Pile Of Balls A pile of colourful balls are arranged to make a square based pyramid by having one ball on the top layer four balls on the second layer nine balls on the third layer and so on.Such that consecutive layers will have a number of balls equal to consecutive square numbers. This is the pattern observed for number of balls in consecutive layers 1 4 9 16 25 ... .If there were one-thousand balls in the pile used to make the pyramid not all of them would be needed.How many balls would be left over?
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A layer of equal spheres is in the form of a square. The spheres are arranged so that each sphere is tangent to every one adjacent to it. In the spaces between sets of 4 adjacent spheres, other spheres rest, equal in size to the original. These spheres form in turn a second layer on top of the first. Successive layers of this sort form a pyramidal pile with a single sphere resting on top. If the bottom layer contains 16 spheres, what is the height of the pile in terms of the common radius r of the spheres?
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