Assignment #1

Due: 28 September, 1998

The Josephus Problem

In this predicament his resourcefulness did not fail him. Putting his trust in divine protection he staked his life on one last throw. 'You have chosen to die,' he exclaimed; 'well then, let's draw lots and kill each other in turn. Whoever draws the first lot shall be dispatched by number two, and so on down the whole line as luck decides. In this way no one will die by his own hand --- it would be unfair when the rest were gone if one man changed his mind and saved his life.' The audience swallowed the bait, and getting his way Josephus drew lots with the rest. Without hesitation each man in turn offered his throat for the next man to cut, in the belief that a moment later his commander would die too. Life was sweet, but not so sweet as death if Josephus died with them! But Josephus --- shall we put it down to divine providence or just to luck? --- was left with one other man. He did not relish the thought either of being condemned by the lot or, if he was left till last, of staining his hand with the blood of a fellow Jew. So he used persuasion, they made a pact, and both remained alive.

Another version of the work quoted above replaces the sentence 'But Josephus --- shall we...' by 'After saying this he counted the numbers cunningly and so deceived them all.' What might that mean? According to a different author's account of the same incident *, Josephus was with forty other men, one other of whom also wished to surrender instead of committing suicide. Josephus is supposed to have suggested that they should arrange themselves in a circle, and then, going around the circle, kill every third (living!) man until only one was left, who would then commit suicide. Of course, Josephus then made sure that he and the other man who wanted to surrender stood in just the right positions to survive...

* The use of the third person notwithstanding, Josephus himself was the author of the quoted passage. The other author, supposedly one Hegesippus, might well have been embellishing Josephus' account.


1. Suppose the second author's version is correct. Which positions would Josephus and the other man opposed to committing suicide have had to occupy to ensure their survival? [5]

2. In general, if there were n men and every kth one was to be killed, how can we determine which position to stand in to be the sole survivor? [5]

Bonus. Who was Josephus and when was this incident supposed to have occurred? [1]

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