The Josephus Problem
Given that a group of 50 men are arranged in a circle under the verdict that every 7th man will be killed going around the circle until only one man is alive, find the position in which one should stand in order to be the last survivor.
General Mathematical Concepts involved in the Josephus Problem
➢ Inverse Permutation:
An inverse permutation is a permutation in which each number and the number of the place which it occupies are exchanged. For example, this set of numbers 1,2,3,4 and this set 4,1,3,2 are inverse permutations. In the case of the Josephus Problem, when each man is eliminated in the circle, another man takes his place etc. when the man in the 7 position is killed the man in the 6th position moves up to take his place as 7. Each time a man is killed, his place would be occupied by another moving to exchange places. Thus, inverse permutation is one of the mathematical concepts used in the Josephus Problem.
Multiplication is an essential concept in the first round of eliminating the people in the circle. For example, if the verdict says that every 3rd man in the circle will be killed, then any men who are standing in position of the multiples of 3 will be eliminated within the first round. Etc. positions 3, 6, 9, 12, 15, 18, 21, 24, 27, 30….
➢ Number Sequence
Application and Possible Extensions
The Josephus Problem is used today to build recursive reasoning skills through exploration and experimenting. To emphasize the importance of reasoning as a problem solving problem, we can consider the following task:
❖ After finding out the position you should stand in the 50-man circle with every 7th man eliminated to be the last survivor, explore and find another set of numbers that the last survivor will be in the same position as the survivor in the 50-man circle.
Weisstein, Eric W. (1999-2008). Josephus Problem. Accessed September 3, 2008,...