Art_of_Programming_Contest_Part10

# Art_of_Programming_Contest_Part10 - APPENDIX A ACM...

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APPENDIX A ACM PROGRAMMING PROBLEMS 183 The Input The input consists of a sequence of integer pairs n and p with each integer on a line by itself. For all such pairs , and there exists an integer k , such that . The Output For each integer pair n and p the value should be printed, i.e., the number k such that . Sample Input 2 16 3 27 7 4357186184021382204544 Sample Output 4 3 1234 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?

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APPENDIX A ACM PROGRAMMING PROBLEMS 184 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 proceeeds from the person to his immediate left, with the k th 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
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Art_of_Programming_Contest_Part10 - APPENDIX A ACM...

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