The surface of a cube is painted red and blue, 3 sides of each color. the cube is then cut into 27 equal sized smaller cubes.
A: How many cubes have at least 2 sides painted blue?
B: How many cubes have at least 1 side painted red?
A: it depends how the cube is painted. If the 3 blue sides of the original cube share a common corner then 7 of the smaller cubes have at least 2 sides that are blue. If 2 opposite faces of the original cube are blue then 6 of the 27 have at least 2 faces that are blue.
B: 19 or 21, again depending on how the original cube is painted.
To find the answers to these questions, we need to break down the problem step by step.
Let's start with question A: How many cubes have at least 2 sides painted blue?
Step 1: Visualize the original cube. We know that 3 sides of the original cube are painted blue and 3 sides are painted red. Each side of the cube consists of one unit square.
Step 2: Divide the original cube into smaller cubes. Since the original cube is divided into 27 equal-sized smaller cubes, it forms a 3x3x3 cube made up of smaller cubes.
Step 3: Count the cubes with at least 2 sides painted blue. We know that each smaller cube has 6 sides. Since 3 sides of the original cube are painted blue, each smaller cube can have at most 3 sides painted blue.
To count the cubes with at least 2 sides painted blue, we need to consider two cases:
Case 1: Cubes with exactly 2 sides painted blue:
Since each smaller cube can have at most 3 sides painted blue, the number of smaller cubes with exactly 2 sides painted blue is the same as the number of smaller cubes with exactly 1 side painted red. This can be found by counting the number of smaller cubes with exactly 1 side painted red and multiplying it by 2.
Case 2: Cubes with exactly 3 sides painted blue:
Since each smaller cube can have at most 3 sides painted blue, the number of smaller cubes with exactly 3 sides painted blue is the same as the number of smaller cubes with exactly 1 side painted red.
To get the total number of cubes with at least 2 sides painted blue, we need to add the number of cubes from case 1 and case 2.
Now let's move on to question B: How many cubes have at least 1 side painted red?
Step 4: Count the cubes with at least 1 side painted red. Since 3 sides of the original cube are painted red, each smaller cube can have at most 3 sides painted red.
To count the cubes with at least 1 side painted red, we need to consider three cases:
Case 1: Cubes with exactly 1 side painted red: Each smaller cube can have at most 3 sides painted red, so the number of smaller cubes with exactly 1 side painted red is the same as the number of smaller cubes with exactly 3 sides painted blue.
Case 2: Cubes with exactly 2 sides painted red: Since each smaller cube can have at most 3 sides painted red, the number of smaller cubes with exactly 2 sides painted red is the same as the number of smaller cubes with exactly 1 side painted blue.
Case 3: Cubes with exactly 3 sides painted red: Since each smaller cube can have at most 3 sides painted red, the number of smaller cubes with exactly 3 sides painted red is the same as the number of smaller cubes with exactly 1 side painted blue.
To get the total number of cubes with at least 1 side painted red, we need to add the number of cubes from all three cases.
Please note that the numbers in each case will depend on the specific arrangement of the colors on the original cube. I have explained the general approach to solving this problem, but without knowing the specific color arrangement, I cannot give you the exact numbers.