模拟约瑟夫环

 

 Roman Roulette 

The historian Flavius Josephus relates how, in the Romano-Jewish conflict  of 67 A.D., the Romans took the town of Jotapata which he was commanding.   Escaping, Jospehus found himself trapped in a cave with 40 companions.  The  Romans discovered his whereabouts and invited him to surrender, but his  companions refused to allow him to do so.  He therefore suggested that they  kill each other, one by one, the order to be decided by lot.  Tradition has it  that the means for effecting the lot was to stand in a circle, and, beginning  at some point, count round, every third person being killed in turn.  The  sole survivor of this process was Josephus, who then surrendered to the  Romans.  Which begs the question: had Josephus previously practised quietly  with 41 stones in a dark corner, or had he calculated mathematically that he  should adopt the 31st position in order to survive?

Having read an account of this gruesome event you become obsessed with  the fear that you will find yourself in a similar situation at some time in  the future.  In order to prepare yourself for such an eventuality you decide  to write a program to run on your hand-held PC which will determine the  position that the counting process should start in order to ensure that you  will be the sole survivor.

In particular, your program should be able to handle the following variation  of the processes described by Josephus.  n > 0 people are initially  arranged in a circle, facing inwards, and numbered from 1 to n.  The  numbering from 1 to n proceeds consecutively in  a clockwise direction.   Your allocated number is 1.  Starting with person number i, counting  starts in a clockwise direction, until we get to person number k (k > 0),  who is promptly killed.  We then proceed to count a further k people in a  clockwise direction, starting with the person immediately to the left of the  victim.  The person number k so selected has the job of burying the  victim, and then returning to the position in the circle that the victim had  previously occupied.  Counting then proceeds from the person to his  immediate left, with the kth person being killed, and so on, until only one  person remains.

For example, when n = 5, and k = 2, and i = 1, the order of execution is  2, 5, 3, and 1.  The survivor is 4.

 

Input and Output

 

 

Your program must read input lines containing values for n and k (in  that order), and for each input line output the number of the person with  which the counting should begin in order to ensure that you are the sole  survivor.  For example, in the above case the safe starting position is 3.   Input will be terminated by a line containing values of 0 for n and k.

Your program may assume a maximum of 100 people taking part in this  event.

 

Sample Input

 

 

 

1 11 50 0

 

 

 

Sample Output

 

 

 

11