A 3 ×3 ×3 cube is made up of 27 unit cubes (a unit cube has...

A $3 \times 3 \times 3$ cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure. (FIGURE CANNOT COPY) (a) Determine how many unit cubes of the $3 \times 3 \times 3$ cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces. (b) Repeat part (a) for a $4 \times 4 \times 4$ cube, a $5 \times 5 \times 5$ cube, and a $6 \times 6 \times 6$ cube. (c) Write formulas you could use to repeat part (a) for an $n \times n \times n$ cube.

A $3 \times 3 \times 3$ cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure.
(FIGURE CANNOT COPY)
(a) Determine how many unit cubes of the $3 \times 3 \times 3$ cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces.
(b) Repeat part (a) for a $4 \times 4 \times 4$ cube, a $5 \times 5 \times 5$ cube, and a $6 \times 6 \times 6$ cube.
(c) Write formulas you could use to repeat part (a) for an $n \times n \times n$ cube.

Key Concepts

Cube Geometry

This concept deals with the structure and properties of cubes, including how a large cube can be divided into smaller unit cubes. Understanding the spatial arrangement of the unit cubes in a three-dimensional grid is key to analyzing which of them have surfaces that are visible (or painted) and which are hidden inside.

Categorization of Unit Cubes

In problems like this, the unit cubes are classified based on their location in the larger cube—specifically, into corners (with three exposed faces), edges (excluding corners, having two exposed faces), face-centers (with one exposed face), and interior cubes (with no exposed faces). This classification is essential for determining how many cubes fall into each category.

Combinatorial Counting

This concept involves applying combinatorial methods and systematic counting techniques to determine the number of unit cubes in each category. By identifying patterns and symmetries in the cube's structure, one can develop formulas to count the cubes with a certain number of painted faces without listing each cube individually.

Generalization to n × n × n Cubes

Generalizing the problem to an n × n × n cube requires expressing the counts of unit cubes in each category as functions of n. For instance, interior cubes (with zero painted faces) can be counted as (n-2)^3, while similar expressions can be derived for cubes with one, two, and three painted faces by considering the geometry of faces, edges, and vertices respectively.

Spatial Visualization

This key concept refers to the ability to mentally visualize the three-dimensional structure of the cube and the location of each unit cube within it. Spatial visualization enables one to understand how many faces of each unit cube are exposed based on its position, which is crucial for solving the problem correctly.

Symmetry and Invariance

Leveraging the inherent symmetry of the cube simplifies the counting process. By recognizing that each face, edge, or corner of the cube is equivalent under rotations and reflections, the problem can be broken down into simpler, repeatable parts. This symmetry allows for the derivation of generalized formulas that work for any n × n × n cube.

Transcript

00:02 Now, in this problem of series, we have given that a three -sided cube that is here, 3 is 2 3 is 2 3.

00:11 Is 2 .7 unit cube.

00:13 So these are made of 27 unit cube.

00:17 So the total number of cubes in the bigger cube is 27.

00:21 A unit cube has length, width and height of one unit.

00:26 So each one is of height of here the side of one unit.

00:29 And only the faces of each cube that are visible are painted blue.

00:35 So only here faces that are visible are painted blue.

00:49 So that means say here i'm drawing it.

00:55 So this is a cube and all the visual faces are painted blue.

01:00 So we have to determine how many unit cubes cubes of the 3 is to 3 is to 3 have 0 blue faces so that means the cube the 3 is 2 3 have 0 blue faces so that means here that means all the cubes here are have been painted except at the center one that means this one so here we can write it as 0 blue face is equals to one cube and now one blue faces so that means only one blue face so that means one blue face so each side this this this this and this so total six faces so this whole total six cubes and now two blue faces so that means here two blue faces that means this one this one so that means in each phase there are a number of two blue faces.

02:14 That means that have two blue faces.

02:18 So the total faces are six.

02:19 So this will be in 12 way number.

02:23 And now three.

02:24 So here each one, two three, one to three at each corner.

02:29 So there are total eight corners.

02:32 So this would be here.

02:35 Here, total eight cube...