CH. i] ARITHMETICAL RECREATIONS 25 



you are getting old. We ought to settle who shall be your 

 heir. Let us arrange our 30 children in a circle, and counting 

 from one of them remove every tenth child until there remains 

 but one, who shall succeed to your estate." The proposal 

 seemed reasonable. As the process of selection went on, the 

 farmer grew more and more astonished as he noticed that the 

 first 14 to disappear were children by his first wife, and he ob- 

 served that the next to go would be the last remaining member 

 of that family. So he suggested that they should see what 

 would happen if they began to count backwards from this lad. 

 She, forced to make an immediate decision, and reflecting that 

 the odds were now 15 to 1 in favour of her family, readily 

 assented. Who became the heir? 



In the general case n men are arranged in a circle which is 

 closed up as individuals are picked out. Beginning anywhere, 

 we continually go round, picking out each mth man until only 

 r are left. Let one of these be the man who originally occu- 

 pied the pih place. Then had we begun with n + 1 men, he 

 would have originally occupied the (p + m)th place when 

 p + m is not greater than rc+1, and the (p + m — n — l)th 

 place when p + m is greater than n + 1. Thus, provided there 

 are to be r men left, their original positions are each shifted 

 forwards along the circle m places for each addition of a single 

 man to the original group*. 



Now suppose that with n men the last survivor (r = 1) 

 occupied originally the pth place, and that with (n + x) men 

 the last survivor occupied the yih place. Then, if we confine 

 ourselves to the lowest value of x which makes y less than m, 

 we have y = (p + mx) - (n + x). 



Based on this theorem we can, for any specified value of n, 

 calculate rapidly the position occupied by the last survivor of 

 the company. In effect, Tait found the values of n for which 

 a man occupying a given position p, which is less than m, 

 would be the last survivor, and then by repeated applications 

 of the proposition, obtained the position of the survivor for 

 intermediate values of n. 

 * P. Q. Tait, Collected Scientific Papers, Cambridge, vol. n. 1900, pp. 432—435. 



