734 



SCIENTIFIC RECREATIONS. 



A French paper once proposed to give a prize of 500 francs to any 

 individual who would solve the following problem : 



Throw the numbers out of the box, replace them at hazard, then in 

 arranging them place them in the following order (A fig. 864). 



Fig. 864. 1 he Sixteen Puzzle. 



Now nobody solved this problem, because in nine cases out of ten it is 

 impossible to do so. The first twelve numbers will come correctly into their 

 places, and even 13 can be put in its place without much trouble; but, 

 instead of getting the last row right we shall find it will come out like B, 

 viz., 14, 15, 13, in the large majority of instances. So any case can be 

 solved in one of the two results given above, and we can tell in advance, 

 without displacing a number, in which way the puzzle will eventuate. 



Fig. 865. Example i. 



Fig. 866. Example a. 



Let us give this problem our attention for a few minutes, and we shall 

 not find it difficult. 



Take the first example. We will throw the cubes out of the box and 

 put them back in the order shown in fig. 865. 



We see now that I occupies the place of 1 1, 1 1 that of 7, 7 that of 8, 

 8 that of 6, 6 of 1 5> 15 of i. This much is evident without any study. 

 We formulate these figures as follows, beginning with I and working from 

 figure to figure till we are led to i again, and so on. 



1st. Series. i, 1 1, 7, 8, 6, 15, I . (6) even. 



Counting the number of different cubes we have 6; and we put 

 (6) in a parenthesis. We call the first series even because 6 is an even 

 number. 



We now establish, by the same formula, a second series commencing 

 with 2, and going back to it, thus 



