It was conjectured by Fermat that if n > 2, the equation has no solution in positive integers. If n is composite, say n = lm, the equation implies Am + Bm = Cm, where A = al, etc. So the insolubility of the equation with exponent m ensures the insolubility of the equation with exponent n.

Suppose you are trying to tile an infinite grid of unit squares with 2 x 1 rectangles so that each rectangle covers two squares. You place two of the rectangles without thinking. Can the tiling now be accomplished? As the answer is “yes” I call the rectangle an “idiot-proof tile”. In general a polyomino will be called idiot proof if the infinite grid can be tiled irrespective of the positions of two of the tiles (the idiotic tiles), except that the idiotic tiles shall not be overlapping and shall not completely surround an area that is less than that of the tile (not even an idiot would do that!). The first interesting idiot-proof tile (IPT) is the L-tromino.

In a previous article I wrote about the remarkable theorem of Holditch, about areas generated by a sliding or sweeping motion. While thinking further on this topic I discovered some relations concerning the areas within closed curves and the lengths of the perimeters of curves.

In two articles in the Mathematical Gazette, Pedal Triangles and Pedal Circles, July 1919, and Some Properties Relative to a Tetrastigm, Oct. 1920, J. H. Lawlor gave proofs of a number of properties of the pedal circles of the four vertices of a plane quadrangle ABCD with respect to the triangles formed by the remaining three vertices. The articles concluded with the statement of ten results, the first nine of which were fairly easily deducible from those already proved, but No. (10) was given as a conjecture. This was that the common chord of the pedal circles of A and D bisects BH1 and CH2, where H1and H2 are the orthocentres of the triangles BAD and CAD, and hence that BH2, CH1 meet on the common chord of the pedal circles of A and D. It can be seen that the second result would imply the former, for if h is the rectangular hyperbola through ABCD, which is unique unless the points are orthocentric, and if O is the centre of h, then H1 and H2 also lie on h, and it is known that the pedal circles pass through O. The meet of BH2, CH1 is a diagonal point of the quadrangle BH1H2C inscribed in h, and its join to the centre of h is the diameter conjugate to the parallel chords BH1CH2 and therefore bisects them.

A selection process used by some compilers of big-prize competitions is to present a triangle (usually equilateral) divided into (say) nine congruent sub triangles by the lines joining corresponding trisection points of the sides, and to ask how many triangles there are in the figure. It is amusing to obtain the general result when each side of the original triangle A is divided into n equal parts and pairs of dividing points are joined by parallels to the sides in all possible ways.

If L, M, N are the feet of the perpendiculars from P to the sides of a triangle ABC, the circle through L, M, N is the pedal circle of P with respect to the triangle ABC.

If a series of points be pricked at random through a strip of paper, and the strip be held horizontally, certain of the points will appear as peaks, namely those which happen to be higher than either of their immediate neighbours. These we call maximal points. Similarly there are others, minimal points, which are lower than either of their immediate neighbours. Between each maximal point and an adjacent minimal point is a “run”, either a “run up” or a “run down”, the points in a run forming a steadily ascending (or descending) series. These runs will naturally be of various lengths, the length of a run being defined as the number of points in the run, including the first and the last. The shortest runs will thus be of length 2, others may be of length 3, 4, or some greater number, up to m, where m is the total number of points on the strip. Clearly where the points form a run of m there is only one run throughout the whole strip, the greatest possible number of runs is m-1, in which case each run is of length 2.

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.