232 MISCELLANEOUS PROBLEMS [CH. XI 



To take off the first 2 rings requires 2 steps ; 

 .'. to take off the first 4 rings requires 2x4 additional steps ; 



• R O yii 



• • » it tt u it n * A1 it » 



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

 off the first 2w rings is 2 + (2 x 4) + (2 x 4 2 ) + ... + (2 x 4"- 1 ), 

 which is equal to £ (2 2n+1 - 2). 



If we take off or put on the first two rings in one step 

 instead of two separate steps, these results become respectively 

 2 m and 2 M - 1 -1. 



I give the above analysis because it is the direct solution 

 of a problem attacked unsuccessfully by Cardan in 1550 and by 

 Wallis in 1693, and which at one time attracted some attention. 



I proceed next to give another solution, more elegant 

 though rather artificial. This, which is due to Monsieur Gros*, 

 depends on a convention by which any position of the rings 

 is denoted by a certain number expressed in the binary scale of 

 notation in such a way that a step is indicated by the addition 

 or subtraction of unity. 



Let the rings be indicated by circles : if a ring is on the 

 bar, it is represented by a circle drawn above the bar ; if the 

 ring is off the bar, it is represented by a circle below the bar. 

 Thus figure i below represents a set of seven rings of which the 

 first two are off the bar, the next three are on it, the sixth is off 

 it, and the seventh is on it. 



Denote the rings which are on the bar by the digits 1 or 

 alternately, reckoning from left to right, and denote a ring 

 which is off the bar by the digit assigned to that ring on the 

 bar which is nearest to it on the left of it, or by a if there is 

 no ring to the left of it. 



Thus the three positions indicated below are denoted re- 

 spectively by the numbers written below them. The position 

 represented in figure ii is obtained from that in figure i by 

 putting the first ring on to the bar, while the position repre- 

 sented in figure iii is obtained from that in figure i by taking 

 the fourth ring off the bar. 



* TMorie du Baguenodier, by L. Gros, Lyons, 1872. I take the account of 

 this from Lucas, vol. ±, part 7. 



