i post 4 month ago but no one gave a soln

Maybe it's time to post a solution? In plane, I think it's not true, but it space it seems true. What about other dimensions? I mean , I know a problem like this: Given a system of a) Circles b) Spheres is it always possible to move far away from the system one of the pieces ?

Well, it's clear that we can block a lamp in the plane with circles, but the spacial analog seems really tricky.

Well consider all lattice points in $2n*2n*2n$ cube with center the lamp. Now consider a sphere of radius $\frac13$ of center of each lattice point. Now use Minkowski theorem for big $n$

what's Minovski theorem for big n?

If $A$ is a convex set of points in space with volume more than 8 and if $(x,y,z) \in A$ then $(-x,-y,-z) \in A$ then $A$ has a lattice point except $(0,0,0)$

Now that you said, I remember a solution.... The great idea is to take all the lattice points, then build circles of equal radiuses $R$ in those latice points and with Minkowski you show that for a quite big radius , at the distance $\frac{1}{R}$ there will be no light. Am I right? Or maybe left? Just kidding ....

Sorry to bump such an old problem but I have a comment here. I encountered this problem(labeled as Iran 2003) in the form: Prove that there does not exist a finite set of spheres of equal radius such that no two intersect and that the ray never goes to infinity. I have been trying to solve this problem for a while with no success. Does anyone have a solution?

Without using Minkowski, it's enough to use compactness. The claim can be also generalized to $\mathbb R^d$, for any finite $d$. Place the light in the origin, and place equal open $d$-discs of any radius $r< 1/2$ centered at every lattice point except the origin. Then put a $(d-1)$-sphere $S$ of the same radius $r$ centered at the origin, and consider the projections of all discs on $S$. Every disc projects to an open subset of $S$, while the projections of all lattice points on $S$ form a dense subset of $S$. Therefore the projections form a cover of $S$ and, by compactness of $S$, there's a finite subset of projections which still covers it. Keep the corresponding discs, and remove all the others. Now you're blocking every ray of light with a finite number of arbitrarily small equal discs in $\mathbb R^d$. @RaduSorici: either the above proof is flawed, or you misread the problem. Maybe there were more constraints on the placement of your spheres?

MindFlyer wrote: ... the projections of all lattice points on $S$ form a dense subset of $S$. Therefore the projections form a cover of $S$ and ... Take the interval $[0,1]$; enumerate the rationals in it by $(r_n)_{n\geq 1}$. Choose $\varepsilon > 0$ and open intervals $I_n = (r_n - \varepsilon/2^{n+1}, r_n + \varepsilon/2^{n+1})$. Then the union of the $I_n$ has measure at most $\varepsilon$, so it does not cover $[0,1]$. Thus just qualitative density and compactness arguments are not enough.

Oh right, I don't know why I said that. However the proof can be fixed quite straightforwardly. Any straight line through the origin passes arbitrarly close to some other lattice point, so the projections of the disks indeed cover the sphere.

A similar problem appears in Engel and he claims six sphere is sufficient. Consider a rectangular prism with three different sized faces and the lamp is the center point of the prism. Then consider 6 infinite cones that orgininate at the center of the prism and each contain one face of the prism. Then these cones intersect and cover the light source. We can inscribe spheres into these cones. If the spheres intersect then we can vary the dimensions of the prism so that they no longer intersect.

Any complete solution to this problem? I am unable to understand the application of Minkowski's theorem mentioned in one of the posts above.Also, the initial constraint includes the fact that no two spheres touch each other. I am pretty sure that the problem can be translated in terms of circles instead of spheres and the plane $\mathbb{R}^2$. Any complete solutions using Minkowski's theorem??