puzzle 46 curious village

Puzzle 46: The Vanishing Cube (from Professor Layton and the Curious Village) Problem: A large cube is painted on all surfaces and then cut into 27 smaller, identical cubes (like a 3x3x3 Rubik's Cube). How many of the small cubes have 0 painted faces? How many have 1 painted face? How many have 2 painted faces? How many have 3 painted faces? Solution: 0 painted faces: The inner cube (not on any outer layer). - Only 1 small cube (the very center). 1 painted face: Cubes at the center of each face. - There are 6 faces, each with 1 center cube 6 cubes. 2 painted faces: Cubes on the edges but not corners. - There are 12 edges on a cube, and each edge has 1 non-corner cube 12 cubes. 3 painted faces: Corner cubes. - A cube has 8 corners 8 cubes. Verification: Total small cubes = 0-face + 1-face + 2-face + 3-face = 1 + 6 + 12 + 8 = 27 (matches the 3x3x3 cube). Final Answer: 0 faces: 1 cube 1 face: 6 cubes 2 faces: 12 cubes 3 faces: 8 cubes This problem tests spatial reasoning by visualizing how the larger cube is divided and counting painted faces based on position (interior, face, edge, or corner).