The Josephus Problem

The Josephus Problem November 2, 2016

The above video was drawn to my attention by a friend. It is nice to know that there are people for whom the “Josephus problem” is not something related to the Testamonium Flavianum or the mention of James the brother of Jesus, but something with real world practical implications. Can you guess what the “Josephus problem” is in mathematics before watching the video, and why it is called that?

UPDATE: Jim Davila pointed out that there is an article in Popular Mechanics about this.

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  • Jeff Cate

    I was expecting a math problem about numeric exaggerations… 🙂

    Interesting math problem. The story is told in Josephus Wars 3.387-391. Josephus wasn’t quite as good at math or counting as the math guy in the video. At Jotapata, Josephus ended up trapped in a deep pit with 40 other Jewish men, so there must have been 41 of them total. When they decided to make their death pact, they drew lots. It plays out opposite of the math problem in the video, but the process would’ve been the same.

    Instead of #1 killing #2 as in the math video, Josephus says lot #1 would be killed off by lot #2. In Josephus, the odd numbers would have been killed off first… not vice versa as in the math problem with the even numbers being killed off first.

    As the first round ended, #41 would’ve been killed off by #2. Round 2 begins with #4 killed off by #6, #8 killed off by #10, etc. At the end of round 2, #40 would’ve been killed off #2, then #6 killed off by #10, #14 killed off by #18, #22 killed off by #26, and #30 killed off by #34.

    Round 3 begins with #38 killed off by #2, then #10 killed off by #18 and #26 killed off by #34.

    At this point, only three survive. Round 4 is simply #2 killed off by #18.

    Only two are left at this point. #18 should be killed off by #34. Josephus needed to be lot #34 to be the last survivor. He must have been lot #18 because #34 was supposed to kill him at the end. But Josephus persuaded the man not to commit bloodshed against his countryman. The guy spared Josephus, and they both survived the death pact.

    Nonetheless, fascinating little math problem inspired by the dramatic events in the pit at Jotapata… but history is usually kinda messy… not quite as clean and neat as a math problem… especially in times of war… 🙂

    • Jeff Cate

      I should add that Josephus doesn’t actually clarify–and I’m no expert in lot-casting–if all 41 of them drew lots only once numbered 1 to 41 (aleph to mem aleph)… or if they only had 2 lots that pairs of men kept drawing repeatedly.

      If the former, Josephus’ odds of surviving were 1 out of 41. If the latter, are his odds still 1 out of 41?… or something different because he would’ve had to survive 5 rounds of 1 out of 2 odds… essentially, 1 out of 32 (= 2 to the 5th power)?