Math Tricks

CHAPTER XVI PROBLEMS OF ARRANGEMENT In these problems of arrangement we shall not deal with games such as chess, checkers, halma or others in which the winner defeats his adversary by a more ingenious arrangement of his pieces or men. In this chapter we are interested only in problems which permit a more or less mathematical treatment.

To be sure, many such problems are rather close to the domain of games. Some of these, such as the Problem of the Knight, the Problem of the Five Queens, the Fifteen Puzzle, and so on, have been considered, because of their intermediary position, worthy of scientific treatment by great mathematicians. Thus, in 1758, Leonhard Euler submitted to the Berlin Academy of Sciences a paper on the Problem of the Knight.

The most prominent problem of arrangement since ancient times has been the magic square. It fascinated great thinkers in the India of Sanskrit times and is by no means scorned by modern mathematicians. Many of you are probably already familiar with the structure of a magic square. It is a square array of n times n distinct positive integers, often consecutive ones, arranged in such a way that the sum of the n numbers in any horizontal, vertical or main diagonal line is always the same. Construction of such magic squares has developed into a real sciences and ingenious methods have been devised for their formation.

However, we doubt that the whole magic squares business is worth all the time and labor spent on this pastime. We have, therefore, no intention of describing the methods of their construction, the less so because once any of the well-known methods for solving them are understood, only a mechanical application of these methods is required. However, there are a number of quite different problems of arrangement which 96