Cubes Quiz 1

Ans - P

Pair of opposite sides are (L, O), (M, P) and (N, Q)

Ans - (C)
I and II have same pairs of opposite sides

Ans - (B)
II and IV have same pairs of opposite sides

Ans = 8
For a cube of side N x N x N painted on all sides which is uniformly cut into smaller cubes of dimension 1x1x1,

Number of cubes with 0 side painted= (N-2) ^{3}
Here N = 4

Number of cubes with 0 side painted = (4-2) ^{3}
= (2) ^{3}
= 8

For whatever size of large cubes, if 3 sides of a smaller cube is to be painted, that cube must belong to a corner piece of the larger cube. A cube has only 8 corners. So the number of smaller cube with three sides painted = 8

For a cube of side N x N x N painted on all sides which is uniformly cut into smaller cubes of dimension 1x1x1,

Number of cubes with 2 side painted= 12(N-2)
Here N = 3
SoNumber of cubes with 2 side painted= 12(3-2)
= 12

For a cube of side N x N x N painted on all sides which is uniformly cut into smaller cubes of dimension 1x1x1,

Number of cubes with 2 side painted = 6 \times (N-2)^{2}

Here N = 5
So
Number of cubes with 2 side painted = 6 \times (5-2)^{2} = Number of cubes with 2 side painted = 6 \times (3)^{2}
= 6 x 9
= 54

Ans - (C)

II and IV can’t form a closed volume with 6 faces

Number of cubes with at least two side painted = Number of cubes with at two side painted + Number of cubes with at three side painted

For a cube of side N x N x N painted on all sides which is uniformly cut into smaller cubes of dimension 1x1x1,
Number of cubes with 2 side painted = 6 \times (N-2)^{2}

Here N = 4

Number of cubes with 2 side painted = 6 \times (N-2)^{2} = 6 \times (4-2)^{2}
= 6 \times (2)^{2} = 24

No of cubes with three sides painted = 8 (from each vertex of larger cube)

Number of cubes with at least two side painted = 24 + 8
= 32

Ans - (A) 3

I, V and VII are standard dice

For a standard dice, sum of values on opposite sides will be 7
So 2 must appear on base of standard cube