Cubes - Cutting a Painted Cube
Cubes - Cutting a Painted Cube
Cubes - Cutting a Painted Cube: If a person cuts a piece of cube along any of its dimension then he gets two pieces. Now if someone makes ‘N’ number of cuts along its length then there will ‘N + 1’ number of pieces coming out of the original cube.
Now let us try with different number of cuts along different dimensions, if we have ‘L’ number of cuts along length, ‘B’ number of cuts along breadth and ‘H’ number of cuts along height then the number of pieces along these dimensions would be ‘L + 1’, ‘B + 1’ and ‘H + 1’ respectively.
Hence, total number of pieces would be (L + 1) (B + 1) (H + 1).
Let us start this with three cuts. Now we know for 3 cuts along a particular dimension there will be (3 +1) = 4 pieces.
But if we make 2 cuts along one dimension and 1 cut along another dimension then the number of pieces according to the formula given above will be (2 + 1) (1 + 1) = 6.So we can see that number of pieces would increase if we distribute the number of cutes along different dimensions.
Now here we can also consider one cut along each dimension. Where the number of pieces would be (1 + 1) (1 + 1) (1 + 1) = 8
Hence, to maximize the number of cuts here we should try to distribute the total number of cuts along each dimension as uniformly as possible.
Let us try some examples
Example : What’s the maximum number of pieces that can be obtained with 17 cuts without putting the pieces one above another?
A. 294
B. 296
C. 298
D. 300
Answer : 294
Explanation : Here, we can split 17 as 5, 6 and 6 which is the most equally distributed case. So the number of pieces will be (5 + 1) (6 +1) (6 + 1) = 294.
Example : A cube is coloured red on all faces. It is cut into 64 smaller cubes of equal size. Now, answer the following questions based on this statement :How many cubes have no face coloured?
A. 24
B. 16
C. 8
D. 10
Answer : C
Explanation : Since, there are 64 smaller cubes of equal size, therefore, n = no. of divisions on the face of undivided cube = 4
No. of cubes with no face coloured = (n – 2)³ = (4 – 2)³ = 8
A solid cube of each side 8 cms, has been painted red, blue and black on pairs of opposite faces. It is then cut into cubical blocks of each side 2 cms.
How many cubes have only one face painted red and all other faces unpainted ? | |||
| Right Option : B | |||
| View Explanation | |||
A solid cube of each side 8 cms, has been painted red, blue and black on pairs of opposite faces. It is then cut into cubical blocks of each side 2 cms.
How many cubes have two faces painted black ? | |||
| Right Option : D | |||
| View Explanation | |||
A solid cube of each side 8 cms, has been painted red, blue and black on pairs of opposite faces. It is then cut into cubical blocks of each side 2 cms.
How many cubes have two faces painted red and black and all other faces unpainted ? | |||
| Right Option : B | |||
| View Explanation | |||
Students / Parents Reviews [20]
My experience with Abhyas academy is very nice or it can be said wonderful. I have been studying here from seven class. I have been completing my journey of three years. I am tinking that I should join Abhyas Academy in tenth class as I am seeing much improvement in Maths and English
Hridey Preet
9thThird consective year,my ward is in Abhyas with nice experience of admin and transport support.Educational standard of the institute recumbent at satisfactory level. One thing would live to bring in notice that last year study books was distributed after half of the session was over,though study ...
Ayan Ghosh
8thMy experience with Abhyas is very good. I have learnt many things here like vedic maths and reasoning also. Teachers here first take our doubts and then there are assignments to verify our weak points.
Shivam Rana
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Hiya Gupta
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Manish Kumar
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Cheshta
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8thIn terms of methodology I want to say that institute provides expert guidence and results oriented monitering supplements by requsite study material along with regular tests which help the students to improve their education skills.The techniques of providing education helps the students to asses...
Aman Kumar Shrivastava
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Aman Kumar Shrivastava
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Shreya Shrivastava
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Barkha Arora
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Shreya Shrivastava
8thAbhyas is a complete education Institute. Here extreme care is taken by teacher with the help of regular exam. Extra classes also conducted by the institute, if the student is weak.
Om Umang
10thAbhyas Methodology is very good. It is based on according to student and each child manages accordingly to its properly. Methodology has improved the abilities of students to shine them in future.
Manish Kumar
10thI have spent a wonderful time in Abhyas academy. It has made my reasoning more apt, English more stronger and Maths an interesting subject for me. It has given me a habbit of self studying
Yatharthi Sharma
10thAbhyas institute is one of the best coaching institute in the vicinity of Ambala Cantt area. The teachers of the institute are well experienced and very helpful in solving the problems of the students.The good thing of the institute is that it is providing extra classes for the students who are w...
Aman Kumar Shrivastava
10thMy experience with Abhyas Academy has been very good. When I was not in Abhyas whenever teacher ask questions I could not speak it confidently but when I came in Abhyas, my speaking skills developed and now I am the first one to give the answer of teachers question.
Upmanyu Sharma
7thIt was good as the experience because as we had come here we had been improved in a such envirnment created here.Extra is taught which is beneficial for future.
