CUBES
Note: Question numbers are numbers from the actual exam in the respective years mentioned below
- Each of the six different faces of a cube has been coated with a different colour
i.e., V, I, B, G, Y and O. Following information is given :
- Colours Y, O and B are on adjacent faces.
- Colours I, G and Y are on adjacent faces.
- Colours B, G and Y are on adjacent faces.
- Colours O, V and B are on adjacent faces.
Which is the colour of the face opposite to the face coloured with O?(2015)
(A) B
(B) V
(C) G
(D) I
Answer -C
Let's analyze the given information to determine the color of the face opposite to the face colored with O:
From statement 4, we know that colors O, V, and B are on adjacent faces. Therefore, the face opposite to O cannot be colored with V.
From statements 1, 2, and 3, we have the following information:
Colors Y, O, and B are on adjacent faces.
Colors I, G, and Y are on adjacent faces.
Colors B, G, and Y are on adjacent faces.
Combining these statements, we can conclude that Y, O, B, I, G must form a chain of adjacent faces. Therefore, the face opposite to O cannot be colored with Y, B, or I.
This leaves us with the remaining color G. The face opposite to O must be colored with G.
Therefore, the correct answer is (c) G.
- A cube has all its faces painted with different colours. It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube.The number of small cubes with only one of the sides painted is(2016)
(A) 32
(B) 24
(C) 16
(D) 8
Answer - B
To solve this problem, let's analyze the dimensions of the cube.
Let's assume that the side length of the big cube is "x." The small cubes obtained by cutting the big cube will have a side length of (1/4)x.
The number of small cubes along each side of the big cube is 4, as the side length of each small cube is one-fourth of the big cube's side length.
Since each side of the big cube is painted with a different color, there are 6 different colors in total.
Now, let's consider the small cubes that have only one side painted.
For each face of the big cube, there is a 2x2 square of small cubes with that face painted. Thus, each face contributes 4 small cubes with only one side painted.
Since there are 6 faces on the big cube, the total number of small cubes with only one side painted is 6 x 4 = 24.
Therefore, the correct answer is (b) 24.
- The outer surface of a 4 cm × 4 cm x 4 cm cube is painted completely in red. It is sliced parallel to the faces to yield sixty four 1 cm x 1 cm x 1 cm small cubes. How many small cubes do not have painted faces?(2017)
(A) 8
(B) 16
(C) 24
(D) 36
Answer - A
To solve this problem, we need to analyze the cube and determine the number of small cubes that do not have any painted faces.
The original cube has dimensions of 4 cm × 4 cm × 4 cm, which means it has 4 layers in each dimension.
When the cube is sliced parallel to the faces to yield 64 small cubes, each layer will be divided into 4 small cubes along each dimension. So, each layer will have a total of 4 x 4 = 16 small cubes.
Since the cube has 4 layers, the total number of small cubes is 4 x 16 = 64 small cubes.
Out of these 64 small cubes, the ones that do not have any painted faces are the ones that are located at the center of the larger cube.
The center of the larger cube forms a smaller cube with dimensions of 2 cm × 2 cm × 2 cm. This smaller cube has 2 layers in each dimension, resulting in a total of 2 x 2 x 2 = 8 small cubes.
Therefore, the correct answer is (a) 8 small cubes that do not have painted faces.
- A solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side. How many of the small cubes will have exactly two painted faces?(2018)
(A) 12
(B) 8
(C) 6
(D) 4
Answer - A
To solve this problem, let's first determine the total number of small cubes that can be created from the large cube.
The original solid cube has a side length of 3 cm, so it consists of 3 × 3 × 3 = 27 small cubes with a side length of 1 cm each.
Therefore, the total number of small cubes that can be created from the large cube is 27.
Now let's consider the small cubes that will have exactly two painted faces.
A small cube will have exactly two painted faces if and only if it lies on an edge of the large cube.
The large cube has 12 edges, and each edge will have one small cube lying on it.
Therefore, the number of small cubes that will have exactly two painted faces is equal to the number of edges of the large cube, which is 12.
Hence, the correct answer is (a) 12.
- Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted ?(2019)
(A) 9
(B) 10
(C) 11
(D) 12
Answer - B
Total 10 different ways the cube can be painted
- All 6 sides are black.
- 1 side is white and the remaining 5 sides are black.
- 2 adjacent sides are white and the other 4 sides are black.
- 2 opposite sides are white and the other 4 sides are black.
- 3 sides have the same corner white and the other 3 sides are black.
- 2 opposite sides and 1 center side are white, and the other 3 sides are black.
- 2 adjacent sides are black and the other 4 sides are white.
- 2 opposite sides are black and the other 4 sides are white.
- 1 side is black and the remaining 5 sides are white.
- All 6 sides are white.
- A solid cube is painted yellow, blue and black such that opposite faces are of same colour. The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are big. None of the faces of the bigger cubes is painted blue. How many cubes have only one face painted?(2019)
(A) 4
(B) 6
(C) 8
(D) 10
Answer - C
Let us first find the number of big cubes. Since the cube is cut into 36 cubes, we have:
36 = 4 + 32
So, there are 4 big cubes and 32 small cubes.
Now, let us consider the big cubes. Since none of the faces of the bigger cubes is painted blue, each big cube has exactly one pair of opposite faces painted yellow, and the other pair of opposite faces painted black.
Therefore, each big cube has exactly 2 faces painted. Thus, the 4 big cubes contribute a total of 8 faces that are painted.
Now, let us consider the small cubes. Each small cube has exactly one face painted. Moreover, since opposite faces are of the same color, exactly half of the small cubes have one face painted yellow and the other face painted black, while the other half have one face painted either yellow or black.
Therefore, the number of small cubes with one face painted yellow or black is:
(1/2) × 32 = 16
Hence, the total number of cubes with one face painted is:
16 + 8 = 24
However, we have counted each face of the big cubes twice, since each big cube has two faces painted. Therefore, we need to subtract the number of faces of the big cubes from the total:
24 - 8 = 16
Thus, there are 16 small cubes that have only one face painted. However, we need to divide this number by 2, since each small cube has one face painted either yellow or black. Therefore, the number of cubes with only one face painted is:
16/2 = 8
So, the answer is (c) 8.
