stingc

This is a really neat problem. The thing that everyone has missed is that it says the *maximum* height is 1+sqrt(n). This is a maximization problem.

I'll get you started on the n=2 case.

Start at the centerline of the big bubble. Then define an angle a up to the bottom of the second bubble. Cos a=R2. the radius of the second bubble, and Sin a=h, the vertical distance from the center of bubble 1 to the bottom of bubble 2.

h=Sin a=sqrt(1-(Cos a)^2)=sqrt(1-R2^2)

The total height of the tower is

H=1+h+R2=1+R2+sqrt(1-R2^2)
(the 1 is from the bottom half of the big bubble)

Maximizing H gives H=1+sqrt(2). Now repeat for arbitrary n.

Sorry that probably sounds like gibberish without a picture to go along.