A geometry problem by Adarsh Adi
Geometry
Level
4
How many spheres of equal radii are needed to shield a point source of light completely?
Note that the source of light is not inside any sphere.
The answer is 6.
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1 solution
Suppose the source of light is in O . We construct a regular tetrahedron ABCD . with centre O . Consider the 4 infinite circular cones each containing strictly four pyramids OBCD , OACD , OABC , OABD , and common vertex O !!!! The cones partly intercept so that every light ray from O lies inside some cone . Let us inscribe four spheres into the cone so that they do not intersect. This is easy to achieve if the radii of spheres differently greatly from each other. This cannot be achieved with four spheres of equal radius. It can be proved that 6 spheres of equal radius are needed to Shield the light completely. Try to find such a distribution of equal spheres.