Recursively defined sequences (in other words difference equations) appear in modelling systems changing in discrete time steps. The asymptotic behaviour of such sequences tells us information about the state of the modelled system after a long time. The simplest difference equations are linear, but the standard solving method of even these simple equations involves eigenvalue and eigenvector computation, complex numbers and linear algebra.

Geometry in a Mathematical Syllabus seems to me to occupy much the same position that Greek does in the Classical. The two have similar merits and similar defects, and I am doubtful whether either deserves its place except in specialist classes.

Every reader knows about the Golden Rectangle (see [1, pp. 85, 119], [2, 3]), and that it can be subdivided into a square and a smaller copy of itself, and that this process can be continued indefinitely, converging towards the intersection point of diagonals of any two successive rectangles in the sequence. The circumscribed logarithmic spiral passing through the vertices and converging to the same point is also familiar (see [3, 4]), and is analogous to the circumcircle of a regular polygon or a triangle. The approximate logarithmic spiral obtained by drawing a quarter-circle inside each of the squares is equally well known [3, p. 64]. Perhaps slightly less familiar is the inscribed spiral, which is tangential to a side of every rectangle, like the incircle of a triangle or a regular polygon. It does not (quite) coincide with the spiral passing through the point of subdivision of each side, as discussed in [3, pp. 73-77]. The Golden Rectangle, its subdivisions, and the circumscribed and inscribed spirals are illustrated in Figure 1.

The game ‘toss up’, manufactured by Patch Products (http://www.patchproducts.com/). is played with ten fair, six-sided dice whose sides are coloured green, yellow and red. In particular, each die has 3 green sides, 2 yellow sides and I red side. Ordinary fair six-sided dice may be used instead, of course, provided three sides are identified as ‘green’ (e.g. 1, 2, 3), two sides are identified as ‘yellow’ (e.g. 4,5) and one side is identified as ‘red’ (e.g. 6). Players take turns with each turn consisting of (potentially) several rolls of dice. All the dice are used on the first roll of a turn. Players accumulate points as they play and the first player to reach 100 points wins.

The Lucas-Lehmer (LL) test is the most efficient known for testing the primality of Mersenne numbers, i.e. the integers Ml = 2l − 1, for l ≥ 1. The Mersenne numbers are so-called in honour of the French scholar Marin Mersenne (1588-1648), who in 1644 published a list of exponents l ≤ 257 which he conjectured produced all and only those Ml which are prime, for l in this range, namely l = 2,3,5,7, 13, 17, 19,31,67, 127 and 257 [1]. Mersenne's list turned out to be incorrect, omitting the prime-producing l = 61, 89 and 107 and including the composite-producing l = 67 and 257, although this was not finally confirmed until 1947, using both the LL test and contemporary mechanical calculators [2]. The LL test is based on the following theorem.

A well-known exercise in classical differential geometry [1, 2, 3] is to show that the set of all points (x, y, z) ∈ ℝ3 which satisfy the cubic equation

is a surface of revolution.

The standard proof ([2], [3, p. 11]), which, in principle, goes back to Lagrange [4] and Monge [5], is to verify that (1) satisfies the partial differential equation (here written as a determinant)

which characterises any surface of revolution F (x, y, z) = 0 whose axis of revolution has direction numbers (l, m, n) and goes through the point (a, b, c). This PDE, for its part, expresses the geometric property that the normal line through any point of must intersect the axis of revolution (this is rather subtle; see [6]). All of this, though perfectly correct, seems complicated and rather sophisticated just to show that one can obtain by rotating a suitable curve around a certain fixed line. Moreover, to carry out this proof one needs to know a priori just what this axis is, something not immediately clear from the statement of the problem. Nor does the solution give much of a clue as to which curve one rotates.

A search of the literature failed to turn up a treatment of the problem which differs significantly from that sketched above (although see [1]).

The polynomial (1) is quite famous and has been the object of numerous algebraical and number theoretical investigations. See the delightful and informative paper [7].

The human mind, by nature, is curious and enjoys dealing, both independently and competitively, with intellectual challenges. Throughout time, mathematical tasks, riddles, and puzzles have offered such challenges. In today's modern era, marketing companies and the media even offer prizes and rewards for the successful solution of puzzles, expanding the audience exposed to the various challenges. In addition, the styles, subjects and depths of the puzzles have become diversified. Experience gained in solving puzzles aids in developing reasoning abilities and deepening thought.