MAGIC SQUARES. 731 



take the ratio --, and multiply it by the ratio *gfg and the product is 

 3*1421. The true value is 3-1415. We have an approximation of yoch^j. 



The dimensions indicated in this experiment are those which present 

 in a given number of attempts the most chances of obtaining the greatest 

 possible approximation. We will conclude these remarks on games by some 

 observations borrowed from Laplace. 



The mind has its illusions like the sense of sight ; and just as the 

 sense of touch corrects the latter, reflection and calculation correct the 

 former, The probability founded on an every-day experience, or exaggerated 

 by fear or hope, strikes us as a superior probability, but is only a simple 

 result of calculation. 



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.* 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. 86 1. 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 pro- 

 duct when multiplied. It is therefore evident from this definition, that there 



* From " La Nature." Notice of M. Ch. Boutemps. 



