Tetris is a computer game which has obsessed many computer users and attracted much attention, despite the simplicity of its rules. This paper addresses the question: ‘can you “win” the game Tetris?’

Designed by Soviet mathematician Alexey Pazhitnov in the late eighties and imported to the United States by Spectrum Holobyte, Tetris won a record number of software awards in 1989. Versions of Tetris are sold for most personal computers. There are Tetris arcade games, Tetris Nintendo cartridges, and hand-held Tetris games; Tetris has been played on machines ranging from mainframes to calculators. The game's success has prompted the invention of several similar games, including Hextris, Welltris, and Wordtris.

We propose an interpretation of the integers as operations on permutations. Addition and multiplication of integers represent operations on these operations. It follows that an equation involving integers, addition and multiplication is valid if and only if the corresponding operations on permutations are the same. In our interpretation, the validity of the fundamental laws of calculation is an immediate consequence of well-known properties of permutations. For example, the ‘Law of signs’: (–1) x (–1) = 1, is valid because the inverse of the inverse of any permutation is the permutation itself.

The succession of enlargements from ℤ (the system of the integers) to ℚ (that of the rational numbers) and thence to ℝ (the real numbers) and then ℂ (the complex numbers) is generally seen as one of increasing richness of structure. But gains are often accompanied by losses: for instance the step from ℝ to ℂ leads to the glories of complex analysis but away from ordinal properties once thought essential to any concept of number.

As far as group structure is concerned there are gains and losses at each stage of the progression. The following exposition is intended to demonstrate this without assuming more than a minimal knowledge of groups and of the theoretical foundations of analysis. Beyond some elementary arithmetical algebra, and an acquaintance with standard notation and vocabulary for maps in general, essentials taken for granted include the idea of a homomorphism (of groups), the existence and fundamental properties of the exponential function (of a real variable), and the Fundamental Theorem of Algebra (which is actually a theorem in complex analysis).

The first part of this article covers some well-known ground in preparation for the second part on the Archimedean honeycomb duals. Mathematically speaking some of this second part is, so far as I know, no more than conjecture: I do not myself know of a proper systematic approach to such honeycombs.

As far as possible I have used names for the polyhedra from standard references (e.g. [1]). But references vary, and I have not found all of them described. In such cases I have tried to choose names which are appropriate and technically correct.

If you enjoy magic squares and their unusual and fascinating numerical properties, you are in good company. Indeed, many a professional mathematician, scientist and amateur has shared the same interest. Technical research work on the properties of magic squares is available aplenty in the literature or in books concerned with the recreational aspects of mathematics. Studies of magic squares naturally lead to some elements of group theory, of lattices, of Latin squares, of partitions, of matrices, of determinants, and so on …

By definition, the general n x n magic square is a square array of n2 numbers {aij; i, j = 1,2,... , n) whose n rows, n columns, and two main diagonals have the same sum s. It is therefore natural to think of a magic square as a matrix and we shall do this in the present note, simply referring to it as a magic matrix.

The need for clarity in the teaching of mechanics in the initial stages is vital if a firm grasp of principle is to be fostered. I had been teaching A level students about locomotion in a conventional way for some years when it occurred to me that something was basically unsound about the way in which textbooks treat the subject. At best they appear unclear and at worst they seem to flout basic principles. Examination papers mirror the textbook treatment.

We will discuss in this article a fractal-like structure made from a flat piece of paper. What will motivate most people to want to make the structure is that it is pretty. In fact, the exercise has two rather obvious uses. First, greeting card companies may want to use the idea to manufacture interesting 3-dimensional cards which fit conveniently into envelopes; and, second, teachers may wish to teach students how to make it (and this process will also involve teaching them some beautiful mathematics). The structure itself is a model for a stage in a self-similarity process leading to a fractal. Building the model involves scoring a flat piece of paper in a prescribed manner, cutting along some of the score lines, and then folding some lines as ‘mountain’ folds and others as ‘valley’ folds.