Page:Popular Science Monthly Volume 26.djvu/464

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. Fig. 12.—The Tower of Hanoi. I. The tower at the beginning of the game; II. Illustrating the process of the removal of the stories in the rebuilding of the tower; III. The tower rebuilt. The tower was composed of successive stones, decreasing in size as they rose, and represented by pawns, or buttons slipped upon a peg. The game consists 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 removed at a time, and a larger one shall never be put upon a smaller one (Fig. 12). The game is always possible, but demands twice as many removals and twice as much time for each story that is added to the tower. Having 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 remove the eight upper stories to the second 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 minutes. If the tower consists of sixty-four stories, the readjustment will