puzzle 005 curious village

Puzzle 005 from Professor Layton and the Curious Village is titled "The Vanishing Cube." Here's how to solve it: Problem Statement: You have a large cube made up of smaller cubes (a 4x4x4 cube). The large cube is dropped into a bucket of paint, coating all its outer surfaces. After the paint dries, the cube is disassembled into the smaller cubes. Question: How many of the smaller cubes have no paint on them at all? Solution: Total Number of Small Cubes: The large cube is 4x4x4, so there are (4 imes 4 imes 4 = 64) small cubes in total. Cubes with Paint: - Faces: The outermost layer of cubes will have paint. - Inner Cubes (No Paint): These are the cubes completely inside, not touching any outer face. - A 4x4x4 cube has a 2x2x2 inner core (since we remove one layer from each side). - So, (2 imes 2 imes 2 = 8) small cubes have no paint. Verification: - Alternatively, subtract the painted cubes from the total: - Cubes with paint on at least one face: - Total cubes: 64 - Inner unpainted cubes: 8 - So, (64 - 8 = 56) cubes have paint. Final Answer: 8 small cubes have no paint on them. Explanation: The key is realizing that only the completely internal* cubes (not touching any outer layer) remain unpainted. For an (n imes n imes n) cube, the number of unpainted cubes is ((n-2)3). Here, (n=4), so ((4-2)3 = 8). Would you like help with another puzzle?