Directions (2 of 3 questions): A cube of 5 cm has been painted on its surfaces in such a...
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Directions (2 of 3 questions):**
A cube of 5 cm has been painted on its surfaces in such a way that two opposite surfaces have been painted blue and two adjacent surfaces have been painted red. Two remaining surfaces have been left unpainted. Now the cube is cut into smaller cubes of side 1 cm each.
How many cubes will have at least red color on its surfaces?
Show answer & explanation
Two adjacent 5x5 faces are painted red. Each face has 25 cubes, and they share a common edge of 5 cubes. By Inclusion-Exclusion, 25 + 25 - 5 = 45 cubes have red paint.
Step-by-step Derivation:
Step 1: The cube is 5 x 5 x 5.
Step 2: Face 1 has 5 * 5 = 25 cubes painted red.
Step 3: Adjacent Face 2 has 5 * 5 = 25 cubes painted red.
Step 4: The intersection of these two adjacent faces is an edge containing 5 cubes (counted twice).
Step 5: Total cubes with at least one red face = 25 + 25 - 5 = 45.