Cubes
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 alongside an edge (or side) of a room, all the exposed surfaces( in this case there are 4) are painted.
 How many of the cubes have 0 faces painted?

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Out of 6 faces of 4 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 2
Number of vertices with 2 faces exposed (Painted) is 4
Number of vertices with 1 faces exposed (Painted) is 2
Number of vertices with 0 faces exposed (Painted) is 0
Number of sides with 2 sides exposed (Painted) is 5
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 1
From the above observation:
Number of cubes with 3 faces Painted is 2
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and required number of cubes is 6 x 4 + 1 x 5 = 29 since there are 4 edges will give us 6 cubes from 1 edge and 1 edge (between two vertices which are painted or exposed from 3 sides) will give us only 5 cubes.
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and required number of cubes is 36 x 2 + 30 x 2 = 132
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  2  29  132 = 180
In other words number of cubes with 0 painted is 6 x 6 x 5 =180
From the above explanation number of the cubes with 0 faces painted is 180.Correct Option: D
Out of 6 faces of 4 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 2
Number of vertices with 2 faces exposed (Painted) is 4
Number of vertices with 1 faces exposed (Painted) is 2
Number of vertices with 0 faces exposed (Painted) is 0
Number of sides with 2 sides exposed (Painted) is 5
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 1
From the above observation:
Number of cubes with 3 faces Painted is 2
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and required number of cubes is 6 x 4 + 1 x 5 = 29 since there are 4 edges will give us 6 cubes from 1 edge and 1 edge (between two vertices which are painted or exposed from 3 sides) will give us only 5 cubes.
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and required number of cubes is 36 x 2 + 30 x 2 = 132
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  2  29  132 = 180
In other words number of cubes with 0 painted is 6 x 6 x 5 =180
From the above explanation number of the cubes with 0 faces painted is 180.
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 at the corner of a room, all the exposed surfaces are painted then:
 How many of the cubes have 3 faces painted?

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Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
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  1 18  108 = 216
In other words number of cubes with 0 painted is (7  1)^{3} = 216.
From the above explanation number of the cubes with 3 faces painted is 1.Correct Option: D
Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
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  1 18  108 = 216
In other words number of cubes with 0 painted is (7  1)^{3} = 216.
From the above explanation number of the cubes with 3 faces painted is 1.
 How many of the cubes have at least 2 faces painted?

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Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
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  1 18  108 = 216
In other words number of cubes with 0 painted is (7  1)^{3} = 216.
From the above explanation number of the cubes with at least 2 faces painted is 18 + 1 = 19.Correct Option: A
Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
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  1 18  108 = 216
In other words number of cubes with 0 painted is (7  1)^{3} = 216.
From the above explanation number of the cubes with at least 2 faces painted is 18 + 1 = 19.
 How many of the cubes have at most faces painted?

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Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
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  1 18  108 = 216
In other words number of cubes with 0 painted is (7  1)^{3} = 216.
From the above explanation number of the cubes with at most 2 faces painted is 216 + 108 + 18 = 342.
Or else 343  1 = 342Correct Option: C
Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
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  1 18  108 = 216
In other words number of cubes with 0 painted is (7  1)^{3} = 216.
From the above explanation number of the cubes with at most 2 faces painted is 216 + 108 + 18 = 342.
Or else 343  1 = 342
 How many of the cubes have 2 faces painted?

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Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
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  1 18  108 = 216
In other words number of cubes with 0 painted is (7  1)^{3} = 216.
From the above explanation number of the cubes with 2 face painted is 18.Correct Option: B
Out of 6 faces of 3 faces are exposed and those were painted.
Number of vertices with three faces exposed (Painted) is 1
Number of vertices with 2 faces exposed (Painted) is 3
Number of vertices with 1 faces exposed (Painted) is 3
Number of vertices with 0 faces exposed (Painted) is 1
Number of sides with 2 sides exposed (Painted) is 3
Number of sides with 1 sides exposed (Painted) is 6
Number of sides with no sides exposed (Painted) is 3
From the above observation
Number of cubes with 3 faces Painted is 1
Number of cubes with 2 faces Painted is given by sides which is exposed from two sides and there are 3 such sides and from one we will get 6 such cubes hence required number of cubes is 6 x 3 = 18
Number of cubes with 1 face Painted is given by faces which is exposed from one sides and there are 3 such faces hence required number of cubes is 36 x 3 = 108
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  1 18  108 = 216
In other words number of cubes with 0 painted is (7  1)^{3} = 216.
From the above explanation number of the cubes with 2 face painted is 18.