Now you have only cubes, of edge length . For simplicity we’ll suppose they go in increasing order, so is the smallest and 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?

8. What is the total volume of this tower?

9. What is the maximum possible total surface area for this tower?

STEP 1. Calculate the total surface area of all the cubes

STEP 2. Move between and obtaining

Move between and obtaining

Repeat STEP 2 until the cubes are stacked in the pattern of alternating smaller ones and bigger ones.

STEP 3. Calculate 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.Â Calculate the maximum possible total surface area for this tower

EXAMPLE: Build a tower of height using cubes

STEP 1. Calculate the total surface area of all the cubes

STEP 2. Move between and obtaining

STEP 3. Calculate the hidden surface areas

STEP 4. Calculate the maximum possible total surface area for this tower

10. What is the minimum possible total surface area?

STEP 1. Stack the cubes in decreasing order of size

STEP 2. Calculate the hidden surface area of the bottom and the hidden surface areas between the cubes

STEP 3. Calculate the minimum possible total surface area

EXAMPLE: Build a tower of height using cubes

STEP 1. Stack the cubes in decreasing order of size

STEP 2. Calculate the hidden surface areas

STEP 3. Calculate the minimum possible total surface area

**
Answer**: Given in solution.