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Shortened version of the question: How many zombies would it take to climb a 1 kilometer pole?

Extra parameters:

  1. I am the last survivor of the human race. I don't need to eat, sleep or drink. I am nearly immortal. There is only one thing that can kill me: zombies.

  2. I don't live on Earth; I live on an infinite flat world. I live on the top of a steel cylinder exactly one kilometer high and one meter in diameter, and the pole cannot be moved. I am stuck to the top of the pole, and cannot leave (unless pushed by a zombie).

  3. There is an army of zombies trying to kill me. The army is infinite in size.

  4. The zombies can't technically climb the pole, since it's far too smooth and slippery. They can only get up by piling on top of each other.

So, taking these things into account, what is the minimum amount of zombies that the army would need to use to kill me?

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closed as off-topic by sphennings, Amadeus, Hohmannfan, dot_Sp0T, Vincent Jun 17 '17 at 0:20

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  • $\begingroup$ I can expand on my answer if you tell us something about the zombies... $\endgroup$ – Pak Jun 16 '17 at 20:36
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    $\begingroup$ Can the zombies climb each other if they're stacked vertically? Can the zombie at the bottom support the weight of all of the zombies on top of him/her if they stand on each others' shoulders? How well do the zombies balance on each other? Can they stand as still and as solid as a rock? If they get up there, do they automatically kill you? $\endgroup$ – Pak Jun 16 '17 at 20:55
  • $\begingroup$ @Pak Those are all things you need to consider, but I don't see how I could add these as parameters. $\endgroup$ – Snowshard Jun 16 '17 at 21:40
  • $\begingroup$ That's more of a zombie math question that worldbuilding. It's like asking the volume of a huge cone made of corpses. $\endgroup$ – Vincent Jun 17 '17 at 0:20
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    $\begingroup$ This is more a physics problem than a math problem. Many answers have approached it as math but the real issue is the strength of materials. Flesh can't take the forces involved, the zombies at the bottom go squish. Since zombie squish has no cohesive force (individual zombies might but they aren't connected to the next zombie) the rate it flows away is a function of viscosity. To pile higher you need to add more zombies than are flowing away underneath. I strongly suspect the required incoming speed >>> zombie movement rates, thus you are safe. $\endgroup$ – Loren Pechtel Jun 17 '17 at 22:30
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Zombie would have to form a zombie pyramid. I assume that zombies do not have superhuman strength or toughness, so most of the pyramid would be just compacted bodies.

Next (a very important assumption) is the question of how steep this pyramid can get. Since the material is soft, I assume not very steep, let's say its base should be equal to its height. So, we have a pyramid 1 km high with a base 1 km wide. What would be its volume? Strictly speaking, this shape is a cone, not pyramid. Given the parameters above, volume comes up at 261,800,000 cubic meters.

Now, how big is the average zombie body? I assume it's 70 liters, excluding lung volume. So, if bodies are unsqueezable, you will need 3,740,000,000 zombies to get to you.

However, real bodies would be squeezed beyond my assumption above. Blood and other bodily fluids would ooze out, so the pyramid would be shrinking over time. That means that 3,74 billion zombies can do it only if they are really quick. Any passing hour would make the required number higher.

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It's a good thing you have infinite zombies, because if each can support the weight of one additional, standing on his not-too-slumped shoulders, then you'll need one grabbing your ankles, standing on another's shoulders, who is standing on two, who are on four, etc. If they are tall, 2m at the shoulders, then the 498 layers below the first two will take 1.6367x10^150 zombies.

Yertle had this problem, too.

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  • $\begingroup$ I don't think a triangle would distribute the weight evenly among all the zombies, would it? $\endgroup$ – PyRulez Jun 16 '17 at 22:06
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Tower method(1) - Each zombie stands on the zombie below its shoulders or head. Using heads this takes 1000/1.74 zombies which is roughly 575 zombies. Standing on shoulders brings it up to 700 zombies.

Tower method(2) - They all lie down on top of each other until they reach the correct height. Working off a width of 20cm (8") this would take 1000/0.2 zombies or 5000 zombies.

Ramp method - The zombies form a huge ramp leading up to the tower. The ramp would be increasing by one zombie each step and would be made from a series of the towers mentioned above. The second version would probably be used as the first would create a large gap at each step. This method uses loads of zombies. The final step is 5000 zombies with each step taking one zombie off. In fact, no we'll take two off each step to give roughly the gradient of stairs. This means the second step is 4998 the third 4996 and so on. This is equal to 6252500 zombies whit 1 extra zombie to walk up the ramp and kill you.

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1 zombie who can climb the pole and kill you before you kill them.

Given that we're free to come up with parameters, based on zombies not being able to climb poles, we need to come up with some basic assumptions.

Let's assume zombies are 2m tall at the shoulder. That means that the zombie tower is 500 zombies tall.

Infinitely strong zombies with adequate balance and steadiness

If the zombies are infinitely strong, have adequate balance, and can stand still long enough, they can just form a zombie tower of 1 zombie wide, resulting in 500 zombies to reach the top of the tower.

Finitely strong zombies with adequate balance and steadiness

If the zombies are not infinitely strong, but still have adequate balance and steadiness, then we can assume that they carry the weight of n zombies on their shoulders.

From the top, we can have a stack of n+1 zombies, 1 zombie wide. The total weight of this stack is n+1 zombies, with the bottom zombie holding up the n zombies above it.

Beneath this layer, we have a stack of zombies, 2 wide. This stack can hold, at most, 2n zombies above it, so its height is $\lfloor n-\frac{n+1}{2}\rfloor$ with 1 extra layer propping them up. The total weight of the top 2 stacks is 2n+2 zombies.

Beneath this, we have a stack of zombies, 3 wide. This stack can hold, at most, 3n zombies above it, so its height is $\lfloor 3n-\frac{2n+2}{3}\rfloor = \lfloor\frac{n-2}{3}\rfloor$. If n is less than 5, this layer won't exist. The total weight of the top 3 stacks is $2n+2+3*\lfloor\frac{n-2}{3}\rfloor$.

This continues down, with stacks getting wider and wider, with some stacks not existing. (I'm still working out the general form of the equation, but the outline of it is here.)

Zombies with imperfect balance or steadiness

If the zombies can't carry a single zombie on their shoulders, the equations should still work; it's just that every layer will be 1 high at most, and wider than it is tall.

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  • $\begingroup$ This answer is currently in the low-quality review queue. Probably because of its length. This looks more like a witty comment than an answer. You might want to expand your idea and edit your post to explain why this is the correct answer in the scenario the OP proposed. Otherwise this answer could be deleted. $\endgroup$ – Secespitus Jun 16 '17 at 21:24
  • $\begingroup$ I'm waiting for clarifications on the question...see the comments above. $\endgroup$ – Pak Jun 16 '17 at 21:26
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    $\begingroup$ It's good that you asked the OP, but normally you should wait for the feedback before answering. Nobody can rate your answer based on what you might write in the future, so people look at your answer in its current state. And someone flagged it as "low quality". $\endgroup$ – Secespitus Jun 16 '17 at 21:29

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