Imagine eight spheres of radius 1 that are at $(\pm1,\pm1,\pm1)$. Place sphere A with its center at the origin externally tangent to all of the other spheres. Then place sphere B externally tangent to sphere A and four of the original eight spheres. Find the radius of sphere B.
I can barely visualize this problem, and I have no idea how to solve it. Any suggestions?
Start by visualizing the first sphere. It's centred at the origin. Let its radius be $r_1$. Now since this sphere and the original $8$ spheres are tangent, the distance between their centres must be $$1+r_1$$ But the distance between their centres is also $$\sqrt{(1-0)^2+(1-0)^2+(1-0)^2}=\sqrt{3}$$ Therefore $$1+r_1=\sqrt{3}$$ or $$r_1=\sqrt{3}-1$$ Now you need to determine the second sphere. By symmetry we can say its centre will be on a coordinate axis. Let's the assume the coordinate of the centre to be then $(d,0,0)$. Clearly this point's distance to the origin is $$r+(\sqrt{3}-1)$$ where r is the radius we want. Also its distance to $(1,\pm1,\pm1)$ will be $$r+1$$ Solve for $r$ and you've got your radius.