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A large cube has a unique whole number, smaller than or equal to 5, written on each of its faces such that the sum of the numbers on opposite faces is equal to 5. Any face that has an odd natural number written on it is painted red, any face that has an even natural number written on it is painted green and any remaining face(s) is/are painted blue. The large cube is then cut into 125 small identical cubes.
Q.How many of the smaller cubes don’t have any of their faces painted?
- a)15
- b)27
- c)12
- d)24
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A large cube has a unique whole number, smaller than or equal to 5, wr...
The number of smaller cubes that do not have any of their faces painted = (n - 2)3 where n = 5. number of required cubes = (5 - 2)3 = 27
Hence, option 2.
Most Upvoted Answer
A large cube has a unique whole number, smaller than or equal to 5, wr...
To solve this problem, let's start by analyzing the given information:
1. The sum of the numbers on opposite faces is equal to 5.
2. Odd numbers are painted red.
3. Even numbers are painted green.
4. Remaining faces are painted blue.
Let's consider the six faces of the large cube and their respective numbers:
- Face 1: Number X
- Face 2: Number 5 - X (opposite face of Face 1)
- Face 3: Number Y
- Face 4: Number 5 - Y (opposite face of Face 3)
- Face 5: Number Z
- Face 6: Number 5 - Z (opposite face of Face 5)
We know that the numbers on the faces are smaller than or equal to 5, and they are whole numbers. Therefore, the possible values for X, Y, and Z are as follows:
X = 1, 2, 3, 4 or 5
Y = 1, 2, 3, 4 or 5
Z = 1, 2, 3, 4 or 5
Now, let's analyze the possible combinations of X, Y, and Z:
- If X is odd, then the opposite face of Face 1 is even (5 - X), and Face 1 is painted red.
- If X is even, then the opposite face of Face 1 is odd (5 - X), and Face 1 is painted green.
- Similarly, the same logic applies to Faces 3 and 5.
Based on the above analysis, we can determine the colors of each face of the large cube:
- Face 1: Red if X is odd, green if X is even.
- Face 2: The opposite face of Face 1, so it will have the opposite color.
- Face 3: Red if Y is odd, green if Y is even.
- Face 4: The opposite face of Face 3, so it will have the opposite color.
- Face 5: Red if Z is odd, green if Z is even.
- Face 6: The opposite face of Face 5, so it will have the opposite color.
Now, let's count the number of cubes with each color:
- Red cubes: There are 3 options for the odd numbers (1, 3, or 5) for each of the three faces (1, 3, and 5). Therefore, there are 3 * 3 = 9 red cubes.
- Green cubes: There are 2 options for the even numbers (2 or 4) for each of the three faces (2, 4, and 6). Therefore, there are 2 * 3 = 6 green cubes.
- Blue cubes: The remaining cubes will have two opposite faces with the same color, so they will be painted blue. The number of such cubes is 125 - 9 - 6 = 110.
Therefore, the number of cubes that don't have any of their faces painted is 110.
1. The sum of the numbers on opposite faces is equal to 5.
2. Odd numbers are painted red.
3. Even numbers are painted green.
4. Remaining faces are painted blue.
Let's consider the six faces of the large cube and their respective numbers:
- Face 1: Number X
- Face 2: Number 5 - X (opposite face of Face 1)
- Face 3: Number Y
- Face 4: Number 5 - Y (opposite face of Face 3)
- Face 5: Number Z
- Face 6: Number 5 - Z (opposite face of Face 5)
We know that the numbers on the faces are smaller than or equal to 5, and they are whole numbers. Therefore, the possible values for X, Y, and Z are as follows:
X = 1, 2, 3, 4 or 5
Y = 1, 2, 3, 4 or 5
Z = 1, 2, 3, 4 or 5
Now, let's analyze the possible combinations of X, Y, and Z:
- If X is odd, then the opposite face of Face 1 is even (5 - X), and Face 1 is painted red.
- If X is even, then the opposite face of Face 1 is odd (5 - X), and Face 1 is painted green.
- Similarly, the same logic applies to Faces 3 and 5.
Based on the above analysis, we can determine the colors of each face of the large cube:
- Face 1: Red if X is odd, green if X is even.
- Face 2: The opposite face of Face 1, so it will have the opposite color.
- Face 3: Red if Y is odd, green if Y is even.
- Face 4: The opposite face of Face 3, so it will have the opposite color.
- Face 5: Red if Z is odd, green if Z is even.
- Face 6: The opposite face of Face 5, so it will have the opposite color.
Now, let's count the number of cubes with each color:
- Red cubes: There are 3 options for the odd numbers (1, 3, or 5) for each of the three faces (1, 3, and 5). Therefore, there are 3 * 3 = 9 red cubes.
- Green cubes: There are 2 options for the even numbers (2 or 4) for each of the three faces (2, 4, and 6). Therefore, there are 2 * 3 = 6 green cubes.
- Blue cubes: The remaining cubes will have two opposite faces with the same color, so they will be painted blue. The number of such cubes is 125 - 9 - 6 = 110.
Therefore, the number of cubes that don't have any of their faces painted is 110.
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A large cube has a unique whole number, smaller than or equal to 5, written on each of its faces such that the sum of the numbers onopposite faces is equal to 5. Any face that has an odd natural number written on it is painted red, any face that has an even natural number written on it is painted green and any remaining face(s) is/are painted blue. The large cube is then cut into 125 small identical cubes.Q.How many of the smaller cubes dont have any of their faces painted?a)15b)27c)12d)24Correct answer is option 'B'. Can you explain this answer?
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A large cube has a unique whole number, smaller than or equal to 5, written on each of its faces such that the sum of the numbers onopposite faces is equal to 5. Any face that has an odd natural number written on it is painted red, any face that has an even natural number written on it is painted green and any remaining face(s) is/are painted blue. The large cube is then cut into 125 small identical cubes.Q.How many of the smaller cubes dont have any of their faces painted?a)15b)27c)12d)24Correct 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 large cube has a unique whole number, smaller than or equal to 5, written on each of its faces such that the sum of the numbers onopposite faces is equal to 5. Any face that has an odd natural number written on it is painted red, any face that has an even natural number written on it is painted green and any remaining face(s) is/are painted blue. The large cube is then cut into 125 small identical cubes.Q.How many of the smaller cubes dont have any of their faces painted?a)15b)27c)12d)24Correct 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 large cube has a unique whole number, smaller than or equal to 5, written on each of its faces such that the sum of the numbers onopposite faces is equal to 5. Any face that has an odd natural number written on it is painted red, any face that has an even natural number written on it is painted green and any remaining face(s) is/are painted blue. The large cube is then cut into 125 small identical cubes.Q.How many of the smaller cubes dont have any of their faces painted?a)15b)27c)12d)24Correct answer is option 'B'. Can you explain this answer?.
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