Painted-Cube Count from a Coordinate List
A structure is built by gluing identical unit cubes face-to-face on an integer grid. The occupied coordinates (x, y, z) are: (0,0,0), (1,0,0), (2,0,0), (0,1,0), (0,2,0), (0,0,1), (1,0,1), (0,1,1), and (0,0,2). No other cubes are present. The entire structure is dipped into paint so that every face not touching another cube receives paint. A student counts the cubes with exactly three painted faces by looking only at the top layer and gets a different number than a student who counts only the bottom layer. The instructor asks for a systematic face-by-face count, not a visual estimate. The class must also determine the total number of painted faces on the whole structure, because that number will be used to estimate paint volume.
1. How many cubes have exactly three painted faces? Show a neighbor count for each cube.
Read the worked answer
For a unit cube in the structure, painted faces = 6 − number of face-sharing neighbors. Count neighbors from the coordinate list.
(0,0,0): neighbors (1,0,0), (0,1,0), (0,0,1) → 3 neighbors → 3 painted. (1,0,0): neighbors (0,0,0), (2,0,0), (1,0,1) → 3 → 3 painted. (2,0,0): neighbor (1,0,0) only → 1 → 5 painted. (0,1,0): neighbors (0,0,0), (0,2,0), (0,1,1) → 3 → 3 painted. (0,2,0): neighbor (0,1,0) only → 1 → 5 painted. (0,0,1): neighbors (0,0,0), (1,0,1), (0,1,1), (0,0,2) → 4 → 2 painted. (1,0,1): neighbors (0,0,1), (1,0,0) → 2 → 4 painted. (0,1,1): neighbors (0,0,1), (0,1,0) → 2 → 4 painted. (0,0,2): neighbor (0,0,1) only → 1 → 5 painted.
The cubes with exactly three painted faces are (0,0,0), (1,0,0), and (0,1,0). That is 3 cubes.
2. What is the total number of painted faces on the structure? Use the face counts and check that no exposed face is counted twice.
Read the worked answer
Sum the painted faces for all nine cubes: 3 + 3 + 5 + 3 + 5 + 2 + 4 + 4 + 5 = 34. Each painted face belongs to exactly one cube and is not shared with another cube, so summing per-cube exposed faces counts every painted face once. A useful cross-check is to count neighbors in pairs: each shared face removes two exposed faces from the hypothetical total of 9 × 6 = 54. There are 10 face-sharing contacts in the structure: (0,0,0)-(1,0,0), (1,0,0)-(2,0,0), (0,0,0)-(0,1,0), (0,1,0)-(0,2,0), (0,0,0)-(0,0,1), (1,0,0)-(1,0,1), (0,1,0)-(0,1,1), (0,0,1)-(1,0,1), (0,0,1)-(0,1,1), and (0,0,1)-(0,0,2). Each contact hides 2 faces, so 54 − 2(10) = 34 painted faces.