CUBES AND DICE

Understanding cubes and dice is crucial for solving problems in the UPSC CSAT. These problems test spatial visualization and logical reasoning skills. This content will help you grasp the fundamental concepts and techniques to tackle cube and dice-related questions efficiently.

Cubes

Cubes are three-dimensional geometric figures with six faces, all of which are squares of the same size. Questions about cubes in the CSAT often involve counting, visualizing cuts, and understanding patterns.

Key Concepts

1. Basic Properties:

   • A cube has 6 faces, 12 edges, and 8 vertices.
   • Opposite faces are parallel and equal.

2. Cutting a Cube:

   • When a cube is cut into smaller cubes, understanding the number of smaller cubes formed is essential.
   • If a cube of side length 'a' is cut into smaller cubes of side length 'b', the number of smaller cubes formed is (a/b)³

3. Painted Cubes:

   • Often, problems involve a cube painted on some or all of its faces and then cut into smaller cubes.
   • Key points include determining how many smaller cubes have paint on zero, one, two, or three faces.
     • Cubes with 3 faces painted: Only the corner cubes have three painted faces.
     • Cubes with 2 faces painted: These are along the edges, excluding the corners.
     • Cubes with 1 face painted: These are on the faces but not along the edges or corners.
     • Cubes with no faces painted: These are entirely inside the larger cube.

Example Problems

1. Problem: A cube with a side of 4 cm is painted on all its faces and then cut into 1 cm smaller cubes. How many smaller cubes will have exactly two faces painted?
   • Solution: The original cube is cut into $64$ smaller cubes.
     • Cubes with 2 faces painted are along the edges, excluding the corners. Each edge of the cube has (4−2)=2 such smaller cubes.
     • Since a cube has 12 edges, the total number of such smaller cubes is 12×2=24

Dice

Dice problems involve understanding the layout and numbering on a six-faced die. Standard dice have faces numbered from 1 to 6, with opposite faces summing up to 7.

Key Concepts

1. Standard Die:

   • Opposite faces: 1-6, 2-5, 3-4.
   • The sum of numbers on opposite faces is always 7.

2. Dice Configurations:

   • Single Die: Determining the opposite face when one face is known.
   • Two Dice: Understanding possible outcomes when rolling two dice and calculating sums, differences, or other conditions.

3. Rotations and Views:

   • Dice can be rotated, and problems may involve determining which face is opposite, adjacent, or not visible given a particular view.
   • Understanding rotations is essential for problems involving multiple views of the same die.

TYPE 1 – SINGLE COMMON FACE

The diagram below illustrates the same die from three different perspectives:

Now, let's consider a question: Determine the face of the die opposite to the one with 1 dot. Notice that in Figure (1) and Figure (2), the face with 5 dots is the only shared face. To address this question, we represent 5 in two rows, one below the other, as follows (since the face with 5 dots is the sole common face in Figure (1) and Figure (2)):

Row 1: 5 __ __

Row 2: 5 __ __

In Figure (1), moving clockwise from the face with 5 dots, we list the numbers in the first row. We note that clockwise from the face with 5 dots, we encounter the face with 4 dots, followed by the face with 6 dots. Hence, we record these numbers in Row 1:

Row 1: 5 4 6

Row 2: 5 __ __

In Figure (2), we again begin moving clockwise from the face with 5 dots. As we progress, we come across the face with 3 dots, succeeded by the face with 1 dot. Sequentially, we list them in Row 2:

Row 1: 5 4 6

Row 2: 5 3 1

In Row 1 and Row 2, the numbers in the second and third columns will be opposite to each other, meaning the face with 4 dots will be opposite the face with 3 dots, and the face with 6 dots will be opposite the face with 1 dot.

Applying the same method, you can utilize Figure (2) and Figure (3) to determine the face opposite to the one with 2 dots. Hint: Figure (2) and Figure (3) share a common face with 3 dots

 

 

TYPE 2 – TWO COMMON FACES

 

Let's examine another cube displaying two distinct views:

 

Now, let's address a query: Determine the face opposite to the one with 5 dots.

Upon observing Figure (4) and Figure (5), we note that the faces with 1 dot and 3 dots are shared in both views. When two views have 2 common faces, the third faces are invariably opposite. Therefore, in Figure (4) and Figure (5), the faces with 1 and 3 dots serve as the common faces, while the faces with 4 and 5 dots will be opposite each other

 

Practice Questions

 

Cubes

1. Question: A cube of side 6 cm is painted red on all its faces and then cut into smaller cubes of side 1 cm. How many smaller cubes will have exactly one face painted red?

2. Question: If a cube has 64 smaller cubes of equal size within it, what is the length of each side of the smaller cubes?

3. Question: A cube is painted blue on all its faces and then cut into 125 smaller cubes. How many smaller cubes will have exactly two faces painted blue?

Dice

4. Question: If two standard six-faced dice are rolled, what is the probability of getting a sum of 8?

5. Question: In a standard die, if the face with 2 dots is adjacent to the face with 5 dots, what is the number on the face opposite to the face with 2 dots?

6. Question: Three views of a cube are shown below:

   Based on the given views, which number is opposite the face with 1 dot?

7. Question: Four views of a cube are shown below:

   Determine the face opposite the face with 4 dots.