Frazer.] ^UO [March 5, 



First Method. 



Arrangement of numbers in consecutive lines reaching from left to right 

 or from right to left. 



Number one heads the column of outside numbers. 

 Solutions only possible when 12 and 13 are diagonally adjacent. The 

 motion of the head of the column must be past 15-12 in the order named, 

 the 1 stopping on the square diagonally adjacent to that occupied by 12. 

 Only those combinations permit solution when the 12 and 13 are diagonally 

 adjacent. After the first row is complete, the 5 with its following series 

 pass on the next line pushing the 15-12 before them, and the 9 and follow- 

 ing three numbers pass on the third line pushing the 13-14 before them. 

 When this third row is complete the numbers are in order. If after the 

 outside series is complete the middle numbers occur as in any of the com- 

 binations of Table I (which are resolvable into each other by moving one 

 of the blocks temporarily to the vacant space on the outside, rotating the 

 others and then replacing it and if necessary repeating the operation by 

 thus temporarily moving one of the blocks which is in its right place and 

 rotating again till the desired combination is effected) the following solu- 

 tions are possible: 



I. The middle numbers occur as in some combination of Table I. 



1. The increase of outside numbers is with the motion of the hands of 



a watch ; a 1, and b 1 (turn box half round). 



2. The increase of outsiders is opposite to the above ; c 2, and d 3. 



II. The middle numbers occur in some combination of Table II. 



1. Increase of outside numbers with motion of watch hands c 1, and d 3. 



2. Increase of outside numbers opposite to motion of watch hands a 3 



and b 2. 



Number fifteen heads the column of outside numbers. 

 Solutions are only possible when the 4 and 3 are diagonally adjacent. 

 The 15 passes the 1 and 4 in this order and stops at the square diagonally 

 adjacent to the 4. The 11 passes to the next line pushing the 1-4 before it; the 

 7 to the third line pushing the 3-2, and when this line is filled the numbers 

 are arranged. 



III. The middle numbers are one of the combinations of Table III. 



1. The decrease of outside numbers with motion of watch hands, c 1 



and d 2. 



2. The decrease of outside numbers against motion of watch hands, b3 



and al. 



IV. The middle numbers are some combination of Table IV. 



1. Decrease of outsiders witli motion of watch hands, a 2 and b 1. 



2. Decrease of outsiders against motion of watch hands, c 3 and d 1. 



