4️⃣ A cube is painted red on all faces. It is then cut into 27 smaller cubes of equal size. How many smaller cubes have three faces painted?
Step-by-Step Explanation
When a cube is cut into 27 smaller cubes, it means each side is divided into 3 equal parts (3x3x3).
Cubes with three faces painted are located at the corners.
A cube has 8 corners.
Answer: 8 ✅
5️⃣ A cube is painted blue on all faces. It is then cut into 64 smaller cubes of equal size. How many smaller cubes have no faces painted?
Step-by-Step Explanation
When a cube is cut into 64 smaller cubes, it means each side is divided into 4 equal parts (4x4x4).
Cubes with no faces painted are located in the inner core.
The inner core will be a cube of size (4-2) x (4-2) x (4-2) = 2x2x2 = 8.
Answer: 8 ✅
6️⃣ A cube is painted green on all faces. It is then cut into 125 smaller cubes of equal size. How many smaller cubes have only one face painted?
Step-by-Step Explanation
When a cube is cut into 125 smaller cubes, it means each side is divided into 5 equal parts (5x5x5).
Cubes with only one face painted are located at the center of each face.
Each face has (5-2) x (5-2) = 3x3 = 9 cubes with one face painted.
There are 6 faces, so there are 9 x 6 = 54 cubes with one face painted.
Answer: 54 ✅
7️⃣ A cube is painted yellow on two opposite faces, red on two other opposite faces, and blue on the remaining faces. It is then cut into 64 smaller cubes of equal size. How many smaller cubes have two faces painted, one red and one yellow?
Step-by-Step Explanation
When a cube is cut into 64 smaller cubes, it means each side is divided into 4 equal parts (4x4x4).
Cubes with two faces painted are located at the edges connecting red and yellow faces.
There are 4 such edges.
Each edge has (4-2) = 2 cubes with two faces painted.
Total cubes = 4 x 2 = 8.
Answer: 8 ✅
8️⃣ A cube is painted with different colors on each face. It is then cut into 216 smaller cubes of equal size. How many smaller cubes have at least two faces painted?
Step-by-Step Explanation
When a cube is cut into 216 smaller cubes, it means each side is divided into 6 equal parts (6x6x6).
Cubes with at least two faces painted include those with two faces painted and three faces painted.
Cubes with three faces painted are at corners: 8.
Cubes with two faces painted are at edges: 12 edges, each with (6-2) = 4 cubes. Total: 12 x 4 = 48.
Total cubes with at least two faces painted = 8 + 48 = 56.
Answer: 56 ✅
9️⃣ A cube is painted red on two opposite faces, green on two other opposite faces, and blue on the remaining faces. It is then cut into 125 smaller cubes of equal size. How many smaller cubes have one face painted red and one face painted green?
Step-by-Step Explanation
When a cube is cut into 125 smaller cubes, it means each side is divided into 5 equal parts (5x5x5).
Cubes with one face painted red and one face painted green are located at the edges connecting red and green faces.
There are 4 such edges.
Each edge has (5-2) = 3 cubes with two faces painted.
Total cubes = 4 x 3 = 12.
Answer: 12 ✅
🔟 A cube is painted black on all faces. It is then cut into 343 smaller cubes of equal size. How many smaller cubes have two adjacent faces painted?
Step-by-Step Explanation
When a cube is cut into 343 smaller cubes, it means each side is divided into 7 equal parts (7x7x7).
Cubes with two adjacent faces painted are located at the edges (excluding corners).
Each edge has (7-2) = 5 cubes with two faces painted.