CH. XI] MISCELLANEOUS PROBLEMS 231 



the (m + 2)th ring and thus it will require x steps from our 

 initial position to remove the (m+l)th and (m + 2)th rings. 



Suppose that these as steps have been made and that thus 

 the first m + 2 rings are off the bar and the rest on it, and 

 let us find how many additional steps are now necessary to 

 take off the (to + 3)th and (to + 4)th rings. To take these off 

 we begin by taking off the (m + 4)th ring: this requires 1 step. 

 Before we can take off the (to + 3)th ring we must arrange the 

 rings so that the (to + 2)th ring is on and the first to + 1 rings 

 are off: to effect this, (i) we must get the (m + l)th ring on 

 and the first m rings off, which requires x — 1 steps, (ii) then 

 we must put on the (m + 2)th ring, which requires 1 step, 

 (iii) and lastly we must take the (to + l)th ring off, which re- 

 quires x — 1 steps: these movements require in all {2 (x - 1) + 1} 

 steps. Next we can take the (to + 3)th ring off, which requires 

 1 step; this leaves us with the first to + 1 rings off, the 

 (to + 2)th on, the (to + 3)th off and all the rest on. Finally to 

 take off the (m + 2)th ring, (i) we get the (m+ l)th ring on 

 and the first to rings off, which requires x — 1 steps, (ii) we take 

 off the (to + 2)th ring, which requires 1 step, (iii) we take the 

 (to + l)th ring off, which requires x — 1 steps : these movements 

 require {2 (x — 1) + 1} steps. 



Therefore, if when the first to rings are off it requires x 

 steps to take off the (TO+l)th and (to+ 2)th rings, then the 

 number of additional steps required to take off the (m + 3)th 

 and (TO+4)th rings is 1 + {2(x- 1) + 1} + 1 + {2(«-l) + l}, 

 that is, is 4r. 



To find the whole number of steps necessary to take off an 

 odd number of rings we proceed as follows. 



To take off the first ring requires 1 step ; 

 /. to take off the first 3 rings requires 4 additional steps ; 



• • M Jl *•* >* » ^ » 1> 



In this way we see that the number of steps required to take 

 off the first In + 1 rings is 1 + 4 + 4 2 + . .. + 4", which is equal 

 to \ (2 m +* - 1). 



To find the number of steps necessary to take off an even 

 number of rings we proceed in a similar manner. 



