Assume that the galaxy is a large cube made up of 200 billio

Assume that the galaxy is a large cube, made up of 200 billion stars, each star positioned on the corner of a small cube arranged together in a lattice. The entire lattice of small cubes makes up the larger cube representing the entire galaxy. Let the distance between each neighboring star be D = 5 LY, where D is the length of the side of one of the small cubes. If 100 currently signaling civilizations are randomly distributed somewhere in this ‘cubic’ galaxy, how far away in LY is the nearest civilization likely to be?

Solution

let N be the number of stars in cubic galaxy; [N = 200x10^9] . If each star is separated by a distance D = 5LY, then the total volume of the cubic galaxy is [Vgal= ND^3] . Now the number of civilizations in the galaxy is [n_{civ}] = 100. Let each one of them be surrounded by an \"empty\" cube of volume [V_{civ}] = [x^3] in which there are no other civilizations--on the avg. So x is the avg distance between these civilization.

Vgal= [(2*10^9)(125)= 500*10^9= 5*10^11] which is NOT [2^{13}] so even that is wrong.

Divide that by 100 to get [5*10^9] as the average volume of a \"civilization\". x is the cube root of that number.