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THE POPULAR SCIENCE MONTHLY. 



To return to our binary abacus : suppose the first pawn on the 

 right weighs a gramme, the second two, the third four, and so on, 

 doubling to the twelfth, which will weigh 512 grammes ; with these 

 twelve weights we can weigh all the whole numbers of grammes to 

 1023, the sum of all the preceding numbers. The principle is the same, 

 fundamentally, as that of the well-known ring-puzzle. A game was 

 published toward the end of last year, professedly of Indo-Chinese 

 origin, which was called the Tower of Hanoi. The tower was com- 

 posed of successive stones, decreasing in size as they rose, and repre- 

 sented by pawns, or 

 buttons slipped upon a 

 peg. The game con- 

 sis ts in taking the 

 pawns off from the 

 peg and arranging 

 them upon one of two 

 other pegs in such a 

 manner that only one 

 of them shall be re- 

 moved at a time, and 

 a larger one shall never 

 be put upon a smaller 

 one (Fig. 12). The 

 game is always possi- 

 ble, but demands twice 

 as many removals and 

 twice as much time for 

 each story that is add- 

 ed to the tower. Hav- 

 ing learned to rebuild 

 the tower for eight 

 stories by removing it 

 from the first peg to 

 the second one, the 

 problem is made easy 

 enough for one of nine 

 stories ; we first re- 



FiG. 12.— The Tower OP Hanoi. I. The tower at the beginning ., • i,* 



ofthegame; II. lilustraiing the process of (he removal of the mOVC tne eigUt Upper 

 stories in the rebuilding of the to wer ; ni. The tower rebuilt. ^^^^.^^ ^^ ^^^ ^ecoudi 



peg, then put the ninth story upon the third peg, and rebuild the eight 

 stories upon it. To perform the operation, we must make, for a tower 

 of two stories, at least three removals ; for one of three, 7 ; for one of 

 four, 15 ; for one of five, 31 ; for one of six, 63 ; for one of seven, 127 ; 

 for one of eight stories, 255 removals. If it takes a second to make one 

 removal, the rebuilding of a tower of eight stories will require four min- 

 utes. If the tower consists of sixty-four stories, the readjustment will 



