puzzle 45 curious village

Puzzle 45: The Vanishing Cube from Professor Layton and the Curious Village is a classic dissection puzzle. Here's how to solve it: Problem: You have a 4x4x4 cube made up of smaller 1x1x1 cubes. The cube is dipped in paint, then disassembled into its smaller cubes. How many of the small cubes have exactly two painted faces? Solution: Understand the Cube Structure: - A 4x4x4 cube has 64 small cubes (4 4 4). - When dipped, the outer layer is painted, while the inner 2x2x2 cubes (8 cubes) remain unpainted. Breakdown by Painted Faces: - Cubes with 3 painted faces: These are the corners of the large cube. A cube has 8 corners 8 cubes. - Cubes with 2 painted faces: These lie on the edges but not corners. Each edge has 2 such cubes (since the corners are excluded), and there are 12 edges 12 2 = 24 cubes. - Cubes with 1 painted face: These are on the faces but not edges. Each face has 4 such cubes (center), and there are 6 faces 6 4 = 24 cubes. - Cubes with 0 painted faces: The inner 2x2x2 cube 8 cubes. Answer: - The number of small cubes with exactly two painted faces is 24. Verification: - Total painted cubes = 8 (3 faces) + 24 (2 faces) + 24 (1 face) = 56. - Unpainted cubes = 8. - Total = 56 + 8 = 64 (matches the original cube). Thus, the correct answer is 24.