CH. XI] MISCELLANEOUS PROBLEMS 233 



It follows that every position of the rings is denoted by a 

 number expressed in the binary scale: moreover, since in going 

 from left to right every ring on the bar gives a variation (that 

 is, 1 to or to 1) and every ring off the bar gives a continu- 

 ation, the effect of a step by which a ring is taken off or put on 

 the bar is either to subtract unity from this number or to add 

 unity to it. For example, the number denoting the position of 

 the rings in figure ii is obtained from the number denoting that 

 in figure i by adding unity to it. Similarly the number de- 

 noting the position of the rings in figure iii is obtained from 

 the number denoting that in figure i by subtracting unity 

 from it. 



I o ooo jo o o o o jo o r> 



o o o J o o i| o o o o 



1101000 1101001 1100111 



Figure i. Figure ii. Figure iii. 



The position when all the seven rings are off the bar is 

 denoted by the number 0000000: when all of them are on 

 the bar by the number 1010101. Hence to change from one 

 position to the other requires a number of steps equal to the 

 difference between these two numbers in the binary scale. The 

 first of these numbers is 0: the second is equal to 2' + 2 4 + 2* + 1, 

 that is, to 85. Therefore 85 steps are required. In a similar 

 way we may show that to put on a set of 2n + 1 rings requires 

 (1 + 2 2 + ... + 2 m ) steps, that is, £ (2 an+!1 - 1) steps ; and to put 

 on a set of 2n rings requires (2 + 2' + ... + 2 an_1 ) steps, that is, 



£(2*"+i_2)steps. 



I append a table indicating the steps necessary to take off 



the first four rings from a set of five rings. The diagrams in 

 the middle column show the successive position of the rings 

 after each step. The number following each diagram indicates 

 that position, each number being obtained from the one above 

 it by the addition of unity. The steps which are bracketed to- 

 gether can be made in one movement, and, if thus effected, the 

 whole process is completed in 7 movements instead of 10 steps: 

 this is in accordance with the formula given above. 



