Directions: The following prism is made up of 27 identical cubes. What is the greatest possible surface area the prism can have after removing 1 or more cubes from the outside?

### Hint

How many cubes (and which ones) would you remove? How could you record the information to help you see a pattern?

### Answer

Removing the center cube in each face of the prism results in a surface area of 78 square units.

Source: Brian Lack

That’s a cool problem, but I don’t understand the title. Isn’t the point NOT to keep the surface area constant? Puzzled…

Darn good point Julie, as always. I’ll change the title.

A colleague has found an additional solution with a surface area of 78 square units:

Remove four cubes on each face – the center cube on the top and bottom row; the two cubes on the ends of the middle row.

This additional solution prompted conversations about the comparison of surface area to volume, as well as real-world instances where maximizing (or minimizing) the ratio of surface area to volume is important.

Could a student think that the phrase “removing cubes from the outside” means removing a layer therefore the student would remove 9 cubes.

I tweaked the language a bit to say “removing 1 or more cubes from the outside.” Does that tighten it up a bit?

Weird how the title says maximizing when your not supposed to change the surface area…

Hi Cameron. What makes you think that is says that you’re not supposed to change the surface area?

Can you remove the X on all sides and still get 78? So the middle and all corners are gone. I believe it is the same solution and could set up a greatest number of blocks you can remove and still have the largest surface area possible.

i dont understand how you got 78

can you help me understand

Imagine one face of the cube, not all six faces. If you remove the center cube from one of the faces, the surface area for that face is 13 square units. You multiply 13 by 6, then you will get 78 square units.

If you multiply 13by 6 you will get78 square units.