Name: The Josephus Problem - Numberphile
Uploaded: 2020-03-27T18:05:19.000Z
Duration: 13 min 58 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Past simple

There was a group of Jewish soldiers who were surrounded by the Roman army

and they didn't want to get captured, so they decided to come up with a system-

So they'd sit in a circle, and the first man would kill the guy to the left of him

The next remaining living person would kill the next remaining living person with their sword.

And they'd go around like this, till there's only one person left,

and the last person would have to commit suicide of course, rather than get captured.

And the story - at least the story I was told, I'm not sure if this is historically accurate -

Was that Josephus preferred captured than suicide,

but he worried that if he said this, the other soldiers would turn on him.

And so he wanted to figure out: WHERE should he sit within the circle-

And then he would surrender, rather than kill himself.

It's a little tricky, so let's do a smaller example.

And so just to be clear, the way it works is: person number one kills number two.

But now things gets a little of harder to kind of see in advance

Because seven, there is no eight, so seven kills one,

So seven's the winner; seven is the last one left over.

For Josephus, there were 41 people. He needed to figure out where to sit.

The mathematical problem is if there were n people.

When I learned this problem I was in high school,

And it was one of the first problems that got me to understand how you approach-

Math problems where you don't know the answer in advance, or someone hasn't shown it to you in advance.

who played a big role in, kind of, leading me to math.

And he suggested: what we should do is we should gather data.

Take various values of n and just do it by hand

I don't know, maybe we should just do that?

n is the number of seats, and w of n will be the winning seat.

And what we know so far is that: n is seven, w of n it turns out is also seven.

And so what I would do is I'd start doing some other values

I guess there was no reason for me to skip six, so why don't we fill in six.

One kills two, three kills four, five kills six...

If you're watching you might already start to notice some patterns.

For instance, they're always odd. I mean, that's maybe the first thing you notice

And you can start asking "why are they always odd?" And maybe you can already see that-

- in the first loop around, all the even people get killed.

So you can - even with a tiny amount of experiment - start to see some patterns.

- *Brady* So you DON'T want to sit in an even seat? - Don't sit in an even seat, no no no.

And maybe we're even getting some glimmers of other patterns

But before I really try to phrase those, I'd like to fill in the table a little bit more.

That person's the winner, so that one was easy.

If there's two people: one kills two. One is the winner

If there's three people: one kills two, three kills one. Three is the winner

If there's four people: one kills two, three kills four, one kills three.

If there's eight people: one kills two, three kills four, five kills six, seven kills eight,

one kills three, five kills seven, one kills five.

Alright so it was one, one, three, one, three, five, seven.

And you could keep going, but maybe you can see the pattern now?

- *Brady* Is thirteen a one? - NO! Good guess actually!

But it is not a one. So let's do thirteen.

*Brady* This is good, I don't know then, I can't see the pattern.

So as you see it's jumping by two each time, but then it resets at some point

And I want to go back to this, so we'll see what thirteen is quickly

So we didn't reset that, and I claim we won't reset it the next one either.

and here, by the way, we're be starting to make predictions.

Even though we maybe can't say exactly what like 178 would be-

- we're starting to see well enough so we can make a prediction.

so you could guess. Is it really true that fourteen is going to give you thirteen?

We could do it out, and you'll see in fact that's what you'll get

But I want to go back to this point you had, you asked if it's going to reset and go back to one there

And the answer was no, and it's worth looking.

Because this is the next thing I was going to do.

So if you look for a sec, there's something very special about the numbers that give you ones.

And so maybe you can now guess that if i put in a sixteen, I would get a one back.

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The Josephus Problem - Numberphile

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