Cubes
Direction: 216 cubes of similar size are arranged in the form of the bigger cube (6 cubes on each side, i.e.., 6 x 6 x 6).Its all the 6 faces are painted with Green, Red, blue, black, white, orange colours.
- Which of the following statement is correct
(i) At least 1 cube is painted with red, green and blue.
(ii) At most 1 cube is painted with red, green and blue.
(iii) At most 6 cubes are painted with red and green.
(iv) At least 6 cubes are painted with red and green.
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Here on each face 6 x 6 = 36 cubes that are painted with one colour.
From solution of previous questions statements (ii) and (iii) are correct.Correct Option: B
Here on each face 6 x 6 = 36 cubes that are painted with one colour.
From solution of previous questions statements (ii) and (iii) are correct.
- How many cubes are painted with red, blue, green and black?
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Here on each face 6 x 6 = 36 cubes that are painted with one colour.
None of the cubes can be painted in four faces.Correct Option: D
Here on each face 6 x 6 = 36 cubes that are painted with one colour.
None of the cubes can be painted in four faces.
Direction: 343 cubes of similar size are arranged in the form of a bigger cube (7 cubes on each side, i.e., 7 x 7 x 7) and kept on the surface of a room, all the exposed surfaces( in this case there are 5) are painted.
- How many of the cubes have 0 faces painted?
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Out of 6 faces of 5 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 4
Number of vertices with 2 faces exposed (Painted) is 4
Number of vertices with 1 faces exposed (Painted) is 0
Number of vertices with 0 faces exposed (Painted) is 0
Number of sides with 2 sides exposed (Painted) is 8
Number of sides with 1 sides exposed (Painted) is 4
Number of sides with no sides exposed (Painted) is 0
From the above observation:
Number of cubes with 3 faces Painted is 4
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides, out of 8 such edges 4 vertical edges will give us 6 cubes per edge and 4 edges from top surface will give us 5 such cubes from each edge and required number of cubes is 6 x 4 + 4 x 5 = 44.
Number of cubes with 1 face Painted is given by faces which is exposed from one sides four vertical faces will give us 6 x 5 = 30 cubes per face and top face will give us 5 x 5 = 25 and required number of cubes is 30 x 4 + 25 x 1 = 145
Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 4 - 44 - 145 = 150
In other words number of cubes with 0 painted is 6 x 5 x 5 = 150
From the above explanation number of the cubes with 0 faces painted is 150.Correct Option: B
Out of 6 faces of 5 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 4
Number of vertices with 2 faces exposed (Painted) is 4
Number of vertices with 1 faces exposed (Painted) is 0
Number of vertices with 0 faces exposed (Painted) is 0
Number of sides with 2 sides exposed (Painted) is 8
Number of sides with 1 sides exposed (Painted) is 4
Number of sides with no sides exposed (Painted) is 0
From the above observation:
Number of cubes with 3 faces Painted is 4
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides, out of 8 such edges 4 vertical edges will give us 6 cubes per edge and 4 edges from top surface will give us 5 such cubes from each edge and required number of cubes is 6 x 4 + 4 x 5 = 44.
Number of cubes with 1 face Painted is given by faces which is exposed from one sides four vertical faces will give us 6 x 5 = 30 cubes per face and top face will give us 5 x 5 = 25 and required number of cubes is 30 x 4 + 25 x 1 = 145
Number of cubes with 0 face Painted is given by difference between total number of cubes - number of cubes with at least 1 face painted = 343 - 4 - 44 - 145 = 150
In other words number of cubes with 0 painted is 6 x 5 x 5 = 150
From the above explanation number of the cubes with 0 faces painted is 150.
Direction: 125 cubes of similar size are arranged in the from of a bigger cube (5 cubes on each side, i. e., 5 x 5 x 5). From on corner of the top layer of this cube, four smaller cubes (2 x 2 x 1) are removed. From the column on the opposite side, two cubes (1 x 1 x 2) are removed, and from the third corner,three cubes (1 x 1 x 3) are removed and from the fourth column four cubes (1 x 1 x 4) are removed. All exposed faces of the block thus formed are coloured red.
- How many cubes have only two coloured faces?
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Only two faces are coloured is when cubes are at the edges (baring the corner cubes)
If no cubes have been removed then on each edges we will get 3 cubes that has exactly 2 faces coloured, hence total number of such cubes = 12 x 3 = 36, because we have 12 edges.
Out of these 3 cubes are removed hence required number of cubes = 36 - 3 = 33Correct Option: A
Only two faces are coloured is when cubes are at the edges (baring the corner cubes)
If no cubes have been removed then on each edges we will get 3 cubes that has exactly 2 faces coloured, hence total number of such cubes = 12 x 3 = 36, because we have 12 edges.
Out of these 3 cubes are removed hence required number of cubes = 36 - 3 = 33
- How many cubes do not have any coloured face?
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In any plane, leave 4 sides cube and select (3 x 3 x 3) inter section. But
the cubes 2 x 2 x 1 give 2 less cube because that part we are already removed.
No. of cubes = (3 x 3 x 3) - 2 = 25.Correct Option: C
In any plane, leave 4 sides cube and select (3 x 3 x 3) inter section. But
the cubes 2 x 2 x 1 give 2 less cube because that part we are already removed.
No. of cubes = (3 x 3 x 3) - 2 = 25.
