168 
Proceedings of Royal Society of Edinburgh. [sess. 
Successive applications of this process give, 
in order 
70 
1 
13,655 
2 
105 
1 
20,482 
1 
158 
2 
30,723 
1 
237 
2 
46,085 
2 
355 
1 
69,127 
1 
533 
2 
103,691 
2 
799 
1 
155,536 
1 
1,199 
2 
233,304 
1 
1,798 
1 
349,956 
1 
2,697 
1 
524,934 
1 
4,046 
2 
787,401 
1 
6,069 
2 
1,181,102 
2 
9,103 
1 
1,771,653 
2 
provided the (merely arithmetical) work is correct. And, of course, 
we can at once interpolate for any intermediate value of n. 
Thus, in 799 men, or in 30,723, the first is safe : — in 1000 the 
604th; in 100,000 the 92,620th, and in 1,000,000 the 637,798th. 
The earlier steps of this process, which lead at once to Bachet’s 
number for 41 (assumed above), are 
1 1 
9 
1 
2 2 
14 
2 
3 2 
21 
2 
4 1 
6 1 
31 
1 
so that the method practically deals with millions, when we reach 
them, more easily than it did with tens. 
Unfortunately the cycles become shorter as the radix, and with 
it the choice of remainders increases ; so that a further improve- 
ment of process must, if possible, be introduced when every 
hundredth man (say) is to be knocked out. 
From the data above given, it appears that up to two millions 
the number of cases in which the first man is safe is 19, while that 
in which the second is safe is only 16. (The case of one man, 
only, is excluded.) As these cases should, in the long run, be 
equally probable, I extended the calculation to 13,059,835,455,001, 
1, with the result of adding 20 and 19 to these numbers respectively. 
But the next 15 steps appear to give only 2 cases in favour of the 
first man ! 
