# A bit of a morbid math question.

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A friend and I were recently watching "The Purge", and the topic came up that if everyone in the world killed only 1 person that the world's population would be halved. I thought about this for a minute and realised mostly it would be reduced by more than half and that in the extreme people pairing off and simultaneously killing each other could reduce it to 0, while still fulfilling the criteria.

My question is what would a probability distribution of this scenario look like. What would be the most likely percentage remaining alive?

What is this kind of maths problem called?

Edit: Feel free to use 7 billion as the world population size or go with any smaller size to show it. Because I'm lazy I used 3 people and it ends with 1/3 and you never want to shoot first lol.

Edited by Sorcerer

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Odd question; at first I wondered how you would get more than half killed then I realised that if you lined up everyone in the world and put them in a really big circle then , if each one stabbed the person in front of them everyone would get killed.

I doubt there's a well defined distribution. It would depend on the model of how people selected their victim.

However, The "everyone get's killed" scenarios need a lot of planning while the "half the people get killed" scenarios can be arranged by more or less random- kill the first person you meet strategy.

So I suspect the answer would be nearer half than all dead.

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I don't follow your logic. If everyone kills one person and (presumably) nobody can be killed twice, then everyone must get killed. How do you figure otherwise?

If the population is 10, say, and each person kills one person, then the total number of people killed is ten. There's no probability distribution, no pairing. Under the hypothesis that everyone in the world kills one person, it's not possible for anyone to survive. They can each kill themselves, or they can pair up and kill each other, or person n can kill person n+1 and the last person kills the first (cyclic homicide). No matter how you arrange this, everyone gets killed.

Can you walk through your idea that it's possible for anyone to not be killed?

Do you mean perhaps that each person has some probability of killing one person? That's a totally different question than what you asked.

Edited by wtf

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I think the idea is that some people get killed before they get a chance to kill.

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I think the idea is that some people get killed before they get a chance to kill.

Tricky little problem then, just to figure out what the rules are. For example, it seems reasonable to say that "Nobody can kill anyone if they themselves have been killed."

That leads to a logical contradiction. Say you have ten people, 1 through 10, and person n kills person n+1, so everyone is dead except 1.

Ok, think about 10. He was killed by 9. Ok, fine, but ... 9 was killed by 8. So by our rule, 9 did not kill 10.

Working backwards, nobody can get killed. But if nobody gets killed, then somebody MUST get killed. Contradiction.

We need to formalize the rules of this game, and that seems trickier than it seems.

(ps) -- If we introduce time, and say that only one person at a time can kill; and we weaken our rule to "Nobody can kill anyone if they have ALREADY been killed," this might be a workable model of the problem

In other words if they're going to get killed in the FUTURE, then they can kill someone in the present.

Edited by wtf

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Of course there's time.

The rule is that: everyone who is alive must kill only one other person or die in the process without killing anyone.

In your 10 people problem time solves the conundrum, by use of the term "then". ie 9 kills 10 then 8 kills 9. But there's as many permutations, ie 10 kills 1 then 5 kills 10, or 2 and 3 kill 4 and 5 and 4 and 5 kill no one etc (or is there? are the permutations restricted?).

In a n=10 population what is the most frequent number of survivors?

Edited by Sorcerer

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I have another idle thought.

Bringing in probability, say that each person has one "shot" that will kill someone with probability 1/2. I would not be surprised if every variation of rules reduces to the scenario where each person shoots themselves. Half will live, half will die. I don't see why this would be any different than any other scenario, under the assumption that if you are killed you can't kill someone (after you get killed).

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As was shown in your sequential killing chain all that is needed is for someone to kill another who has already killed for the end population to be less than 50%.

You can either kill a killer or a non-killer. The two extreme possibilities as I said in the OP are:

1) Half the population kill one other person who has not yet killed another. Result = 50% (minus 1 for odd numbered population)

2) The entire population kills another simultaneously. Be it lining up in pairs or in a circle, or any other crazy pattern.

So you could imagine with n=10, all could from a line single file, 1*10, here 1 being the number of people in the row and 10 being the number of rows and the killing done in order from first to last, with r(resulting number of survivors)=1. Or go in pairs, 2*5, r=2. And then all other positive interfere equations which equal 10, 3*3+1 r=2 (3 rows of 3 with 1 remaining), 4*2+2 r=2 etc.

Edited by Sorcerer

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It's a matter of framing the problem. Ever every person kills one person, everyone dies. If you kill at most one person, then the answer will be different. Without more details, you can't really come up with an answer.

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Doesn't it boil down to the following?

If everybody kills one person simultaneously (as stated above with everybody in the world in a circle) then everybody dies. 100%

If everybody kills one person sequentially, then 50% die.

These are the limits, and any other arrangement will lie between them.

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If everybody kills one person sequentially, then 50% die.

I think all but one could die in this scenario, depending on whether that person you kill has already killed someone.

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I think all but one could die in this scenario, depending on whether that person you kill has already killed someone.

Ah, I see what you mean. What I meant with sequentially was if everybody were allocated a number, then the highest number kills someone, then the next highest surviving number kills someone, and so on. That would continue until everybody alive had killed somebody. That must result in 50% (I think)

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I think the only way you could come up with solutions rather than artificial boundaries (everyone in a circle = all dead -vs- everyone pairs off and the one on the right kills the one on the left = half dead) would be by simulations

- At beginning you have x people on the surface of a sphere.

- Each takes a step in a random direction or no step at all

- When two persons are in reach of each other they

i if both have killed - they carry on walking

ii if one has killed and the other hasn't - the already-killer is killed and the other now walks away

iii if neither has already-killed a coin is tossed to work out which dies / which kill - and one walks away

- If more than two people meet then pairwise checks as above are selected randomly until one or more walks away

- everyone takes another step

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Yes and there are vastly more scenarios. Could anyone do an example of all the possible outcomes for a small set of people, as in 10 which I started above.

There's some mistakes in my examples above. The 6 outcomes are.

1. Everyone kills a partner or forms a circle and kills simultaneously = 0 alive.

2. Everyone lines up single file 10*1 and shoots the person in front = 1 alive

3. They pair up and as above 5*2 = 2 alive

4. There's 3 rows of 3 and 1 single 3*3+1 = 3 alive

5. Two rows of 4 and one of 2, 2*4+2 = 4 alive

6. Two rows of 5, 2*5 = 5 alive.

It seems to me the answer lies in finding all the possible interger equations for the population and then working out a way to quickly calculate the resulting number of survivors.

That'd be one way anyway.

It's a matter of framing the problem. Ever every person kills one person, everyone dies. If you kill at most one person, then the answer will be different. Without more details, you can't really come up with an answer.

That is a better way of stating it.

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Imagine you look back on this from "some time later".

The sensible assessment of the "success rate" in killing is

Number of dead / number of killers,

What's the probability distribution (for a given hit rate for killers).

How can it differ from the hit rate?

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