Josephus Problem: Counting-off Puzzle

This week's lesson is the historic math problem that is the predecessor for the game Duck, Duck, Goose. However, it uses very sophisticated mathematical algorithms. The story of Josephus, the Jewish historian/mathematician, is that he was trapped in a cave with 40 other soldiers under siege by the Romans during the Battle of Yodfat 2000 years ago. No one wanted to be captured but they decided to surrender one-by-one and negotiated with the Romans that the last two people remaining would be set free. Josephus and his friend were not happy with the plan so they suggested a circle with every third man surrendering. The soldiers not eliminated would continue to count off 1, 2, 3 until the last man was left. Of course, the circle gets smaller and smaller. Josephus and his friend knew which person would be the last two remaining and put themselves in that position.

For example, take a circle of 10 people who count off by 3. Who would be the last person remaining? The answer is the 4th person. The challenge for the children is to be able to concentrate enough to follow the whole way around the circle until the last person is standing. I created two spreadsheets for the children to record their solutions. The first one in the pdf is easier where they only work with groups of 8, 10, and 12 people counting off by first 2, then 3, then 4 and so on until the last person in the group. The more difficult spreadsheet allows them to look at groups of up to 21.

Finally, when they feel they have mastered this process, they can try the actual Josephus problem with 41 people around the circle (page 9 of the pdf). They should record the last two people standing since both Josephus and his friend went free. To take this one step further, try 41 by eliminating every 4th person, every 5th, and so on.

I will review their work the next class.