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

Answer
: Given in solution.

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , , , . Bookmark the permalink.

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