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A cuboid is to be painted with red, blue and green colours.
Conditions:
(1) A surface has only one colour.
(2) Each of the three colors must be used.
(3) All six surfaces must be painted.
The cuboid is then cut by 5, 6 and 7 equally spaced planes parallel to xy, yz and xz planes respectively.
Q.What is the minimum number of cubes with exactly two of the three colors on them?
- a)32
- b)38
- c)42
- d)46
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A cuboid is to be painted with red, blue and green colours.Conditions:...
In order to have minimum number of cubes with exactly two of the three colors on them, let’s paint the 2 adjacent surfaces with 42 and 48 squares with green and blue respectively and remaining 4 surfaces with red colour.
All cubes along the perimeter of green surface will have exactly two colors except for cubes at 2 corners = 2 2 - 2 = 20
All cubes along the perimeter of green surface will have exactly two colors except for cubes at 2 corners = 2 2 - 2 = 20
All cubes along the perimeter of blue surface will have exactly two colors except for cubes at 2 corners = 24 - 2 = 22 But 4 cubes along the edge joining blue and green surfaces are common.
None of the remaining cubes can have only 2 colors
None of the remaining cubes can have only 2 colors
Minimum number of cubes with exactly two of the three colors on them = (20 + 22 - 4) = 38
Hence, option 2.
Hence, option 2.
Most Upvoted Answer
A cuboid is to be painted with red, blue and green colours.Conditions:...
To find the minimum number of cubes with exactly two of the three colors on them, we need to consider the different possibilities of painting the cuboid and then calculate the number of such cubes in each case. Let's break down the problem into steps:
Step 1: Determine the possible combinations of coloring the cuboid
Since each surface of the cuboid can only have one color and all three colors must be used, there are only two possible combinations for coloring the cuboid:
Combination 1: Two opposite faces have the same color, while the remaining four faces have different colors.
Combination 2: Three adjacent faces have the same color, while the remaining three faces have different colors.
Step 2: Calculate the number of cubes with exactly two of the three colors for each combination
Combination 1:
In this case, there are three pairs of opposite faces with the same color. Let's consider each pair separately:
Pair 1: The top and bottom faces have the same color. Since the cuboid is cut by 5 equally spaced planes parallel to the xy plane, there will be 6 cubes with this color on their top and bottom faces.
Pair 2: The front and back faces have the same color. Similarly, there will be 6 cubes with this color on their front and back faces.
Pair 3: The left and right faces have the same color. Again, there will be 6 cubes with this color on their left and right faces.
Therefore, the total number of cubes with exactly two of the three colors in this combination is 6 + 6 + 6 = 18.
Combination 2:
In this case, there are four sets of three adjacent faces with the same color. Let's consider each set separately:
Set 1: The top, bottom, and front faces have the same color. Since the cuboid is cut by 6 equally spaced planes parallel to the yz plane, there will be 7 cubes with this color on their top, bottom, and front faces.
Set 2: The top, bottom, and back faces have the same color. Similarly, there will be 7 cubes with this color on their top, bottom, and back faces.
Set 3: The top, left, and right faces have the same color. There will be 7 cubes with this color on their top, left, and right faces.
Set 4: The bottom, left, and right faces have the same color. There will be 7 cubes with this color on their bottom, left, and right faces.
Therefore, the total number of cubes with exactly two of the three colors in this combination is 7 + 7 + 7 + 7 = 28.
Step 3: Calculate the total number of cubes with exactly two of the three colors
Adding the number of cubes from both combinations, we get 18 + 28 = 46.
Therefore, the minimum number of cubes with exactly two of the three colors on them is 46, which corresponds to option (B).
Step 1: Determine the possible combinations of coloring the cuboid
Since each surface of the cuboid can only have one color and all three colors must be used, there are only two possible combinations for coloring the cuboid:
Combination 1: Two opposite faces have the same color, while the remaining four faces have different colors.
Combination 2: Three adjacent faces have the same color, while the remaining three faces have different colors.
Step 2: Calculate the number of cubes with exactly two of the three colors for each combination
Combination 1:
In this case, there are three pairs of opposite faces with the same color. Let's consider each pair separately:
Pair 1: The top and bottom faces have the same color. Since the cuboid is cut by 5 equally spaced planes parallel to the xy plane, there will be 6 cubes with this color on their top and bottom faces.
Pair 2: The front and back faces have the same color. Similarly, there will be 6 cubes with this color on their front and back faces.
Pair 3: The left and right faces have the same color. Again, there will be 6 cubes with this color on their left and right faces.
Therefore, the total number of cubes with exactly two of the three colors in this combination is 6 + 6 + 6 = 18.
Combination 2:
In this case, there are four sets of three adjacent faces with the same color. Let's consider each set separately:
Set 1: The top, bottom, and front faces have the same color. Since the cuboid is cut by 6 equally spaced planes parallel to the yz plane, there will be 7 cubes with this color on their top, bottom, and front faces.
Set 2: The top, bottom, and back faces have the same color. Similarly, there will be 7 cubes with this color on their top, bottom, and back faces.
Set 3: The top, left, and right faces have the same color. There will be 7 cubes with this color on their top, left, and right faces.
Set 4: The bottom, left, and right faces have the same color. There will be 7 cubes with this color on their bottom, left, and right faces.
Therefore, the total number of cubes with exactly two of the three colors in this combination is 7 + 7 + 7 + 7 = 28.
Step 3: Calculate the total number of cubes with exactly two of the three colors
Adding the number of cubes from both combinations, we get 18 + 28 = 46.
Therefore, the minimum number of cubes with exactly two of the three colors on them is 46, which corresponds to option (B).
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A cuboid is to be painted with red, blue and green colours.Conditions:(1) A surface has only one colour.(2) Each of the three colors must be used.(3) All six surfaces must be painted.The cuboid is then cut by 5, 6 and 7 equally spaced planes parallel to xy, yz and xz planes respectively.Q.What is the minimum number of cubes with exactly two of the three colors on them?a)32b)38c)42d)46Correct answer is option 'B'. Can you explain this answer?
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A cuboid is to be painted with red, blue and green colours.Conditions:(1) A surface has only one colour.(2) Each of the three colors must be used.(3) All six surfaces must be painted.The cuboid is then cut by 5, 6 and 7 equally spaced planes parallel to xy, yz and xz planes respectively.Q.What is the minimum number of cubes with exactly two of the three colors on them?a)32b)38c)42d)46Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A cuboid is to be painted with red, blue and green colours.Conditions:(1) A surface has only one colour.(2) Each of the three colors must be used.(3) All six surfaces must be painted.The cuboid is then cut by 5, 6 and 7 equally spaced planes parallel to xy, yz and xz planes respectively.Q.What is the minimum number of cubes with exactly two of the three colors on them?a)32b)38c)42d)46Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A cuboid is to be painted with red, blue and green colours.Conditions:(1) A surface has only one colour.(2) Each of the three colors must be used.(3) All six surfaces must be painted.The cuboid is then cut by 5, 6 and 7 equally spaced planes parallel to xy, yz and xz planes respectively.Q.What is the minimum number of cubes with exactly two of the three colors on them?a)32b)38c)42d)46Correct answer is option 'B'. Can you explain this answer?.
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