A cube with edge length 5 is painted on all six faces, then cut into 125 unit cubes (each with edge length 1). How many of the unit cubes…

Cubes with exactly two painted faces sit on an edge of the large cube but not at a corner. A corner cube touches three painted faces. Each edge of the big cube has 5 unit cubes, 2 of which are corners, leaving $5 - 2 = 3$ edge cubes with exactly two painted faces. There are 12 edges, so the total is $12 \times 3 = 36$. A common wrong answer is 60, from counting all 5 cubes on each edge. Another is 54, which counts the face-center cubes (one painted face): $6 \times 9 = 54$. A third is 8, the number of corner cubes, which have three painted faces.