In the extension from the use of natural numbers with zero to integers, i.e. the introduction of directed numbers including negative numbers, much experimental work is now being done with the use of mathematical structure. Some attempts are being made to give this as the initial treatment at about the age of 12, and some teachers are getting their children very interested and consider it a correct method. For other children the formal treatment will come later, when they are more mature and begin to query the rules they are using. This may be when they are starting to work ahead of their O level syllabus, the beginning of a more advanced course with work on groups. It could give work for those doing some mathematics when they are beginning to specialise in some other subjects and want to try to understand more about the mathematics they have learnt.

The purpose of this paper is to consider the question: if p is the probability that a player P will win a given rally against a player Q, what is the probability that P will win a given game? The answer, of course, depends upon the method of scoring; and the methods considered here will be those used in squash and, for comparison table tennis.

In table tennis P earns a point each time that he wins a rally. But in squash he must serve the rally that he wins in order to gain a point; and he relinquishes the service when he loses a rally that he serves.

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.

Three straight lines are needed to enclose a finite region of a plane, a two-dimensional space. The polygon formed is a triangle. Different types of triangle are described by adjectives; scalene, isosceles, equilateral and acute angled, right angled, obtuse angled. All the triangles have a property in common: they are all convex.

Four lines form a quadrilateral. The convex examples are well known. There are two further types: those which are “re-entrant” having an interior angle greater than 180 degrees, a reflex angle; those in which a pair of opposite sides cross within the quadrilateral, a “crossed” quadrilateral.

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.

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.

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).