calculators, 1967

“A realisation that the use of machines has arrived in order to ease any necessary burdens of computation”

Examinations Bulletin No. 1 1963

Recently we bought for our small daughter a “Merit” toy, called “Design Doodler” [1], which consists of a single open chain of 8 flat lozenges, linked end to end, identical in size and shape (see Figure 1), the overall length of each being twice the breadth. The linkages are such that the lozenges can move in their own plane so that, when two ends are so linked, any of the five edges of one can be made coincident with any of the five equivalent edges of the other. No other incidence is possible, and lozenges cannot overlap.

One is sometimes asked for an elementary introduction to the principles of information theory. I suggest that this need may be met by showing its application to the well-known problem of twelve balls. The problem is as follows:

Some time ago I asked what is the average rectilinear distance D between the ends of a string of length L when thrown down at random on a table (The Mathematical Gazette, LII (1968) 165). I have to thank a number of correspondents (J. Clemow, R. 0. Davies, A. R. Everett, M. Fine, J. R. Hunt, G. S. Light and E. Sike) for letters about the problem. Some offered solutions based on simple assumptions regarding randomness. The simplest (Everett) is that all values of D in the range (0, L) are equally probable, and so the answer is \L. I do not find this assumption acceptable, nor the other assumptions put forward.