CH. XI] MISCELLANEOUS PROBLEMS 229 



consists of a transfer of the smallest disc from one peg to another, 

 the pegs being taken in cyclical order : further if the discs be 

 numbered consecutively 1, 2, 3, ... beginning with the smallest, 

 all those with odd numbers rotate in one direction, and all those 

 with even numbers in the other direction. 



Obviously, the discs may be replaced by cards numbered 

 1, 2, 3, ... n ; and if n is not greater than 10 playing cards may 

 be conveniently used. 



De Parville gave an account of the origin of the toy which 

 is a sufficiently pretty conceit to deserve repetition*. In the 

 great temple at Benares, says he, beneath the dome which 

 marks the centre of the world, rests a brass plate in which are 

 fixed three diamond needles, each a cubit high and as thick 

 as the body of a bee. On one of these needles, at the creation, 

 God placed sixty-four discs of pure gold, the largest disc resting 

 on the brass plate, and the others getting smaller and smaller 

 up to the top one. This is the Tower of Bramah. Day and 

 night unceasingly the priests transfer the discs from one diamond 

 needle to another according to the fixed and immutable 

 laws of Bramah, which require that the priest on duty must 

 not move more than one disc at a time and that he must place 

 this disc on a needle so that there is no smaller disc below it. 

 When the sixty-four discs shall have been thus transferred from 

 the needle on which at the creation God placed them to one of 

 the other needles, tower, temple, and Brahmins alike will crumble 

 into dust, and with a thunderclap the world will vanish. 



The number of separate transfers of single discs which the 

 Brahmins must make to effect the transfer of the tower is 

 2 W -1, that is, is 18,446744,073709,551615: a number which, 

 even if the priests never made a mistake, would require many 

 thousands of millions of years to carry out. 



Chinese Rings f. A somewhat more elaborate toy, known 

 as Chinese Rings, which is on sale in most English toy-shops, 



* La Nature, Paria, 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. m, p. 587 ; by Wallis in his Algebra, Latin edition, 

 1693, Opera, vol. n, ohap. oxi, pp. 472 — 478 ; and allusion iB made to it also 

 in Ozanam's RScriations, 1723 edition, vol. rv, p. 439. 



