Is the shortest path from A to B the straight line between them? Your first response might be to think it's obviously so. But in fact you know that it's not quite that straightforward. Your sat-nav knows it's not that straightforward. It asks whether you would like it to find the shortest route or the fastest route, because finding the best path depends on knowing what exactly you mean by ‘long’. Likewise, if you're on a walk in the mountains, there's a good chance you'd rather follow the path around the head of the valley, rather than heading down the steep slope and up the other side.

The same sorts of considerations apply in mathematical worlds. I use the mountainside image because it is my preferred way of thinking of a Riemannian metric. Pick an abstract surface S. A Riemannian metric on S gives a well-behaved distance function. By force of habit I tend to picture S as sitting somehow within the physical world. Probably, I'm looking at it from the outside. But if I change viewpoint, so that I am walking around on S, I can picture how the topography affects the idea of the ‘shortest path’.

A shape is called equable if its area and perimeter are numerically equal relative to some given system of units. For example, if a square is equable, then its side, a, must satisfy 4a = a2. So there is only one equable square, and it has side 4.

It is easy to investigate this idea for other shapes. Though not connected with this problem, the work of Imre Lakatos suggested a generalisation to us. Lakatos showed that Euler's classical formula V + F = E − 2 for polyhedra could be extended when the notion of tunnels was introduced [1].

Curves are given in polar coordinates (r, θ)by equations of the form r = f (θ), where for f (θ) > 0 all θ. Consider curves which are symmetric about the origin O, so that, f(θ + π) = f (θ) for all θ. For such a curve, its interior is the set {(r, θ) : 0 ≤ r ≤ f (θ)}. Further, assume that the curve is convex. Recall that a closed curve is convex if a line segment between any two of its points has no points exterior to the curve [1], [2, pp. 198-203]. We call these curves M-curves, because the curves are fundamental objects in Minkowski geometry, where they are called Minkowski circles or simply circles [3, 4]. That application is briefly discussed in the Section 4 but is not required for our purposes.

Examples of M-curves are displayed in Figures 1 to 6. In order to express these curves as functions in rectangular coordinates, we need axes.

This Article deals with the following problem.

Problem: Find positive integers x, y and z with x < y < z such that

x + y, x + z, y + z and x + y + z are squares. (1)

The interest in this problem arose from a conversation between the first two authors at The Times Listener Crossword Setters’ Dinner in March 2017. The Listener was a weekly magazine published by the British Broadcasting Corporation (BBC) from 1929 until 1991. Its main purpose was to reproduce the text of broadcast talks. However, from 1930 onwards it also contained a crossword, which was acknowledged to be the hardest cryptic crossword in a national weekly. When the magazine ceased publication, the crossword was ‘rescued’ by The Times newspaper and has appeared therein on Saturdays for over 25 years.

The following Japanese Temple geometry theorem appears in Fukagawa and Rigby [1]:

Thoerem 1: The products of the radii of the excircles on each pair of opposite sides of a circumscribed quadrilateral are equal.

See Figure 1. A circumscribed quadrilateral (also called a tangential quadrilateral) is one that possesses an inscribed circle. Now the diagram contains four lines (the sides of the quadrilateral) and one circle S (the inscribed circle) that touches all four of them; this we call the base circle of the theorem. The other four circles each touch just three of the lines. For the moment, assume no two of these lines are parallel, and also assume that no circle apart from S touches all four lines; the exceptional cases will be discussed in Section 4. So any three of the lines are sides of a triangle.

Calculating Fisher's F-ratio is a key step in a number of statistical procedures involving null hypothesis significance testing. This is particularly so in the case of ANOVA (analysis of variance) in its several forms, but even multiple regression includes a test of significance of the overall model which employs an F-ratio. The present paper aims at making the basic ideas behind this common statistic more comprehensible by providing a visual counterpart to, and justification for, its algebraic definition.

As an example of how the definition works, consider the following very simple set of data comprising an independent variable consisting of three groups, where the values of the dependent variable are 1, 2, 3 for the first group, 4, 5, 6 for the second group and 7, 8, 9 for the third group. The groups could represent three drug treatments, and the numbers, a measure of clinical outcome for each of nine participants. One might represent this set of data as a row vector thus: (1, 2, 3, 4, 5, 6, 7, 8, 9).

The well-known Fibonacci numbers Fn are defined by the recurrence relation

Fn = Fn – 1 + Fn – 2. (1)

together with the starting values F0 = 0, F1 = 1, or equivalently F1 = F2 = 1.

We record the first few:

The recurrence relation can also be applied backwards in the form Fn = Fn + 2 – Fn + 1 to define Fn for n < 0. An easy induction verifies that F−n = (−1)n – 1Fn for n > 0.

The following observations are motivated by the facts that the area of a planar figure displayed on a screen can be expressed by a certain number of pixels; and if the figure is drawn by a plotter, then its area can be characterised by the total length of a line which fills it in.

The generalisations of the Pythagorean theorem are of three kinds. Firstly, the squares on the sides of the right triangle are substituted by other geometrically similar planar figures (Euclid's Elements Book VI, Proposition 31 [1]). Secondly, the assumption of the right angle is omitted (the law of cosines), or both of these generalizations occur simultaneously (Pappus’ area theorem [2], see also H. W. Eves [3]). Thirdly, mathematical spaces other than the plane are considered (for example, de Gua-Faulhaber theorem about trirectangular tetrahedra [3], further generalised by Tinseau [4], Euclidean n-spaces, Banach spaces [5], see also [6]).