KA-0090Counting cubes in a stack, including hidden ones
5 points, difficulty 3 of 3Level 5–6about 135sA cube 3 units on each side is painted all over and then cut into unit cubes. How many of them have exactly two painted faces?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Where does a cube meet exactly two outside faces?
2The strategy
Sort the small cubes by position: corners touch three outside faces, edges touch two, face centres touch one, and the core touches none. Count the positions, not the cubes you can see.
3The full solution
The cubes with two painted faces sit along the edges but not at the corners. A cube has 12 edges, and each holds 3 - 2 = 1 such cube: 12 in all.
Solution
The reliable way
Sort the 27 small cubes by where they sit. The 8 corners touch three outside faces. The edge cubes that are not corners touch exactly two: a cube has 12 edges and each edge of a 3-cube holds 3 - 2 = 1 non-corner cube, so 12 of them. The 6 face centres touch one, and the single core touches none. Check the total: 8 + 12 + 6 + 1 = 27. The transferable idea: classify by position — corner, edge, face, core — and the counts follow from the cube's own structure.
The elegant way
Non-corner edge cubes number 12 times (n - 2), which for n = 3 is 12.
Why this is on the test: Sorting the cubes by position is the method, and the check that the four groups add back to n cubed is what catches an error before it is submitted.