In introductory courses in abstract algebra the cyclic groups Zn, the dihedral groups Dn, and the symmetric and alternating groups Sn and An, for various integers n, are often used to provide examples of various concepts. The intent of this note is to argue that another group could readily be added to this list: SL (2, 3), the multiplicative group of two by two invertible matrices of determinant 1 with entries from the field with three elements. Though this non-abelian group of order 24 does not appear often in introductory presentations, it is an excellent source of group theoretic examples and is also accessible to beginners, who at this stage of their study most likely have had some experience with matrices. Thus computations in SL (2, 3) can be implemented almost as easily as in the cyclic groups, and defining relations need not be introduced.

The regular decagon will be dissected into other regular polygons and into Greek and Latin crosses by the methods outlined in my previous article (Math. Gaz. 45, 1961, p. 94).

The first step is to find tessellations. In one of those that are used in this note, shown in 1 and 2, the decagon is cut into three pieces, and in the other only two. A further tessellation element, not used here but worth noting, is shown in 5.

A familiar result is that in any group of six people there are three who either all know each other or are all strangers to each other. This can be reformulated to say that, if six points are marked on a piece of paper and every pair of points is joined by a red or blue line, then there will be a monochromatic triangle, i.e. one which is all red or all blue. This result is related to a theorem which the Cambridge philosopher Frank Ramsey proved in 1930.

When a parent sees a little girl sitting on the floor cutting paper dolls, many thoughts may come to mind: ‘She’s keeping out of trouble’ or ‘She’s making a mess’ or even ‘There go my tax returns’. The thought that should have come to my parent’s mind, however, was ‘One day she’ll be a mathematician’. My grandmother, who worked as a dressmaker, often allowed my sister and me to use her razor sharp scissors on the strips of leftover tracing paper. This paper is inspired by a notebook that I kept in grade school when I ‘studied’ paper dolls, and the figures are based on dolls found pressed between the pages.

The determination of centres of curvature and evolutes of well-known curves still forms an item of interest in the customary first course in Calculus. This note calls attention to an elegant construction for the centre of curvature of the conic based upon a property apparently forgotten [1] and offers an excuse for the consideration of evolutes.

In a Gazette article some years ago [1] Derek G. Ball proved the theorem that the only regular polygon which can be constructed on a square pinboard is a square. This note gives two more proofs of this result.

The fact that a mechanical system so well known to so many in its practical aspect as the swing has been thoroughly neglected in textbooks and problem papers may justify a few remarks on it here.

The only reference to the swing which I have been able to find is the question requiring it to be shown that the amplitude of the swing may be increased by internal action only, if, when the swing passes through its lowest position, the distance of the centre of gravity from the point of suspension is (suddenly) shortened from l to (l – h), the full distance l being restored at the end of the swing.