We consider here the positive integers with respect to their unique decimal expansions, where each n ∈ ℕ is given by for some non-negative integer k and digit sequence αkαk-1 … α0. With slight abuse of notation, we also use n to denote αkαk-1 … α0. For such sequences of digits (as well as for the numbers represented by the corresponding expansions) we write x ⊲ y if x is a subsequence of y, which means that either x = y or x can be obtained from y by deleting some digits of y. For example, 514 ⊲ 352148. The main problem is as follows: Given a set S ⊂ ℕ, find the smallest possible set M ⊂ S such that for all s ∈ S there exists m ∈ M with m ⊲ s.

1. It is well known that if p is a prime number not equal to 2, then exactly half the non-zero residues of p are quadratic residues (that is, squares in the arithmetic mod p). What happens if the modulus m is not prime? We can look at this problem in two ways: we can insist that our residues are relatively prime to m, or we can consider all the residues.

If from a point, regarded as a centre of inversion, a radius is drawn on which are two points at distances r1 and r2 from the centre, satisfying the condition r1r2=r2 = constant, the points defined by r1 and r2 are said to be inverses one of the other.

The cracking or tearing of thin sheets of material is of considerable interest in modern-day engineering. A recent investigation into the behaviour of concrete indicated that it would be of interest to consider a probabilistic model in which a crack is regarded as some form of non-self-intersecting random walk in the plane. This paper considers some simple models and in the Appendix are included some results on paper tearing.

The algebra taught at present in schools and colleges and discussed in recent reports is essentially that of Euler and Chrystal as regards both content and method. Must this tradition continue? It is hoped that this short note will suggest the contrary and indicate the possibility of incorporating the ideas and methods of modern algebra in elementary teaching. To this end attention will be drawn to some of the simpler concepts of the subject.

The increasing use of simple computing languages in schools gives an opportunity for rationalisation of our existing mathematical notation as follows:

It is time we abandoned “÷”, which is used in primary schools and causes great difficulties later on when children have to cope with the fact that or, worse still, If we used “|” from the start, what could be easier than (Nor is there a case for restricting ÷ to ‘sharing’ situations, since + and − are used for both natural numbers and rationals.)

The subject matter of my talk is geometry in the sense that I shall deal with such entities as lines, circles, and squares. But the results I aim at bear little resemblance to the theorems of traditional geometry, which describe properties of given configurations. Our present object is quite different; we shall begin by specifying not configurations but processes (for example processes of selection), and then seek to determine how economically or how effectively these processes can be carried out. Questions of this nature are quantitative and the solution, if it is known, is generally expressed by a number. It is for this reason that I speak of 'arithmetical' geometry. The problems of the type I shall discuss are usually simple to state and require very little in the way of mathematical equipment for their understanding. However, their solution often presents formidable difficulties and is in many cases as yet far from complete. Let us now consider in some detail a few of the characteristic problems in this field.

Smallest defining sets

A standard Sudoku square is a 9 × 9 latin square in which each of the nine 3 × 3 subsquares into which it can be separated contains each of the integers 1 to 9 exactly once.

A current problem is to complete such a square when only some of the cells have been filled. These cells are often called ‘givens’. (Such problems are currently (2005) published daily in British newspapers.) In more mathematical terms, the given filled cells constitute a defining set or uniquely completable set for the square if they lead to a unique completion of the square. If, after deletion of any one of these givens, the square can no longer be completed uniquely, the givens form a critical set. The investigation of critical sets for ‘ordinary’ latin squares is a topic of current mathematical interest. (See [1] for more details.)

In Keynes's contribution to the ‘Newton Tercentenary Celebrations’ there is the following passage :

“Again, there is some evidence that Newton in preparing the Principia was held up almost to the last moment by lack of proof that you could treat a solid sphere as though all its mass were concentrated at the centre, and only hit on the proof a year before publication. But this was a truth which he had known for certain and had always assumed for many years.

“Certainly there can be no doubt that the peculiar geometrical form in which the exposition of the Principia is dressed up bears no resemblance at all to the mental processes by which Newton actually arrived at his conclusions.

“His experiments were always, I suspect, a means, not of discovery, but always of verifying what he knew already.”