Important Concepts: Cubes & Dices - CAT PDF Download

Introduction

• Questions related to the visualization of cubes and dice frequently appear in various management entrance exams and other aptitude tests. Proficiency in solving these questions relies on the ability to mentally visualize different shapes in three dimensions, along with understanding how the faces of a cube or dice would align when cut open.
• To excel in these problems, one must possess skills in conceptualizing and manipulating shapes in their mind, anticipating the placement of faces when the object is unfolded or cut. Additionally, it is essential to comprehend the logical process involved in visualizing how the faces of an open cube would come together to form a complete cube.
• Familiarity with the following visualizations is crucial as part of the foundational knowledge in this subject area.

Visualising the Opening and Closing of Cubes

The accompanying figures represent open cubes, and the accompanying text specifies which faces would align opposite each other when the cube is folded into a closed box.

Cutting a Cube

A cube has 6 faces, 12 edges and 8 vertices.

When 'a' parallel cuts are made along the x-axis, 'b' parallel cuts along the y-axis, and 'c' parallel cuts along the z-axis, the resulting number of pieces for the original cube is given by (a + 1) × (b + 1) × (c + 1). Whether these pieces are cubes or not depends on the equidistance of the cuts and the equality of a, b, and c.

Three cases arise:

• Case 1: All pieces are cubes only when a = b = c, and all cuts are equidistant.
• Case 2: Not all pieces are cubes when a = b = c, and the cuts are not equidistant.
• Case 3: Not all pieces are cubes when a ≠ b ≠ c.

Key Learning Point: To maximize the total number of pieces with the minimum number of cuts, it is essential to minimize the difference between the number of cuts along any two axes.

Additional Learning Point: To minimize the total number of pieces, one should aim to maximize the difference between the number of cuts along any two axes.

Painting and Cutting a cube

When confronted with the task of visualizing the distribution of painted faces on a cube after making cuts along its axes, specific formulae come in handy. Although it is beneficial to memorize these formulae, it is equally important to develop the ability to mentally picture these outcomes when faced with a real problem.

1. Cutting equidistant along each axis with an equal number of cuts

Consider a cube painted black, and if we make n cuts along each axis, the distribution of smaller cubes with varying numbers of painted faces can be determined using specific formulas:

• Number of cubes with 3 faces painted black: 8 (corresponding to all corner cubes).

• Number of cubes with 2 faces painted black: (n - 1) × 12.

• Number of cubes with 1 face painted black: (n - 1)² × 6.

• Number of cubes with no face painted black: (n - 1)³.

For instance, if we make two equidistant cuts on each edge of the cube (n = 2), we would obtain 27 smaller cubes. Out of these 27 cubes, 8 would have three sides painted black, 12 would have two sides painted black ((n - 1) × 12 = (2 - 1) × 12 = 12), 6 would have one side painted black ((n - 1)² × 6 = (2 - 1)² × 6 = 6), and 1 cube would have no sides painted ((n - 1)³ = (2 - 1)³ = 1).

For n = 3, resulting in 64 smaller cubes, the distribution would be as follows:

• Cubes with 3 sides painted: 8.

• Cubes with 2 sides painted: (n - 1) × 12 = (3 - 1) × 12 = 24.

• Cubes with 1 side painted: (n - 1)² × 6 = 24.

• Cubes with no side painted: (n - 1)³ = (3 - 1)³ = 8.

Visualizing these cases aids in better understanding and mastery of the problem.

2. Cutting equidistant along each axis with an unequal number of cuts

In scenarios where the number of cuts differs, for instance, "a" cuts on the X-axis, "b" cuts on the Y-axis, and 'c' cuts on the Z-axis, the resulting smaller pieces are not cubes but cuboids. Consequently, questions may inquire about determining the number of cuboids with specific painted face configurations.

• Number of cuboids with 3 faces painted: Consists of all corner cuboids, totaling 8 cuboids.

• Number of cuboids with exactly 2 faces painted: (a - 2) × 4 + (b - 2) × 4 + (c - 2) × 4.

• Number of cuboids with exactly 1 face painted: 2 × [(a - 2)(b - 2) + (b - 2)(c - 2) + (a - 2)(c - 2)].

• Number of cuboids with 0 faces painted: Corresponds to the inner cuboid and equals (a - 2) × (b - 2) × (c - 2).

It's important to visualize a scenario where the faces of the cuboids are not uniformly painted but have different colors, with each color used to paint two faces.

Key Concepts

Here are the key observations derived from the given information:

• Cubes with all three faces painted in different colors: Two cubes positioned diagonally opposite to each other.

• Cubes with only the same color on two faces: Three edges without corner cubes, each edge colored yellow, totaling 3 × (n - 1).

• Total cubes on the edges: Comprising 4 × (n + 1) cubes on the outer edges and 8 × (n - 1) cubes on the inner edges.

• Cubes with exactly two colored faces: Calculated by subtracting the cubes with all three color faces from the total cubes on edges.

• Cubes with exactly one face painted red: Found on the two faces of the red-painted surface, totaling 2 × (n - 1).

• Cubes with faces only painted red: Sum of the cubes with a single red face and those with two red faces, resulting in (n - 1) + 2 × (n - 1)².

• Cubes with exactly one face painted: Calculated as 6 × (n - 1)².

• Cubes with exactly two surfaces painted with different colors: 9 × (n - 1).