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riddles >> putnam exam (pure math) >> Bright & Dark Stars
(Message started by: James Fingas on Dec 3rd, 2002, 12:38pm)

Title: Bright & Dark Stars
Post by James Fingas on Dec 3rd, 2002, 12:38pm
This puzzle has been stagnant for long enough, so I propose the following answer: 4

How do I get this answer? I reason to myself--what is the maximum angle of view that a dark star can occlude, seen from the surface of the light star? Obviously, a dark star can occlude nearly 1/2 of our 3D angle of view. How big a dark star do we need to do this? Simple: the size depends on how far away from the light star we want to put it. But for any given distance from the light star, there is a dark star of a large enough size to occlude (to within epsilon) half the angle of view of the light star.

Knowing that we occlude exactly half the angle of view of the light star with each dark star (we cannot do better than this, and doing worse than this will not help us), now we just have to figure out where to place the dark stars (ie. in which directions) to accomplish the darkening in the smallest number of stars. With one dark star, obviously we cannot occlude the light star. With two dark stars, we would put them on opposite sides of the light star, but there would still be a gap between the dark stars where light could get out. With three dark stars, we will arrange them in a triangle around the dark star, leaving two rays of light emanating from the arrangement. With four dark stars, we can arrange them tetrahedrally, therefore blocking out the light completely. This uses the fewest dark stars for the job.

You may be wondering: "James, how are you going to do this? I believe that if you make the dark stars big enough, they are like planes, and then you can assemble the planes to form a tetrahedron and block out the light, but we aren't allowed to have the dark stars intersect. How do you keep them from intersecting?" I have already thought of this, and that is where my angle of occlusion principle comes into play.

Start with four dark stars that intersect, are all the same size, located tetrahedrally around the bright star, and just barely block all the bright star's light. Name these stars Dark A, Dark B, Dark C, and Dark D (the Dark Crew). Leave Dark A as it is, and then proceed to grow Dark B, increasing both its size and its distance from the bright star, but keeping the angle of occlusion the same, as measured from the point on the bright star farthest from Dark B (we could do better than this, but why bother?). Stop growing Dark B as soon as it no longer intersects Dark A. Now start growing Dark C in the same way, until it no longer intersects Dark A or Dark B. Now grow the dastardly Dark D until it no longer intersects Dark A, Dark B, or Dark C. Of course the Dark Crew will consist of some very large dark stars, but this is neither here nor there, since the sizes are finite. These four dark stars now completely absorb all the light from the bright star.  8)

Title: Re: Bright & Dark Stars
Post by towr on Dec 4th, 2002, 1:04am
call me daft, but I don't see how that can work..
With just 4 non-intersecting dark stars there has to always be an opening the bright star can shine through..
I'd say you'd need at least 8.. 4 more to occlude the holes is the tetrahedron..

Title: Re: Bright & Dark Stars
Post by James Fingas on Dec 4th, 2002, 10:21am
But that is the cleverness of the solution. You don't put them in a tetrahedron, because then to cover over the holes they would have to overlap.

Think of looking up towards the sky at a large spherical object. If the object is far enough away, the human eye can't percieve its exact distance. It could be 1 km away and 100m across, or 2 km away and 200m across, or 1 light year away and 0.2 light years across.

My method is to start with a tetrahedron of overlapping dark stars, just barely big enough to completely cover the bright star. For instance, if the distance between the centers of the dark stars is initially 1 km, then we make the dark stars 2/sqrt(3) km in diameter. This tetrahedron isn't the final solution--it's invalid because the dark stars overlap.

But considering how a larger dark star that's farther away will cover exactly the same area of sky, we can move each of those stars farther away (while simultaneously making it larger), maintaining the integrity of the original overlapping tetrahedral configuration. If we move each one to a different distance from the bright star, they will no longer overlap each other.

When we're done, we'll find that, for every point on the surface of the bright star, the dark stars cover the same angles of view that the original overlapping tetrahedral configuration covered. That is to say, they cover the whole sky, completely blocking out the light of the bright star.

Title: Re: Bright & Dark Stars
Post by towr on Dec 4th, 2002, 1:38pm
I just don't see how that can be done..
Can't you make a 3D model to show the solution  ;D

hmm.. I'll think about it more tomorrow.. (I'll get me some numbers on them distances and sizes..)

Title: Re: Bright & Dark Stars
Post by James Fingas on Dec 5th, 2002, 9:16am
If you want to come up with the exact numbers, here's the (correct, finally) method for growing a dark star while maintaining (or enlarging) its appearance from the surface of the bright star.

Construct a line going through the centers of the bright star and the dark star you want to grow. The center of the dark star will stay on this line for the entire operation.

Construct a cone, with the line as its axis. The cone must graze the surfaces of both the bright star and the dark star (so that both stars are inside the cone).

Any dark star that is centered on the line and grazingly touches the cone will occlude at least the same area as the original dark star, viewed from any point within the cone. Seeing how the entire bright star is within the cone, this tells us how to grow the dark star. Just move it away from the bright star along the line, while keeping it touching the cone.

Title: Re: Bright & Dark Stars
Post by James Fingas on Dec 9th, 2002, 10:07am
It's kind of funny in a perverse way that when I say that my method for growing the dark stars is finally right, that it's still wrong.

When you pick the cone that grazingly touches both the bright and dark stars, you must pick the cone that has its vertex between the bright and dark stars. That is to say, the bright star is in one arm of the cone, and the dark star is in the other arm. Of course you can't do this when the bright and dark stars are touching each other, so you just make the initial overlapping tetrahedron so that the dark stars don't touch the bright star. You can easily do this by selecting a tetrahedron that would be touching a bright star of twice the diameter. When you put this around the bright star, it won't be touching any of the dark stars, and therefore you can make the cone construction, and grow the dark stars, etc.

I've also come up with some numbers for you. Take a bright star with radius 1, and then put around it four dark stars with radii 100, each one located 150/sqrt(2) away from the bright star (measured center-to-center). These will overlap, but completely cover the bright star. Leave Dark A where it is, but move Dark B to a distance of 1500/sqrt(2) away from the bright star, enlarging it to a radius of (10*101 - 1). Move Dark C to a distance of 15000/sqrt(2) away from the bright star, enlarging it to a radius of (100*101 - 1). Move Dark D to a distance of 150000/sqrt(2) away from the bright star, enlarging it to a radius of (1000*101 - 1). I believe that will do the trick.

Hmm ... maybe I should think about ways to optimize the size of the dark stars ;)

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