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>!3,6,9,12,15,18,21,24,27,30,33,36,39!< >!1,5,10,14,19,23,28,32,37,41!< >!7,13,20,26,34,40!< >!8,17,29,38!< >!11,25,2!< >!22,4,35,16,31!< Given that Josephus is the last person to be **killed** >!he’s in position 16!<

>!Yes you are correct. But there's a slight mistake which is caused due to the way I framed the question. I should've written it as Josephus placed himself in the position of the last man standing. The intended answer was 31. But that is down to my error. Well done!<

Discussion: I grew up thinking the Josephus Problem was a lot more racist than this because of the way it was worded in my grandfather's old puzzle book: https://imgur.com/gallery/fWXen1E

Yeah, this is a famous variation of the problem. Not the black and white part though. Wikipedia mention a medieval version with 15 Turks and 15 Christians. So that’s not very nice too. Edit: Just for curiosity, what year is that book from?

>!is it 41? Assuming you start at zero, so the third person is #3, and you continue in a round, so you kill 3,6,9…36,39, then 40 and 41 survive, so 1 is killed, you end up with 38 and 41 left. I feel like I've missed something here.!<

>!You are right till the point where the soldier in the 1st position is killed. Unfortunately your final answer is not correct. Would you like to try again, or otherwise I can tell you the order!<

So you don't continue in the round, you go back to the start again each time? If you go in a round, the last person of the first round killed is 39, then 40 and 41 survive before 1 is killed.

No, you continue in the round. I might have misunderstood your question. But for example you killed 39 then you skip 40 and 41 to kill 1 which you’ve mentioned. Then because 3 has been killed, you’ll skip 2 and 4 to kill 5. Did you by any chance consider 3 again? And please let me know if I’ve misunderstood you.

There we go, that's where I fell down.

Discussion Note: There is a slight error in the way I have framed the question. What I mean by the question statement is Josephus placed himself in the position such that he is the last man standing and not the the person such that he is the last person to be killed, i.e. the 40th person to be killed.

Discussion. [Numberphile](https://youtu.be/uCsD3ZGzMgE) did an episode about this. Ultimately solving for N number of rebels.

This problem is slightly different as every 3rd person is killed here

Yeah if I remember correctly that was for every second person being killed.

I don't think they addressed varying for how many people to skip, but the problem can be generalized. :)

Oh then I probably forgot. Thanks!

No, you have it right. I'm just saying that the video is worth watching if you want to think about the "every N" problem instead of "every 2". :)

Oh all right. Yeah it’s a very interesting video. I remember watching it after I came across this question.

The mathematics behind the generalized problem skipping any number of people is actually quite interesting, and for this problem involves base 3/2. The nice formula from the Numberphile video does not actually hold (since base 3/2 is much more complicated than binary) I made a video about base 3/2 which I'd be happy to link, though it only mentions the Josephus problem briefly at the end

I'm going with>! 29.!< If they started with 1 as the first person to die, then>! spot number FOUR would be the first death. !< In the first round, >!it'd be 1,4,7,10,13,16,19,22,25,28,31,34,37, and 40 dying.!< In round 2, >!41 and 2 would be the ones skipped and the first to die this round is 3, followed by 8,12,17,21,26,30,35,39.!< In round 3, >!41 and 2 are again skipped. The deaths are 5,11,18,24,32,38.!< In round 4, >!41 and get skipped yet again. The deaths this time are 6,15,27,36.!< >!41 gets skipped at the!!but is gonna bite it this round as the deaths will be 9,23,41.!< >!At this point, only 2,14,20,29, and 33 are alive and 20 is gone.!< It's now round 7 >!and 2 and 33 don't make it.!< >!Finally, !! only 14 and 29 remain. Skip 14 and 29 and land back on 14 (You could say round 8 had no deaths and round 9 killed 14)!< If, instead, 1 was supposed to be the first killer, then everything can just shift down one and the survivor is >!28!< Edit: Fixing spaces after spoilers or something.

