166 
Proceedings of Royal Society of Edinburgh. 
SESS. 
the process were continued, they would he exterminated in the 
order given. And any* similar question, involving only moderate 
numbers, would probably be most easily solved in a similar fashion. 
But, suppose the number of companions of Josephus to have been 
of the order even of hundreds of thousands only, vastly more if 
of billions, this graphic method would involve immense risk of 
error, besides being toilsome in the extreme ; and the ivhole 
process would have to be gone over again if we wished the 
solution for the case in which the total number of men is altered 
even by a single unit. 
It is easy, however, to see that the following general statement 
gives the solution of all such problems : — 
Let n men be arranged in a ring which closes up its ranks as 
individuals are picked out. Beginning anywhere, go continuously 
round, picking out each ??z th man until r only are left. Let one 
of these be the man who originally occupied the p> th place. Then, 
if we had begun with n + 1 men, one of the r left would have been 
the originally (p + m)^, or (if p 4- to >n + 1) the (p + m — rt — l) th . 
In other words, provided there are always to be r left, their 
original positions are each shifted forwards along the closed ring 
by to places for each addition of a single man to the original 
group. 
A third, but even more simple and suggestive, mode of state- 
ment may obviously be based on the illustrations which follow. 
In these the original number of each man is given in black type, 
the order in which he is struck off, if the process be carried out to 
the bitter end, in ordinary type. 
By threes : — 
351742860 
12345678 
971462853 
123456789 
Increase by unit every number in the first line (to which a 0 
has been appended) and write it over the corresponding number in 
the third. We have the scheme 
462853971 , 
97146285 3 . 
