In a long series of events of the same kind, the mere chances of accident sometimes offer these curious veins of good or bad fortune, which many persons do not hesitate to attribute to a kind of fatality. It often happens in games which depend both on chance and the cleverness of the players, that he who loses, overwhelmed with his want of success, seeks to repair the evil by rash playing, which he would avoid on another occasion; he thus aggravates his own misfortune and prolongs it. It is then, however, that prudence becomes necessary, and that it is desirable to remember that the moral disadvantage attaching to unfavourable chances is increased also by the misfortune itself.[41] Mathematical games, formerly so much studied, have recently obtained a new addition in the form of an interesting game, known as the “Boss” puzzle. It has been introduced from America, and consists of a square box, in which are placed sixteen small wooden dice, each bearing a number (fig. 860). No. 16 is taken away, and the others are placed haphazard in the box, as shown in fig. 861. The point is then to move the dice, one by one, into different positions, so that they are at last arranged in their natural order, from one to fifteen; and this must be accomplished by slipping them from square to square without lifting them from the box. If the sixteenth dice is added, the game may be varied, and we may seek another solution of the problem, by arranging the numbers so that the sum of the horizontal, vertical, and diagonal lines gives the number 34. In this form the puzzle is one of the oldest known. It dates from the time of the primitive Egyptians, and has often been investigated during the last few centuries, belonging, as it does, to the category of famous magic squares, the principles of which we will describe. The following is the definition given by Ozanam, of the Academy of Sciences, at Paris, at the end of the seventeenth century. The term magic square is given to a square divided by several small equal or broken squares, containing terms of progression which are placed in such a manner that all those of one row, either across, from top to bottom, or diagonally, make one and the same sum when they are added, or give the same product when multiplied. It is therefore evident from this definition, that there are two kinds of magic squares, some formed by terms of arithmetical progression, others by terms of geometrical progression. We must also distinguish the equal from the unequal magic squares.
Fig. 860.—The sixteen puzzle.
Fig. 861.—The numbers placed at hazard, and No. 16 removed.
We give here several examples of magic squares with terms of mathematical progression, among them the square of 34, giving one of the solutions to the puzzle just described (fig. 862). We also give an example of a magic square composed of terms of geometrical progression. The double progression for examples 1, 2, 4, 8, 16, 32, 64, 128, 256, as here arranged (fig. 863), forms such a square that the product obtained by multiplying the three terms of one row, or one diagonal, is 4,096, which is the cube of the mean term 16. The squares have been termed magic, because, according to Ozanam, they were held in great veneration by the Pythagoreans. In the time of alchemy and astrology, certain magic squares were dedicated to the seven planets, and engraved on a metal blade which sympathized with the planet. To give an idea of the combinations to which the study of magic squares lends itself, it is sufficient to add that mathematicians have written whole treatises on the subject. Frénicle de Bessy, one of the most eminent calculators of the seventeenth century, consecrated a part of his life to the study of magic squares. He discovered new rules, and found out the means of varying them in a multitude of ways. Thus for the magic square, the root of which is 4, only sixteen different arrangements were known.
Fig. 862.—Examples of magic squares formed by terms of arithmetical progression.
Frénicle de Bessy found 880 new solutions. An important work from the pen of this learned mathematician has been published under the title of “Carrés ou Tables Magiques,” in the “Memoirs de l’Académie Royale des Sciences,” from 1666-1699, vol. v. Amateurs, therefore, who are accused of occupying themselves with a useless game, unworthy the attention of serious minds, will do well to bear in mind the works of Frénicle, and better still, to consult them.
