Frazer.] *>vb [March 5, 



The requirements of the popular Fifteen Puzzle are to "move the blocks 

 until in regular order." This regular order may be of three kinds, when 

 the numbers are in consecutive series and the blank space is left either at 

 the beginning or the end : 



1. That usually understood where the numbers follow each other in 

 broken lines like reading matter in type, or in the opposite direction. 



2. Where the numbers follow a single coil from some point on the edge 

 of the box to the centre, or vice versa. 



3. Where the numbers follow a zigzag course across the box, reading 

 from left to right on the first and third lines, and from right to left on the 

 second and fourth, or vice versa. 



It can be shown that the three conditions which render possible one or 

 the other of these solutions are : 1st, the number which heads the outside 

 column, whether 1 or 15 ; 2d, the direction in which the numbers increase 

 along the outside, whether with or against the motion of the hands of a 

 watch ; 3d, the order in which the four middle numbers occur. 



Note. The direction of the motion of the column itself must always 

 be such that the head does not move to a square just vacated by one of the 

 series of which it is the first or last member. 



There are sixteen possible solutions under each of the three methods, 

 eight of them applicable to cases where 1 is at the head, and eight to 

 cases where 15 is at the head. 



Of these eight solutions four only are possible in any given position of 

 the middle blocks in the box, the other four becoming possible when the 

 positions of any two of the middle blocks are exchanged. 



Of each group of four possible solutions in any given position of the 

 middle blocks, two are possible when the outside column of numbers in- 

 crease in magnitude with the motion of the hands of a watch, and two 

 when the increase is the reverse of this. 



Finally of the two possible solutions where the outside numbers are in 

 arithmetical series and the position of the middle blocks is given, one 

 brings the head of the outside column to a given corner, and the other to 

 the diametrically opposite corner. 



The middle numbers must always be the lowest four or the highest four. 



The following four tables comprise all possible groupings of these two 

 sets of four numbers. The dots in the small squares indicate by their num- 

 ber the method by which the solution can be obtained without disturbing 

 the middle blocks, provided the outside numbers have been put in the 

 proper order : 



