How many of the cubes have at least 2 faces painted?

Correct Answer: 342

Explanation:

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

Correct Answer: 18

Explanation:

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 Answer: None of these

Explanation:

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 0 faces painted is 216.

Correct Answer: 144

Explanation:

Since total number of cubes is hence in the formula we will substitute n = 6
At least 2 faces painted means number of cubes with 2 face painted + number of cubes with 3 face painted = 96 + 8 = 104.

Correct Answer: 208

Explanation:

Since total number of cubes is hence in the formula we will substitute n = 6
At most 2 faces painted means number of cubes with 0 face painted + number of cubes with 1 face painted + number of cubes with 2 face painted = 64 + 48 + 96 = 208

Correct Answer: None of these

Explanation:

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 Answer: None of these

Explanation:

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 Answer: 29

Explanation:

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 2 faces painted is 29.

Correct Answer: 341

Explanation:

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 at most 2 faces painted is
180 + 132 + 29 = 341.
Or else 343 - 2 = 341

Correct Answer: 31

Explanation:

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 at least 2 faces painted is 29 + 2 = 31.