Consider the statement (whose truth is obvious):

This is known as a pigeon-hole principle (or Dirichlet’s). It is one of the strategies which is often used for problem-solving. Although it is one of the most powerful tools of combinatorics, the principle can be used successfully in other branches of mathematics. Below you can find a collection of problems which can be solved easily by applying this principle to geometrical objects.

In the geometry of the triangle, the Feuerbach quadrangle comprises the four points of contact between the nine point circle and the four tritangent circles. By omitting each of the four points in turn, we create four Feuerbach triangles. When the basic triangle is isosceles we find that two of the four points of contact coincide and the resultant single Feuerbach triangle is also isosceles. In the spirit of Steiner-Lehmus we address the converse question – if one of the Feuerbach triangles is isosceles, is the basic triangle necessarily isosceles? The answer is negative.

Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

In the preceding discussion of the Galileo-Newton theory, force has been introduced as measured by mass-acceleration, in accordance with the generally accepted procedure. The existence of laws, connecting force with the conditions accompanying it, has been referred to as a fundamental fact which makes the theory possible; though, as a matter of logical order, all details as to such laws have been put into a secondary position. The recognition of the fact that acceleration was the thing, which could be connected by laws with the conditions under which a body moved, has been mentioned as Galileo’s great achievement. This was historically the first and most important, and was probably the most difficult, step to be taken. There is no record of any suggestion of it before Galileo’s time, and no theory of motion of any value had been constructed on any other basis. Perhaps the nearest approach to such a theory was the study of the celestial motions on the basis of circular motion; and this was not capable of progressing beyond the empirical stage.

In 1972 Kenneth Arrow was awarded the Nobel prize for economics. Much of his work was in the theory and philosophy of decision making, in particular the theory of voting, and in this field “Arrow’s theorem” is a truly remarkable result. It is a genuine application to the social sciences of some elementary mathematics and as such I feel that it is a pity that it is not in the repertoire of more teachers. Hence this article.