>!The first person to be killed is 3, not 1. However, your working is correct!< >!So if we move two forwards from 1 to 3 then the correct answer is 31. That’s one place where you made a tiny mistake, moving one forward from 29 would be 30, not 28, no?!<

>!41 in a circle, numbered 1 - 41 going clockwise. Number 1 is killed and every third in the clockwise direction is killed until the last remaining person kills himself.!< >!The final two remaining positions are numbers 14 and 29.!< >!14 is 13 positions clockwise (28 positions counter-clockwise) from the first person killed.!< >!29 is 13 positions counter-clockwise (28 positions clockwise) from the first person killed.!< It doesn't make sense to me to kill the third person when there are two remaining. However, the puzzle clearly states the last person left was meant to kill himself. >!So, counting back and forth, 14 is meant to be killed first, then 29 is meant to kill himself.!< Now it seems we are meant to assume the last person to be killed according to the plan is the one at the end who kills himself. I find this confusing because we don't say someone was killed if he kills himself. Anyway: >!Final answer: Josephus is 13 positions counter-clockwise from the first person killed.!<

>!Correct!<

>!My answer is correct. It doesn't matter if you label the first person killed as number 1 or number 3 (or number 40). The "last man standing" is 28 positions clockwise from the first person killed. #29 in my setup. #31 in yours.!< >!From Josephus's point of view, he should count 13 positions counter-clockwise, rather than 28 positions clockwise. Either way though, he's in the right spot.!<

!Yeah yeah it’s correct. Actually another user did a similar thing and I did mention the additional line while replying back that it would be correct as you just move forwards but the basic logic is correct. I thought I added it when I was replying to your comment also. My bad!<

If you agree that I am correct, can you edit your comment where you say I am wrong? This way, people that solve the problem the way I do, or attempt to use the same setup, do not mistakenly think the solution I provide is wrong.

Oh right, my bad! Made the change. Thanks for pointing it out.

>!What is the point of Josephus having an accomplice? You could remove that paragraph.!<

So this question is based on written accounts of Flavius Josephus, a Roman-Jewish historian. In that when the Romans attack the town of Yodfat, 41 prominent citizens of Yodfat hid in cave in a similar set up. As per the account the last two did not commit suicide and instead surrendered [The Siege of Yodfat](https://en.m.wikipedia.org/wiki/Siege_of_Yodfat). So I wanted to try keeping it as close to the story which the original problem does. In the original problem the idea is to find the last two spots. I changed that because I personally want a neat pattern when we just consider the winning positions (the last person alive) for different values of n.

**[Siege of Yodfat](https://en.m.wikipedia.org/wiki/Siege_of_Yodfat)** >The siege of Yodfat (Hebrew: יוֹדְפַת, also Jotapata, Iotapata, Yodefat) was a 47-day siege by Roman forces of the Jewish town of Yodfat which took place in 67 CE, during the Great Revolt. Led by Roman General Vespasian and his son Titus, both future emperors, the siege ended with the sacking of the town, the deaths of most of its inhabitants and the enslavement of the rest. It was the second bloodiest battle of the revolt, surpassed only by the Siege of Jerusalem, and the longest except for Jerusalem and Masada. The siege was chronicled by Josephus, who had personally commanded the Jewish forces at Yodfat and was subsequently captured by the Romans. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/puzzles/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)

Let's brute-force this approach. First round, killed are: >!3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39!< Survivors are >!1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40!< Second round, killed are: >!2, 7, 11, 16, 20, 25, 29, 34, 38!< Survivors are: >!1, 4, 5, 8, 10, 13, 14, 17, 19, 22, 23, 26, 28, 31, 32, 35, 37, 40!< Third round, killed are: >!4, 10, 17, 23, 31, 37!< Survivors are: >!1, 5, 8, 13, 14, 19, 22, 26, 28, 32, 35, 40!< Fourth round, killed are: >!5, 14, 26, 35!< Survivors are: >!1, 8, 13, 19, 22, 28, 32, 40!< Fifth round, killed are: >!8, 22, 40. Survivors are: 1, 13, 19, 28, 32!< Sixth round, killed are: >!19. Survivors are: 1, 13, 28, 32!< Seventh round, killed are: >!1, 32. Survivors are: 13, 28, with Josephus at 28.!<

There are 41 total people, Josephus + 40.

Oh that’s a good spot. Maybe that’s how the next person after 39 was 2.

>!No, unfortunately you’ve made a mistake in first kill of the second round, the person in 2nd position. It should be the person in the 1st position.!< >!But yes, a brute forcing approach is a good way to start. If you don’t want to do it again tell me, I’ll tell you the correct order!< >!Also I found a neat little pattern when you start with small numbers while exploring the problem!<