Pascal’s triangle is the most famous of all number arrays - full of patterns and surprises. One surprise is the fact that lurking amongst these binomial coefficients are the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers which arose in the Hindu studies of arrangements and selections, together with the Fibonacci numbers from medieval Italy. New identities continue to be discovered, so much so that their publication frequently excites no one but the discoverer.

The practice of long jumping a motor cycle from an inclined ramp over obstacles such as parked buses has become popular with stunt-men. There have been serious accidents, some with fatal results. Some crashes have been caused by the faulty technique of keeping the throttle wide open after leaving the take-off ramp. In some cases this can lead to a backward rotation of the cycle by up to 90° during the jump, with inevitable disaster on landing.

In this paper we analyse a game that was published in the 1996 Burger King Kid’s Club calendar. This calendar included an activity for each month, and for the month of September the activity was called the Mountain Bike Rally. To begin the game, each player places a marker on day 1. A turn consists of each player flipping a fair coin. Each player whose coin comes up heads moves his marker forward one day. If a player’s coin shows tails, then he moves his marker back one day, unless his marker is on day 1, in which case it stays there. The object is to be the first player to reach square 31. We never played the game because it was clear that it would take a long time to complete.

There are numerous arithmetical puzzles concerning the quantities of beer that can be measured if the only available measures are a 3-pint jug and a 5-pint jug, or the exact amounts of money obtainable using only 3p and 5p coins, or the amounts that can be made with only 3p and 5p stamps.

Every geometrical truth has an individual beauty which delights the mind: but the most fascinating theorems, as an eminent mathematician rightly asserts, are those which have simplicity, depth, and obvious generality I propose to show that nearly all the important circles of the geometry of the triangle—the in-circle, the nine-points circle, the circum-circle, Taylor’s circle, Lemoine’s circle, and many other useful but nameless circles—are each and all particular cases of one fundamental general circle.

In this article we show how to obtain by recurrence all primitive Pythagorean triples from the single basic (4, 3, 5) triple. It is all done by repeated application of two integer unimodular transformations that leave the indefinite metric x2 + y2 - z2 invariant, together with the additional transformations (i) that change the sign of x and (ii) that change the sign of y. These alone would restrict the triples to those in which x is even, y is odd and z is positive, so we then include two further transformations (iii) that exchange x and y and (iv) that change the sign of z, thereby accounting for all primitive triples. There is a bonus in extending the method beyond the first objective in that the theory, which we outline, of the Lorentz transformations involved not only provides necessary background explanation of the first part, but it also enables us to show how to solve by recurrence Diophantine equations of the form x2 + y2 = z2 + n, where n is any integer.

Before reading this article, you may like to turn to page 297, where you will find a quiz on ‘famous-name’ mathematics. In each of the questions it seems reasonable to suppose that the mathematician indicated in the title of the result is the discoverer or inventor of the result. But a careful look at mathematical history shows, in each case, that there is a ‘story behind the story’. As you read on, keep in mind that even the answers given below are subject to revision as new historical data is found and interpreted.