How do you know that the equation x3 + x = 5 has one real root? Well, the function x3 + x is monotonic increasing: any increase in the value of x produces an increase in x3 + x. In particular, since 13 + 1 = 2 and 23 + 2 = 10 , we know that, for some x between 1 and 2, x3 + x = 5 . The function must “pass through” 5, and, since it is always increasing, it does this exactly once.

In three earlier articles we drew attention to an astonishing error which appears in several texts in the United States, even some of otherwise very high quality. The error relates to a fallacious procedure for the calculation of the Euler characteristic of a polyhedron, producing, in particular, the extraordinary statement that the Euler characteristic of a torus is 2.

The following is a well known pursuit problem: n bugs are at rest at the vertices of a regular polygon. At a certain instant each bug starts crawling towards its neighbour on the right with a constant speed, always altering its course so as to be headed directly towards its neighbour. How long does it take for the bugs to meet at the center of the polygon and what paths do they trace out in getting there? The earliest reference I could find to this problem was in a collection of mathematical puzzles edited by L.A.Graham. A more recent reference to it occurs in the amusing and informative collection of geometrical curiosities by David Wells. The “n-bug” problem was also discussed briefly in two of Martin Gardner’s columns in Scientific American. The cover of the July 65 issue of that magazine shows the equiangular spirals traced out by four bugs as they “square dance” their way towards their tryst at the centre.

Take any positive integer written in the usual base 10 notation, reverse the digits and add to the original number. If the result is a palindrome, stop. If not, continue. Will a palindrome always occur at some point? Let us do a fe examples to see the sort of thing that happens.

The distribution of the prime numbers is of course a well-trodden path. The casual enquirer, working with small numbers, soon finds that the distribution has no obvious regularities, and sees that the primes thin out. The Prime number theorem (which is not elementary) helps to firm up this observation:

Prime number theorem: If π(n) is the number of primes no greater than n, then i.e. π(n)/(n/ln(n)) → 1 as n → ∞. Thus for the proportion of prime numbers upto to n, π(n)/n, we have that which tends slowly to 0.

This article provides a brief introduction to Evangelista Torricelli and to one aspect his work. It is divided into three parts. Part 1 gives some biographical details and a short survey of his work and achievements. Part 2 is concerned with a geometrical approach towards the study of projectiles as seen in Torricelli's early work De motu. Part 3 shows how Torricelli applied his projectile studies to practical ballistic problems.

In MacKinnon‘s article he makes mention of Gregory of St Vincent‘s method of squaring the circle, giving the impression that this method involved ruler and compass, an impression that is reinforced by the title and extravagant frontispiece of the book. In this article I intend to correct this impression by describing St Vincent‘s method, which used a kind of integration method, to calculate, among other things, the area of the circle.

Can we produce a ratio ψ to rival the golden ratio ϕ? In this article I shall construct a variant of the classical golden rectangle by choosing a different geometrical property. The rectangle described has more advanced mathematical properties than its golden relation, though it is not extravagant. In this sense it could be described as a supergolden rectangle.

Starting with a finite set of numbers {xl, x2, x3, …, xn} we can generate a sequence of numbers {yi} by the rule

so that yi is the average of the previous n terms of the sequence.

The National Lottery is upon us, and as you read this you may be clutching your numbers waiting for the live draw on November 19. This article will discuss the probabilities of winning, how to improve your chances, and expose a loophole that may be worth millions to a determined syndicate.