Mathematical recreations and essays . more elaborate toy, knownas Chinese Rings, which is on sale in most English toy-shops, * La Nature, Paris, 1884, part i, pp. 285—286. t It was described by Cardan in 1550 in his De Subtilitate, bk. xv,paragraph 2, ed. Sponius, vol. in, p. 587; by Wallis in his Algebra, Latin edition,1693, Opera, vol. ii, chap, cxi, pp. 472—478; and allusion is made to it alsoin Ozanams Recreations, 1723 edition, vol. iv, p, 439. 230 MISCELLANEOUS PROBLEMS [CH. XI is represented in the accompanying figure. It consists of anumber of rings hung upon a bar in such a manner tha
Mathematical recreations and essays . more elaborate toy, knownas Chinese Rings, which is on sale in most English toy-shops, * La Nature, Paris, 1884, part i, pp. 285—286. t It was described by Cardan in 1550 in his De Subtilitate, bk. xv,paragraph 2, ed. Sponius, vol. in, p. 587; by Wallis in his Algebra, Latin edition,1693, Opera, vol. ii, chap, cxi, pp. 472—478; and allusion is made to it alsoin Ozanams Recreations, 1723 edition, vol. iv, p, 439. 230 MISCELLANEOUS PROBLEMS [CH. XI is represented in the accompanying figure. It consists of anumber of rings hung upon a bar in such a manner that thering at one end (say A) can be taken off or put on the barat pleasure; but any other ring can be taken off or put ononly when the one next to it towards A is on, and all therest towards A are off the bar. The order of the rings cannotbe changed. Only one ring can be taken off or put on at a time. [Inthe toy, as usually sold, the first two rings form an exceptionto the rule. Both these can be taken off or put on together. Cxg. To simplify the discussion I shall assume at first that only onering 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 thebar, it will be necessary to take a ring off or to put a ring oneither J (2**+^— 1) times or J (2^+^ — 2) times according as n isodd or even. Let the taking a ring off the bar or putting a ring on thebar be called a step. It is usual to number the rings from thefree end A. Let us suppose that we commence with the firstm rings off the bar and all the rest on the bar; and supposethat then it requires x—1 steps to take off the next ring,that is, it requires x—1 additional steps to arrange the ringsso that the first m +1 of them are off the bar and all therest are on it. Before taking these steps we can take off CH. Xl] MISCELLANEOUS PROBLEMS 231 the (m 4- 2)th ring and thus it will require x steps from ourinitial position to remove the (m + l)th and {m + 2
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