is there a name for this?

>>8345571
Yes, it's the principle of "it's impossible lad", at least not the problem you gave us.
As soon as you have two wolves there is a safe place at the equidistant point between them.

>>8345574
That only prevents those two wolves from eating that sheep, it does not prevent that sheep from getting eaten by another wolf which has a unique distance to it.

File: gay.png (10KB, 822x323px) Image search: [Google]

10KB, 822x323px

>>8345574
so the only way for that scenario in the op to work is to have a shape with one point equidistant to all others and fill that with a wolf?

>>8345571
A trivial solution would be to pack the space with wolves. If there is no space for sheep then there cannot be a safe space for sheep.

You need to clarify several things. What space is being filled with wolves and sheep? A line? A plane? How much do space do wolves and sheep the up? Are they points? Do they fill a grid?

Also, if there is two wolves are the closest but equidistant from the sheep, can that sheep be eaten by a wolf which is farther away?

>>8345586
Oh I see, "they" was ambiguous.
lemme think about it

>>8345603
i believe a 3d plane
you don't need to think of them as physical animals with mass, just as individual points (i think)
and yeah, filling a circle/everything was one solution that came to mind

>Also, if there is two wolves are the closest but equidistant from the sheep, can that sheep be eaten by a wolf which is farther away?
no, it will only be eaten by the closest single wolf

If there were infinitely many wolves there would be no space for the sheep to exist in m8

>>8345611
>just as individual points (i think)

Even then, you have the case of an infinite amount of points existing 0.000...1m away.

>>8345571
You should really add that wolves should be countable, if you don't want this thread to be smartasses repeating "but durr just fill the plan" over and over

>>8345600
that will only work as long as no wolf on the shape is also positioned equidistant from the sheep as the central wolf

>>8345633
WAIT NO I'M AN IDIOT. And you don't actually even need the central wolf.

say we have a circle with center at (0,0) and radius 1 and a sheep positioned at point (0,2), and a wolf at (0,0) and wolves positioned around the shape.

clearly the distance from W(0,0) to S(0,2) is longer than the distance from W(0,1) to S(0,2), and the distance from W(0,0)-S(0,2) is shorter than the distance from W(1,0)-S(0,2). so, there must be a point somewhere on the arc from W(0,1) to (1,0) where W(x,y)-S(0,2)==W(0,0)-S(0,2).

so the wolf at W(0,0) clearly isn't helping us at all.

However, there's only one wolf that can go at exactly W(0,1) and W(0,-1), which are the minimum and maximum distances from the sheep at S(0,2). You can put wolves arbitrarily close to those points, but you can't put them exactly there.

My gut says you could extend this to conclude that any closed shape with exactly one minimum or maximum distance from any given S(x,y) would provide a field where no sheep can be safely placed.

>>8345649
oh wait i'm an idiot, you need the central wolf to deal with internal sheep

>>8345611
Another solution would be a straight line of wolves

>>8345571
Just have an infinite line of wolves. If the sheep is on that line, there is a wolf ontop of it that will eat it(unless they can't exist on the same point which case this is still a solution.) If there is a sheep anywhere off of the line, draw an imaginary line perpendicular from the sheep to the line of wolves there will be only one wolf that that line will intersect with. And that wolf wil eat the sheep, while all others have another wolf equidistant from the sheep.
2EZ

>>8345767
>>8345732
darnnit, you posted while I was typing

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68KB, 630x467px

n wolves can kill at most n sheep. Obviously, if you place one wolf and there is one sheep anywhere in the plane, then it will be the unique wolf closest to the sheep, so it will eat it.

However, if you want the wolf arrangement to succeed for any placement of n sheep where 1 < n < infinity then you need infinitely many wolves to succeed at eating all the sheep.

Proof: if there are w < infinity wolves then take the first wolf W0. Now take any wolf W1 that is at least as close to W0 than any other. Then, you can place all of the sheep on the segment W0 W1, and only 2 of them will be eaten. And if n(sheep) < n(wolves)/2, then you can place the nth sheep halfway between the nth wolf and whatever wolf is closest to it, so that none of them will be eaten.

If you have at least three (but still a finite number of) wolves then you can always place them in such a way that you end up killing at least 1 sheep no matter where the sheep are placed (assuming the sheep have distinct positions), and probably 2 if you have enough (4?) so this is about as good as the sheep can do.

Also, clearly if you can have uncountably many wolves then there is no safe place for sheep as >>8345627 said. (This also excludes filling a straight line which is just as silly.)

The only remaining open case is whether we can have a countably infinite placement of wolves that kills any countable configuration of sheep.

>>8347116
(continued)

Ironically the wolves cannot be dense in the plane, because then you just place a sheep at a limit point of the set of wolves where there is no wolf, and then NO wolf is closest to it.

Therefore there must be some nonempty open set U with no wolves in it. If we take a configuration of sheep like this:

. .

.

. .

inside of U, it should always be possible to protect the inner sheep by placing enough sheep on a sphere around it and making the sphere sufficiently small. Therefore it's impossible to always win even with countably many wolves, if your opponent is allowed to place arbitrarily many sheep.

I conjecture that if you fix a coordinate system in which the inner sheep is at 0 and is > 2 unit away from any wolf, and place a sheep at each basis vector and negation of a basis vector, then the inner sheep cannot be eaten. It should be a simple inequality but I'm having trouble proving this. If this is true then you can never eat more than 2N sheep in [math]\mathbb{R}^N[/math], so you can never kill all the sheep.