A wooden block is painted all over and sawn into little cubes. A few come out with three painted faces, most with fewer, some with none at all — and how many of each is fixed entirely by where the cube sat, not by counting them one at a time.
Every little cube's paint is decided by its position, and there are only four kinds. The eight corners carry three painted faces — always eight, whatever the block's size. Along each of the twelve edges, between the corners, sit the cubes with two faces; in the middle of each of the six faces, the cubes with one; and buried inside, touching nothing, the cubes with none. Peel the painted shell off and what is left is a smaller cube, two shorter on every side — and each count is a product built from it.
Why this sheet works
The mistake almost everyone makes is to use the block's whole size where the answer needs the size minus two. Twelve edges times the full side over-counts, because the two corners on each edge were already spoken for; six faces times the full face counts the edges twice. The sheet is built so the shell cannot be skipped — once a child sees that the corners and edges must come off before the middle is counted, all four numbers fall out of the same small cube.
A block is painted all over, then cut into little cubes. Count the ones the question asks for, and show why.
1A solid wooden block, 3 cubes long on every side, is painted red all over its outside and then sawn into 3 × 3 × 3 little cubes.How many of the little cubes have paint on exactly two faces?Why
2A solid wooden block, 6 cubes long on every side, is painted red all over its outside and then sawn into 6 × 6 × 6 little cubes.How many of the little cubes have no paint on them at all?Why
Painted cubeUp to 8 on each edge
Name: Date:
A block is painted all over, then cut into little cubes. Count the ones the question asks for, and show why.
3A solid wooden block, 8 cubes long on every side, is painted red all over its outside and then sawn into 8 × 8 × 8 little cubes.How many of the little cubes have paint on exactly one face?Why
4A solid wooden block, 7 cubes long on every side, is painted red all over its outside and then sawn into 7 × 7 × 7 little cubes.How many of the little cubes have paint on exactly three faces?Why
Painted cube · Up to 8 on each edgeAnswer key
1. Two painted faces means a cube along an edge, between the two corners. Taking the corners off each edge leaves 1 along it, and a cube has 12 edges: 12 × 1 = 12.2. No paint at all means a cube deep inside, never on the surface. Peel one layer off all six sides and the cube left inside is 4 on each edge: 4 × 4 = 16, then 16 × 4 = 64 cubes hidden in the middle.3. One painted face means a cube in the middle of a face, off every edge. Each face's middle is 6 × 6 = 36, and a cube has 6 faces: 6 × 36 = 216.4. Three painted faces means a corner, where three sides meet. A cube has 8 corners no matter how big it is, so the answer is 8 — the size of the block never enters into it.
