Building Blocks (Level 2)

Now you have only $m$ cubes, of edge length $N_1,N_1,N_3,\cdots,N_m$. For simplicity we’ll suppose they go in increasing order, so $N_1$ is the smallest and $N_m$ is the biggest. There may be ties (or they may even be all the same size). You use them all to build a single tower.
7. What is the height of this tower?
8. What is the total volume of this tower?
9. What is the maximum possible total surface area for this tower?
10. What is the minimum possible total surface area?
Source: Julia Robinson Mathematics Festival

SOLUTION
7. What is the height of this tower?
$N_1+N_2+N_3+\cdots+N_m$

8. What is the total volume of this tower?
$N_1^3+N_2^3+\cdots+N_m^3$

9. What is the maximum possible total surface area for this tower?
STEP 1. Calculate the total surface area of all the cubes
$T=6\left (N_1^2+N_2^2+N_3^2+\cdots+N_m^2\right )$
STEP 2. Move $N_1$ between $N_{m-1}$ and $N_m$ obtaining
$N_2,N_3,N_4,\cdots,N_{m-2},N_{m-1},N_1,N_m$
Move $N_2$ between $N_{m-2}$ and $N_{m-1}$ obtaining
$N_3,N_4,N_5,\cdots,N_{m-3},N_{m-2},N_2,N_{m-1},N_1,N_m$
Repeat STEP 2 until the cubes are stacked in the pattern of alternating smaller ones and bigger ones.
STEP 3. Calculate $H$ the hidden surface area of the bottom and the hidden surface areas in between the cubes where one is resting on top of another.
STEP 4. Calculate the maximum possible total surface area for this tower
$T-H$

EXAMPLE: Build a tower of height $15$ using cubes $1,2,4,8$
STEP 1. Calculate the total surface area of all the cubes
$T=6\left (1^2+2^2+4^2+8^2\right )$
$=6\left (1+4+16+64\right )$
$=6\left (85\right )$
$=510$
STEP 2. Move $N_1$ between $N_{m-1}$ and $N_m$ obtaining
$1,2,4,8$
$2,4,1,8$
STEP 3. Calculate $H$ the hidden surface areas
$H=2^2+2\left (2^2\right )+2\left (1^2\right )+2\left (1^2\right )$
$=4+8+2+2$
$=16$
STEP 4. Calculate the maximum possible total surface area for this tower
$T-H=510-16$
$=494$

10. What is the minimum possible total surface area?
STEP 1. Stack the cubes in decreasing order of size $N_m,N_{m-1},N_{m-2},\cdots,N_3,N_2,N_1$
STEP 2. Calculate the hidden surface area of the bottom and the hidden surface areas between the cubes
$H=N_m^2+2\left (N_{m-1}^2+N_{m-2}^2+\cdots+N_2^2+N_1^2\right )$
STEP 3. Calculate the minimum possible total surface area
$T-H=6\left (N_1^2+N_2^2+N_3^2+\cdots+N_m^2\right )-\left [N_m^2+2\left (N_{m-1}^2+N_{m-2}^2+\cdots+N_2^2+N_1^2\right )\right ]$
$=4\left (N_1^2+N_2^2+N_3^2+\cdots+N_{m-1}^2\right )+5N_m^2$

EXAMPLE: Build a tower of height $15$ using cubes $1,2,4,8$
STEP 1. Stack the cubes in decreasing order of size $N_m,N_{m-1},N_{m-2},\cdots,N_3,N_2,N_1$
$8,4,2,1$
STEP 2. Calculate the hidden surface areas
$H=8^2+2\left (4^2\right )+2\left (2^2\right )+2\left (1^2\right )$
$=64+32+8+2$
$=106$
STEP 3. Calculate the minimum possible total surface area
$T-H=510-106$
$=404$