The cube above is made from 27 small cubes, each with an edge of length 1. If the shaded cube is removed what will be the surface area of the remaining solid?

I’m assuming the shaded cube is a corner cube, right? So when you take it away, you’re removing 3 faces from the solid? Like this?

rubiks cube


Look! When you remove that cube, you’re taking away 3 faces, but you’re also exposing 3 more previously unexposed faces! And those newly exposed faces have the exact same areas as the faces you took away. So the surface area, amazingly enough, remains the same.

You’ve got a 3\times 3\times 3 cube there. The original area of each face was 9, and of course all cubes have 6 faces. So the original cube had a surface area of 54.

When you remove the corner piece, you’ve got 54 – 3 + 3 = 54.

Does that help?

Bonus question: what would happen if the removed cube came from the middle of a face, instead of a corner?

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