Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted?
UPSC - 2019
This problem involves determining the number of distinct ways a cube can be painted with two colors (black or white) on its faces. Since a cube is a three-dimensional object, rotations of the cube that result in the same visual pattern are considered the same way of painting. This is a classic problem in combinatorics, specifically using Burnside's Lemma or Polya Enumeration Theorem, but for simple cases like a cube with two colors, it can often be solved by considering the number of faces painted with one color.
A cube has 6 faces. Each face can be painted in 2 colors (black or white).
Without considering rotations, there would be \(2^6 = 64\) ways to paint the cube. However, we need to account for rotational symmetry.
Let's classify the distinct ways based on the number of faces painted black (or white, as the problem is symmetrical for black and white):
0 black faces: All 6 faces are white. There is only 1 way (all white).
1 black face: One face is black, and the other 5 are white. Due to rotational symmetry, all faces are equivalent. So, painting any one face black results in the same configuration. There is only 1 way.
2 black faces: Two faces are black, and the other 4 are white. We need to consider the relative positions of the two black faces:
Adjacent faces: The two black faces share an edge. All pairs of adjacent faces are equivalent by rotation. There is 1 way.
Opposite faces: The two black faces are opposite to each other. All pairs of opposite faces are equivalent by rotation. There is 1 way.
So, for 2 black faces, there are 2 distinct ways.
3 black faces: Three faces are black, and the other 3 are white. We need to consider their relative positions:
Three faces meeting at a vertex: These are mutually adjacent. By rotation, all such configurations are equivalent. There is 1 way.
Three faces forming a line (two opposite, one adjacent to one of them): This configuration is also distinct. Imagine painting the top, bottom, and front faces black. This is distinct from the previous case. There is 1 way.
So, for 3 black faces, there are 2 distinct ways.
4 black faces: This is equivalent to 2 white faces. As established for 2 black faces, there are 2 distinct ways (2 white faces can be adjacent or opposite).
5 black faces: This is equivalent to 1 white face. As established for 1 black face, there is 1 distinct way.
6 black faces: All 6 faces are black. There is only 1 way (all black).
Summing up the distinct ways:
\[ \text{Total ways} = (\text{0 black}) + (\text{1 black}) + (\text{2 black}) + (\text{3 black}) + (\text{4 black}) + (\text{5 black}) + (\text{6 black}) \]\[ \text{Total ways} = 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10 \]B) 10
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:
1. Colours Y, O and B are on adjacent faces.
2. Colours I, G and Y are on adjacent faces.
3. Colours B, G and Y are on adjacent faces.
4. Colours O, V and B are on adjacent faces.
Which is the colour of the face opposite to the face coloured with O?
UPSC - 2015
This problem involves determining the arrangement of colours on a cube's faces based on given adjacency information. We need to identify which colour is opposite to 'O'. We can solve this by listing the adjacent faces for each colour and then deducing the opposite pairs.
Correct Option: 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:
UPSC - 2016
To solve this problem, we need to determine how many smaller cubes have only one face painted. This involves understanding how a larger cube is cut into smaller cubes and identifying the cubes located on the faces but not on the edges or corners of the original cube.
Determine the number of divisions along each side: The side of the small cube is one-fourth the big cube. This means the big cube is cut into \(4\) segments along each of its length, width, and height. So, if the side of the big cube is \(L\), and the side of the small cube is \(l\), then \(l = \frac{1}{4}L\), or \(L = 4l\). This implies that there are \(n=4\) small cubes along each edge of the big cube.
Calculate the total number of small cubes: The total number of small cubes formed is \(n^3\).
\[ \text{Total cubes} = n^3 = 4^3 = 64 \]Identify cubes with only one face painted: Cubes with only one face painted are those located on the faces of the original cube, but not on its edges or corners. For each face of the large cube, the number of small cubes with only one face painted can be found by considering the inner square of cubes on that face. If there are \(n\) cubes along an edge, then removing the cubes along the edges (which have two or three faces painted) leaves \((n-2)\) cubes along each dimension of the inner square.
Calculate cubes with one face painted per face: For one face of the big cube, the number of small cubes with only one face painted is \((n-2)^2\).
\[ \text{Cubes per face} = (4-2)^2 = 2^2 = 4 \]Calculate the total number of cubes with only one face painted: A cube has \(6\) faces. Since each face contributes \(4\) such cubes, the total number of small cubes with only one face painted is:
\[ \text{Total one-face painted cubes} = 6 \times (n-2)^2 = 6 \times 4 = 24 \]B) 24 is the correct number of small cubes with only one side painted, as calculated by \(6 \times (n-2)^2\).
A dice is numbered from 1 to 6 in different ways. If 1 is adjacent to 2, 4 and 6, then which of the following statements is necessarily true?
3 is opposite to 5
2 is opposite to 6
1 is adjacent to 3
3 is adjacent to 5
In a standard die, each face has an opposite face, and four adjacent faces. We are given information about the faces adjacent to '1'. We need to deduce the relationships between other numbers based on this information.
Correct Option: D) 3 is adjacent to 5
Four positions of the same dice are shown. Select the number that will be on the face opposite to the one showing ‘3’.
To determine the number on the face opposite to the one showing '3', we need to analyze the positions of the dice shown in the diagrams. By observing the adjacent faces, we can deduce the number opposite to '3'.
Correct Option: B) 5
Which number is in opposite of 2 ?
To determine which number is opposite to 2 on a standard die, we need to analyze the given diagrams and understand the positioning of numbers on the faces of the die.
Correct Option: C) 3
