Bubble Tower

Lets denote radius of hemispherical bubbles as $r_1$ , $r_2$ , ... $r_{80}$ (from largest to smallest)

distance between centers of two consecutive bubbles $i$ and $i + 1$ will be $h_{i,i+1} = \sqrt{r_i^2-{r_{i+1}}^2}$ ( by Pythagorean theorem, ${r_{i}}^2 = h_{i,i+1}^2 + r_{i+1}^2$ )

so total height of a bubble tower will be $h=r_{80}+\sqrt{r_{79}^2-r_{80}^2}+\sqrt{r_{78}^2-r_{79}^2}+...\ +\sqrt{r_1^2-r_2^2}+\sqrt{1-r_1^2}+1$

then take partial derivatives and compare them to zero (necessary conditions for optima)

$\frac{d h}{d r_1} = \frac{r_1}{\sqrt{r_1^2-r_{2}^2}} - \frac{r_1}{\sqrt{1 - r_1^2}}$

...

$\frac{d h}{d r_i} = \frac{r_i}{\sqrt{r_i^2-r_{i+1}^2}} - \frac{r_i}{\sqrt{r_{i-1}^2 - r_i^2}}$

...

$\frac{d h}{d r_{80}} = 1 - \frac{r_{80}}{\sqrt{r_{79}^2 - r_{80}^2}}$

they will be equal to zero, when

$1 - r_{1}^2 = r_{1}^2 - r_{2}^2$

...

$r_i^2-r_{i+1}^2$ = $r_{i-1}^2-r_{i}^2$

...

$r_{80}^2 = r_{79}^2 - r_{80}^2$

so, numbers $0, r_{80}^2, r_{79}^2, ... r_{2}^2, r_{1}^2, 1$ forms arithmetic progression with difference $d = \frac{1}{81}$

and total height of a bubble tower can be written as

$h=\sqrt{r_{80}^2 - 0}+\sqrt{r_{79}^2-r_{80}^2} +...\ +\sqrt{r_1^2-r_2^2}+\sqrt{1-r_1^2}+1=81\sqrt{d}+1=\frac{81}{9}+1=\boxed{10}$

The height of bubbles is given by for the nth bubble $H_n = 1 + \sqrt{1-r_1^2} + \sqrt{r_1^2 - r_2^2} +\ldots+ \sqrt{r_{n- 1}^2 - r_{n}^2} + r_{n}$ . Using QM-AM maximum height is: $\displaystyle{\sqrt{\frac{1 - r_1^2 + r_1^2-r_2^2 + \ldots + r_{79}^2 - r_{80}^2 + r_{80}^2}{81}}\ \geq \frac{\sqrt{1-r_1^2} + \sqrt{r_1^2 - r_2^2} + \ldots + \sqrt{r_{79}^2 - r_{80}^2} + r_{80}}{81}}$

$81\times \frac{1}{9}\ \geq H_{80} - 1$

$10 \ \geq H_{80}$

Thus maximum height possible is 10. equality occurs when $\sqrt{1-r_1^2} = \sqrt{r_1^2 - r_2^2} =\ldots= \sqrt{r_{79}^2 - r_{80}^2} = r_{80}$

Suppose that the maximum height of a bubble tower of $n$ bubbles, where the largest bubble has radius $1$ is $1 + \sqrt{n}$ . This is certainly true if $n=1$ . If we now consider a tower of $n+1$ bubbles, then then second to $(n+1)$ st bubbles form a tower of $n$ bubbles (apart from the first hemisphere), where the bottom bubble (of this subtower) has radius $0 < x < 1$ (and the height of this subtower will be as great as possible). Thus the height of the part of the tower from the centre of the bottom bubble (of the subtower) to the top will be $x\sqrt{n}$ , and so the height of the whole tower will be $H(x) \; = \; x\sqrt{n} + 1 + \sqrt{1-x^2}$ and so we need to choose $x$ to maximize $H(x)$ . Since $H'(x) = \sqrt{n} - \frac{x}{\sqrt{1-x^2}}$ , we see that $H(x)$ is maximized when $x^2 = \tfrac{n}{n+1}$ . This makes the maximum height of the whole tower $1 + \sqrt{n+1}$ .

Thus we deduce, by induction, that the maximum height of a tower of $n$ bubbles is $1 + \sqrt{n}$ . This makes the answer $1 + \sqrt{81} = \boxed{10}$ .

I though that if is 3 dimension an 80 bubbles, (80)/(2^3)=10

While I never was able to prove it using induction, the solution to the problem can actually be derived using Lagrange’s multiplier method. In particular, let’s say that the entire height is composed of multiple segments h[1], h[2], h[3] … h[n] with associated radii r[1], r[2], r[3] … r[n]. Then clearly h[n] = sqrt(r[n]^2 – r[n+1]^2).

Now let’s square this to get h[n]^2 = r[n]^2 – r[n+1]^2 (this is an expression [a])

The radius of last hemisphere is h[n] = r[n].

From [a], it follows that r[n]^2 = h[n]^2 + r[n+1]^2 = h[n]^2 + h[n+1]^2 + … h[n]^2.

So when r[1] = 1 then

h[1]^2 + h[2]^2 + … h[n]^2 = 1.0

So we’ve established a constraint on the sum of h[n]. The tower height H = F(h[i]) = 1.0 + sum (h[i], 1 <= i <= n)

Lagrange’s multiplier method says that

gF = lamdba x gG (where g is nabla and lambda is a scalar.) So

gF = [1 1 1 … 1]

and gG = [1/2 lamdba h[1], 1/2 lamdba h[2] … 1/2 lambda h[n])

So h[i] = n / (4 lambda^2), so lambda = sqrt(n) / 2. Subsituting lambda in individual h[i] expression we get:

h[i] = 1/2 * 2/sqrt(n) = 1/sqrt(n).

Finally substituting into original equation we get the sought-after result:

H = 1 + sum(h[i], 1 <= i <= n) = 1 + n/sqrt(n) = 1 + sqrt(n)

H=1+81/9 H=1+9 H= *10 *