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 two faces painted red and black 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 three faces painted ? | |||
| 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 three faces painted with different colours ? | |||
| Right Option : C | |||
| View Explanation | |||
Students / Parents Reviews [20]
I 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
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Archit Segal
7thWe started with lot of hope that Abhyas will help in better understnding of complex topics of highers classes. we are not disappointed with the progress our child has made after attending Abhyas. Though need to mention that we expected a lot more. On a scale of 1-10, we would give may be 7.
Manya
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Om Umang
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Aman Kumar Shrivastava
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Aman Kumar Shrivastava
10thAbhyas institute is one of the best coaching institute in the vicinity of Ambala cantt.The institute provides good and quality education to the students.The teachers are well experienced and are very helpful in solving the problems. The major advantages of the institute is extra classes for weak...
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Sharandeep Singh
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Shreya Shrivastava
8thAbhyas 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
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