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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 can have their faces painted with all three colours?
- a)0
- b)1
- c)2
- d)5
Correct answer is option 'C'. 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 blue face is the one with 0 written on it and the face opposite to it is the face with 5 written on it i.e. the red face.
Let 0 correspond to the front face. Hence, 5 corresponds to the back face.
Now, 1 (red) and 4 (green) are opposite each other while 2 (green) and 3 (red) are also opposite each other.
Let 1 and 4 correspond to the right and left face respectively; and let 2 and 3 correspond to the top and bottom face respectively (as shown below).
Let 0 correspond to the front face. Hence, 5 corresponds to the back face.
Now, 1 (red) and 4 (green) are opposite each other while 2 (green) and 3 (red) are also opposite each other.
Let 1 and 4 correspond to the right and left face respectively; and let 2 and 3 correspond to the top and bottom face respectively (as shown below).
The intersection of faces 0, 1 and 2 gives one cube having all three colours. Similarly, the intersection of faces 0, 3 and 4 gives another cube having all three colours.
Thus, there are 2 such cubes.
Hence, option 3.
Thus, there are 2 such cubes.
Hence, option 3.
Most Upvoted Answer
A large cube has a unique whole number, smaller than or equal to 5, wr...
Problem Analysis:
- The large cube has 6 faces, and the sum of the numbers on opposite faces is equal to 5. This means that if a face has a number x on it, then the opposite face will have a number 5-x on it.
- The numbers on the faces of the cube are whole numbers less than or equal to 5. So the possible numbers on the faces can be 1, 2, 3, 4, 5.
- We are given that faces with odd numbers are painted red, faces with even numbers are painted green, and the remaining faces are painted blue.
Solution:
- Let's consider the possible numbers on the faces of the cube one by one and determine the colors of the faces.
- If a face has the number 1 on it, then the opposite face will have the number 5-1 = 4 on it. So, the colors of these faces will be red and green respectively.
- If a face has the number 2 on it, then the opposite face will have the number 5-2 = 3 on it. So, the colors of these faces will be green and red respectively.
- If a face has the number 3 on it, then the opposite face will have the number 5-3 = 2 on it. So, the colors of these faces will be red and green respectively.
- If a face has the number 4 on it, then the opposite face will have the number 5-4 = 1 on it. So, the colors of these faces will be green and red respectively.
- If a face has the number 5 on it, then the opposite face will have the number 5-5 = 0 on it. Since 0 is neither odd nor even, this face will be painted blue.
Determining the Number of Small Cubes:
- Each small cube will have 3 faces painted with different colors if it is located on a corner of the large cube. So, we need to count the number of corners in the large cube.
- A cube has 8 corners, so each corner will contribute 1 small cube with 3 differently colored faces.
- Therefore, the total number of small cubes that can have their faces painted with all three colors is 8.
Conclusion:
- The correct answer is option 'C'.
- The large cube has 6 faces, and the sum of the numbers on opposite faces is equal to 5. This means that if a face has a number x on it, then the opposite face will have a number 5-x on it.
- The numbers on the faces of the cube are whole numbers less than or equal to 5. So the possible numbers on the faces can be 1, 2, 3, 4, 5.
- We are given that faces with odd numbers are painted red, faces with even numbers are painted green, and the remaining faces are painted blue.
Solution:
- Let's consider the possible numbers on the faces of the cube one by one and determine the colors of the faces.
- If a face has the number 1 on it, then the opposite face will have the number 5-1 = 4 on it. So, the colors of these faces will be red and green respectively.
- If a face has the number 2 on it, then the opposite face will have the number 5-2 = 3 on it. So, the colors of these faces will be green and red respectively.
- If a face has the number 3 on it, then the opposite face will have the number 5-3 = 2 on it. So, the colors of these faces will be red and green respectively.
- If a face has the number 4 on it, then the opposite face will have the number 5-4 = 1 on it. So, the colors of these faces will be green and red respectively.
- If a face has the number 5 on it, then the opposite face will have the number 5-5 = 0 on it. Since 0 is neither odd nor even, this face will be painted blue.
Determining the Number of Small Cubes:
- Each small cube will have 3 faces painted with different colors if it is located on a corner of the large cube. So, we need to count the number of corners in the large cube.
- A cube has 8 corners, so each corner will contribute 1 small cube with 3 differently colored faces.
- Therefore, the total number of small cubes that can have their faces painted with all three colors is 8.
Conclusion:
- The correct answer is option 'C'.
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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 can have their faces painted with all three colours?a)0b)1c)2d)5Correct answer is option 'C'. 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 can have their faces painted with all three colours?a)0b)1c)2d)5Correct answer is option 'C'. 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 can have their faces painted with all three colours?a)0b)1c)2d)5Correct answer is option 'C'. 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 can have their faces painted with all three colours?a)0b)1c)2d)5Correct answer is option 'C'. Can you explain this answer?.
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 can have their faces painted with all three colours?a)0b)1c)2d)5Correct answer is option 'C'. 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 can have their faces painted with all three colours?a)0b)1c)2d)5Correct answer is option 'C'. 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 can have their faces painted with all three colours?a)0b)1c)2d)5Correct answer is option 'C'. Can you explain this answer?.
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