Problem
Layer and position fix how many faces show
A large cube built from small cubes is painted all over. We need how many small cubes end up with exactly one painted face.
Geometry
Your answer
How to solve
Strategy Change Focus / Count the Complement — A small cube's painted faces are just the big cube's faces it sits on, so exactly one painted face means it sits on one face and not on any edge or corner. That is the inside of a face -- and its size is easy to count.
1STEP 1
Say where a one-face cube must sit
A cube on an edge of the big cube touches two faces and one at a corner touches three, so a one-face cube must be strictly inside a face.
Not on an edge, not at a corner.
6.G.A.4Change Focus Count The ComplementA small cube's painted faces are exactly the big cube's faces it sits on.
Why?
Paint reaches only the outside, and the outside of the big cube is its six flat faces, so a small cube is painted once for each of those it touches.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
A cube on an edge touches two of those faces and one at a corner touches three, so only a cube strictly inside a face touches just one.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Count the inside of one face
A face is 5 by 5 small squares, and taking off the outer ring leaves the middle 3 by 3.
5 - 2 = 3, 3 × 3 = 9
Each face has 9 such cubes.
5.MD.C.3Visualize Spatial Relationships3STEP 3
Count all six faces
The big cube has six faces, and no cube is inside two of them at once, so nothing is counted twice.
9 × 6 = 54
54 cubes have exactly one painted face.
6.G.A.4Draw A DiagramAnswer
54 cubes
Account for all 125: 8 corners with three painted faces, 36 edge cubes with two, 54 with one, and 3 × 3 × 3 = 27 hidden inside with none -- and 8 + 36 + 54 + 27 = 125.
Takeaway
How many faces of a small cube get painted depends only on whether it sits on a face, an edge or a corner.
- Say where a one-face cube must sit
- Count the inside of one face
- Count all six faces