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 | |||
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