230 



MISCELLANEOUS PROBLEMS 



[CH. XI 



is represented in the accompanying figure. It consists of a 

 number of rings hung upon a bar in such a manner that the 

 ring at one end (say A) can be taken off or put on the bar 

 at pleasure; but any other ring can be taken off or put on 

 only when the one next to it towards A is on, and all the 

 rest towards A are off the bar. The order of the rings cannot 

 be changed. 



Only one ring can be taken off or put on at a time. [In 

 the toy, as usually sold, the first two rings form an exception 

 to the rule. Both these can be taken off or put on together. 



-j , — j — j j — j — , — _ 



To simplify the discussion I shall assume at first that only one 

 ring is taken off or put on at a time.] I proceed to show that, 

 if there are n rings, then in order to disconnect them from the 

 bar, it will be necessary to take a ring off or to put a ring on 

 either £ (2 n+1 — 1) times or ^ (2" +1 — 2) times according as n is 

 odd or even. 



Let the taking a ring off the bar or putting a ring on the 

 bar be called a step. It is usual to number the rings from the 

 free end A. Let us suppose that we commence with the first 

 m rings off the bar and all the rest on the bar; and suppose 

 that then it requires x—\ steps to take off the next ring, 

 that is, it requires x — 1 additional steps to arrange the rings 

 so that the first m+1 of them are off the bar and all the 

 rest are on it. Before taking these steps we can take off 



