26 ARITHMETICAL EECEEATIONS [CH. I 



For instance, take the Josephus problem in which to = 3. 

 Then we know that the final survivor of 41 men occupied 

 originally the 31st place. Suppose that when there had been 

 (41 + x) men, the survivor occupied originally the yth place. 

 Then, if we consider only the lowest value of x which makes 

 y less than m, we have y = (31 + 3x) — (41 + x) = 2x — 10. 

 Now, we have to take a value of x which makes y positive and 

 less than to, that is, in this case equal to 1 or 2. This is x = 6 

 which makes y = 2. Hence, had there been 47 men the man 

 last chosen would have originally occupied the second place. 

 Similarly had there been (47 + x) men the man would have 

 occupied originally the yth place, where, subject to the same 

 conditions as before, we have y = (2 + Sx) — (47 + x) = 2x — 45. 

 If x = 23, y — 1. Hence, with 70 men the man last chosen 

 would have occupied originally the first place. Continuing 

 the process, it is easily found that if n does not exceed 

 2000000 the last man to be taken occupies the first place when 

 n = 4, 6, 9, 31, 70, 105, 355, 799, 1798, 2697, 9103, 20482, 30723, 

 69127, 155536, 233304, 349956, 524934, or 787401 ; and the 

 second place when n = 2, 3, 14, 21, 47, 158, 237, 533, 1199, 4046, 

 6069, 13655, 46085, 103691, 1181102, or 1771653. From these 

 results, by repeated applications of the proposition, we find, for 

 any intermediate values of n, the position originally occupied 

 by the man last taken. Thus with 1000 men, the 604th place ; 

 with 100000 men, the 92620th place; and with 1000000 men, 

 the 637798th place are those which would be selected by a 

 prudent mathematician in a company subjected to trimation. 



Similarly if a set of 100 men were subjected to decimation, 

 the last to be taken would be the man originally in the 26 th 

 place. Hence, with 227 men the last to be taken would be the 

 man originally in the first place. 



Modifications of the original problem have been suggested. 

 For instance* let 5 Christians and 5 Turks be arranged round 

 a circle thus, TGTGGTGTGT. Suppose that, if beginning 

 at the ath man, every Ath man is selected, all the Turks will be 

 picked out for punishment ; but if beginning at the 6th man, 

 * H. E. Dudeney, Tit-Bits, London, Oct. 14 and 28, 1905. 



