Josephus Flavius Problem

Following is an excerpt from
Mathematical Recreations and Essays
by W.W. Rouse Ball and H.S.M.Coxeter

Hegesippus says that Josephus saved his life by such a device. According to his account, after the Romans had captured Jotapat, Josephus and forty other Jews took refuge in a cave. Josephus, much to his disgust, found that all except himself and one other man were resolved to kill themselves, so as not to fall into the hands of their conquerors. Fearing to show his opposition too openly he consented, but declared that the operation must be carried out in an orderly way, and suggested that they should arrange themselves round a circle and that every third person should be killed until but one man was left, who must then commit suicide. It is alleged that he placed himself and the other man in the 31st and 16th place respectively.

The medieval question was usually presented in the following form. A ship, carrying as passengers 15 Turks and 15 Christians, encountered a storm, and, in order to save the ship and crew, one-half of the passengers had to be thrown into the sea. Accordingly the passengers were placed in a circle, and every ninth man, reckoning from a certain point, was cast overboard. It is desired to find an arrangement by which all the Christians should be saved. In this case we must arrange the men thus: CCCCTTTTTCCTCCCTCTTCCTTTCTTCCT, where C stands for a Christian and T for a Turk. The order can be recollected by the positions of the vowels in the following line: From numbers' aid and art, never will fame depart, where a stands for 1, e for 2, i for 3, o for 4, and u for 5. Hence the order is o Christians, u Turks, etc.

If every tenth man were cast overboard, a similar mnemonic line is Rex pappy cum gent bona date sign serene. An oriental setting of this decimation problem runs somewhat as follows. Once upon a time, there lived a rich farmer who had 30 children, 15 by his first wife who was dead, and 15 by his second wife. The latter woman was eager that her eldest son should inherit the property. Accordingly one day she said to him, "Dear Husband, you are getting old. We ought to settle who shall be your heir. Let us arrange our 30 children in a circle, and counting from one of them, remove every tenth child until there remains but one, who shall succeed to your estate." The proposal seemed reasonable. As the process of selection went on, the farmer grew more and more astonished as he noticed that the first 14 to disappear were children by his first wife, and he observed that the next to go would be the last remaining member of that family. So he suggested that they should see what would happen if they began to count backwards from this lad. She, forced to make an immediate decision, and reflecting that the odds were now 15 to 1 in favor of her family, readily assented. Who became the heir?

Reference

  1. W.W. Rouse Ball and H.S.M. Coxeter, Mathematical Recreations and Essays, Dover, 1987

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