In the familiar [32] game of noughts and crosses or ticktacktoe, there are eight ‘winning’ Lines. These are the lines having the maximum number, 3, of Points in a 3 × 3 lattice, as in the figure.

The use of reflecting surfaces to unscramble distorted pictures has been apparent over the past three centuries, at least, in an art form known as anamorphic (ana = back again, morphe = form) art. Of course, working in the opposite sense, the mirrors in fun parlours give distorted images from an undistorted (!) object. A famous portrait of “Bonnie Prince Charlie” in the West Highland Museum, Fort William, Scotland, appears as a series of incomplete concentric circular streaks of colour, but when a cylindrical reflecting surface (the anamorphoscope) is placed at the centre of the painting a recognisable image is seen on looking into it. Niceron and DuBreuil both wrote treatises in the seventeenth century on the techniques of producing anamorphograms, and Gardner has given a recent discussion of the general topic. DuBreuil described a method for generating a distorted object (drawing) that will be transformed into an undistorted image by the use of a conical reflecting surface whose base sits on the plane of the object. The drawings are very easily done by following simple rules of geometry.

These words of John Playfair’s are quoted by R. C. Archibald in his survey of the “minor English mathematical serials” which appeared in the celebratory 200th issue of this Gazette. Archibald describes how, early in the eighteenth century, serials containing elementary mathematical problems began to appear. Thus, during a century or more when mathematics had still to establish itself in the curriculum of the country’s leading endowed schools, numerous attempts were made to interest the ‘reading’ public in the subject and to serve those readers who were already ‘scientific’. An earlier, extensive survey of such publications was undertaken in the Mechanics Magazine (1848-1853) by T. T. Wilkinson [3] (who also wrote, inter alia, on that arcane (to two Yorkists!) subject “The origin and progress of the study of geometry in Lancashire”). Little, however, appears to have been done in recent years to examine these early attempts to diffuse mathematical knowledge: to study the mathematics contained in such periodicals and columns, how it was learned and enjoyed, and the people who set and solved the problems. This, in a small way, is what is attempted in this investigation. As our subject we chose to study the mathematical column which appeared in the York Courant during the early years of the nineteenth century.

Quaternions feature in most introductory textbooks on abstract algebra, often in the form of exercises. They offer an important example of a skew field (or division ring); the subset of quaternions ±i, ±j, ±k, ±l (or ±I, ±J, ±K, ±L according to taste) under multiplication affords an example of the ‘awkward’ group of order 8, which is often known as the quaternion group; they have a convenient representation in the form of matrices, which provides the work with a computational foundation; and, for those who like that kind of thing, they can be introduced as ordered quadruples of real numbers with prescribed laws for addition and multiplication, on lines similar to the number-pair approach to integers, rationals and complex numbers.