The surface of a cube is painted red and blue, 3 sides of each. The cube is cut into 27 equal sized smaller cubes. Find both answers to the question, "How many cubes have at least 2 sides painted blue?" and "How many cubes have at least 1 side painted red?"
I attempted it but still don't get how to do it. Please be very specific when explaining.
To solve this problem, we need to break it down step by step. Let's start by understanding the given information.
We know that the surface of the cube is painted red and blue, with 3 sides of each color. This means that each face of the cube has three red sides and three blue sides. Since a cube has six faces in total, there are a total of six red sides and six blue sides on the cube.
The cube is then cut into 27 equal-sized smaller cubes. This indicates that the larger cube is divided into smaller cubes with the same dimensions.
Now, let's find the answers to the questions one by one:
1. How many cubes have at least 2 sides painted blue?
To determine the number of cubes with at least 2 sides painted blue, we need to consider three scenarios:
- Scenario 1: Cubes with 3 sides painted blue
Since there are six blue sides in total, we can form two of these cubes. Therefore, there are two cubes with 3 sides painted blue.
- Scenario 2: Cubes with 2 sides painted blue
Each face of the larger cube has three blue sides. Since there are six sides in total, we can form two cubes on each face with 2 sides painted blue. Hence, there are 2 x 6 = 12 cubes with 2 sides painted blue.
- Scenario 3: Cubes with 1 side painted blue
For each face, there are three cubes with 1 side painted blue. Since there are six faces, there are 3 x 6 = 18 cubes with 1 side painted blue.
To find the total number of cubes with at least 2 sides painted blue, we add up the numbers from all three scenarios:
2 + 12 + 18 = 32 cubes.
Therefore, there are 32 cubes with at least 2 sides painted blue.
2. How many cubes have at least 1 side painted red?
Similarly, we need to consider three scenarios:
- Scenario 1: Cubes with 3 sides painted red
Since there are six red sides in total, we can form two of these cubes. Therefore, there are two cubes with 3 sides painted red.
- Scenario 2: Cubes with 2 sides painted red
Each face of the larger cube has three red sides. Since there are six sides in total, we can form two cubes on each face with 2 sides painted red. Hence, there are 2 x 6 = 12 cubes with 2 sides painted red.
- Scenario 3: Cubes with 1 side painted red
For each face, there are three cubes with 1 side painted red. Since there are six faces, there are 3 x 6 = 18 cubes with 1 side painted red.
To find the total number of cubes with at least 1 side painted red, we add up the numbers from all three scenarios:
2 + 12 + 18 = 32 cubes.
Therefore, there are 32 cubes with at least 1 side painted red.
In summary, there are 32 cubes with at least 2 sides painted blue and 32 cubes with at least 1 side painted red.