On 10th July 1796, when he was still a teenager, Gauss famously wrote EYPHKA! in his diary when recording the completion of a proof that every positive integer is the sum of at most three triangular numbers.

The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal then the corresponding sides have equal lengths. That is to say if P is the incentre of ΔABC and if BP and CP meet the sides AC and AB at B′ and C′, respectively, then

An elegant proof of this theorem appeared in [1] and is reproduced in [2].

The idea of this work originally arose from a question pertaining to a laboratory experiment on circular motion in our departmental lab manual. The experiment itself involves rotating a bob along a horizontal circle (Figure 1), where the tension in the string attached to the bob provides the centripetal acceleration of the bob, the string itself passing through a smooth vertical pipe. It is assumed that the rotation is fast enough for the effect of gravity to be neglected and therefore the orientation of the part of the string between the top end of the pipe and the bob can be taken to be horizontal. The abovementioned question enquires what happens to the speed of the bob in the case that the bottom end of the string is hand-held and pulled slowly so that the radius of the circular orbit decreases. The answer to the question is straightforward. Either a work-energy argument or an argument involving the conservation of angular momentum provides the same correct answer.

There are various combinatorial questions on rectangular arrays consisting of points, numbers, fields or, in general, of symbols such as chessboards, lattices, and graphs. Many such problems in enumerative combinatorics come from other branches of science and technology like physics, chemistry, computer sciences and engineering; for example the following two very challenging problems from chemistry:Problem 1: Dimer problem (Domino tiling)In chemistry, a large molecule composed repeatedly from monomers as a long chain is called a polymer and a dimer is composed of two monomers (where: mono = 1, di = 2, poly = many and mer = part).