Essays on recreational mathematics

Snooker, billiards, and pool differ in the details of the table, its pockets, and the balls used. But all rely on the same physics (and mathematics) determining the way the balls move and collide. At their basis is the principle of conservation of momentum, which follows from Newton's laws of motion. But various other effects come into play as well.

Draw straight lines from each vertex of a triangle to the mid-point of their opposite sides. Whatever the triangle, these three lines always meet in a single point, the triangle's centroid, corresponding to its centre of gravity. There are many other 'centres' associated with a triangle's geometry, and some beautiful properties are still being found.

In mathematics, the term infinity is used for something which is larger than any given natural or real number. The concept of infinity is used throughout mathematics. However, its philosophical nature has been the subject of consideration and bafflement ever since the time of the ancient Greeks.

Carl Friedrich Gauss (1777-1855) is considered to be the greatest mathematician of his time and, arguably, of all time. His abilities became evident at a very young age, and as a teenager he solved problems that had defeated mathematicians since the time of the ancient Greeks.

Start with any positive integer. If the number is even, divide it by two. If the number is odd, triple it and add one. Perform this simple operation repeatedly. The Collatz conjecture is that no matter what value is used to start, the sequence will always reach 1. A proof appears to lie beyond today's best mathematicians.

Dissecting and re-arranging the pieces of polygons to form other polygons is a fascinating branch of mathematics. While any polygon can be dissected into a finite number of pieces that will form any other polygon of the same area, many dissections involving just a small number of pieces turn out to be particularly elegant.

Music combines many elements: pitch, timbre, texture, and dynamics, as well as rhythm, harmony, tempo, and structure. But our focus here is on the most mathematical element of music, the scale, along with the tuning systems that have been introduced as slight compromises to the intervals of 'just intonation' to meet other needs.

A perfect number, in number theory, is a positive integer that is equal to the sum of its positive divisors (excluding the number itself). For example, the first perfect number, 6, has divisors 1, 2 and 3. Since 1+2+3 = 6, it follows that 6 is a perfect number. The next is 28, since 1+2+4+7+14 = 28. Are there any perfect numbers that are odd?

Paul Erdös was one of the most prolific authors in mathematical history. He wrote more than 1500 mathematical articles in his lifetime, with more than 500 different collaborators. Some examples of his less abstract work are given. His eccentricity is widely recounted. Mathematicians still like to discuss their 'Erdös number'.

Roll a fair dice, and the chance is 1/6 that any chosen number shows up. The answer is obvious. But as problems of probability get more involved, the answers can be more difficult to calculate. Answers to even simple problems can often deviate from our intuitive feeling of what they should be. And quite often, our intuition is simply wrong!

Random sequences of numbers have many uses in mathematics, and across the sciences. But generating sequences which pass rigorous tests for 'randomness' is far from trivial. As the demands have become stricter, old random number generators have been discarded, while much more sophisticated generators have been developed.

Flexagons are folded hexagonal or square polygons, with the intriguing property of changing their exposed faces when 'flexed'. They were discovered by the British mathematician Arthur Stone in 1939, and popularised by Martin Gardner in his Scientific American column in 1956. New types of flexagon are still being discovered today.

In topology, a knot is a figure consisting of a single loop with any number of crossing or knotted elements: a closed curve in space which may be moved around so long as its strands never pass through each other. And any physical knot (or tangle!) in a piece of string can be thought of as a mathematical knot simply by joining the two ends.