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A cube is a three-dimensional structure with six faces, eight corners and twelve edges. A cube is composed of six square faces that meet each other at right angles. Let us see how the six different faces of a cube can be represented:




And finally the cube appears like the following:

Generally, the questions asked regarding cubes in LR pertains to finding out the number of cubelets being formed from the original cube by cutting it into several pieces. However, sometimes we might be asked to find out the total number of cuts formed from the total cubelets by cutting the original cube.

Cutting the Cubes

Before moving on to solving questions, we should be clear with the basics about what happens when we cut a cube:
  • One cut divides the cube into two parts.
  • The second cut divides the cube in either a total of three parts or four parts, depending upon the axis of the cut.
  • The third cut divides the cube in either a maximum of eight parts or a minimum of four parts.
A carpenter had a large wooden cube with a side length 4 inches. He wanted to cut it into 64 smaller cubes with a side length 1 inch. What is the least number of cuts required if
  1. he can rearrange the pieces before each cut.
  2. the rearrangement of the pieces before/after making the cut is not allowed?
Let us understand the difference between the two questions first: In first question (i), we are allowed to move the cube one over the other or we can stack the pieces of the cube side by side or on top of each other, whereas in the second question (ii), we have to assume that the cube is fixed on the horizontal surface and what we can do at best is to make cuts along any of its surfaces.
  1. When rearrangement is allowed, the minimum is found by cutting each edge as nearly in half as possible; putting the pieces together and cutting as nearly in half again until we obtain a solid with a unit dimension. We would start hereby making a cut midway on all the axis.
    And now this is what we will obtain as a unit:
    Now restack the solids into the 4 × 4 × 4 solid and repeat the procedure. After completing the three sides, we will have 1 × 1 × 1 cubes. Therefore, the sum of the cuts which is six is the answer.
  2. When rearrangement is not allowed:
    We know that making n cuts along one axis divides the cube into (n + 1) parts. To obtain 64 cubelets by making a minimum number of cuts, we should be making the cuts along all the axis.
    Assume we have made n, m, p cuts along three axi s.
    So, the number of cubelets formed = (n + 1) (m + 1) (p + 1) = 64
    To minimize the number of cuts, (n + 1) = (m + 1) = (p + 1) = 4
    So, n = m = p = 3, hence, a total of 9 cuts.
    Alternatively, to make the minimum number of cuts, they should be made symmetrically.
    Look at this figure:
    It has 64 cubelets of size 1 × 1 × 1, and the total number of cuts made = 9.

Painting the Cubes and then Cutting the Cubes

If we paint a cube of the dimension n × n × n in any one colour and cut it to get n3 symmetric cubelets; then the number of cubelets with colour on different faces can be categorized as follows:
  • Cubelets with only one face painted = (n - 2)2 × 6
  • Cubelets with two faces painted = (n - 2) × 12
  • Cubelets with three faces painted = 8
  • Cubelets with no face painted = (n - 2)3
Examples 1 to 4 :
Directions for questions 1 to 4: Read the following passage and solve the questions based on it.
64 symmetrical small cubes are put together to form a big cube. This cube is now coloured green on all its surfaces.
How many of the smaller cubes have none of their faces coloured?
From the given 4 × 4 × 4 cube, if we remove one layer from the top making it a 2 × 2 × 2 cube, it will not be coloured.
Hence, 8 small cubes will not be coloured on any of their surfaces.
How many of the smaller cubes have exactly three faces coloured?
In the figure look at the three faces coloured small cubes:
All the corner cubes (blackened) will have exactly three faces coloured.
These are 8 in number.
Remember, for any n × n × n dimension (n $ 2), the number of cubes which have exactly three faces coloured = 8
How many of the smaller cubes have exactly two faces coloured?
In the figure look at the two faces coloured small cubes:
We can see that along every edge, there are two cubes painted with two colours.
So, the total number of small cubes painted on exactly two of their faces = 2 × 12 = 24
How many of the smaller cubes have exactly one face coloured?

It can be seen from the above figure that the total number of cubes coloured on only one of their faces = 4 × 6 = 24
Alternatively, the total number of small cubes = Total number of cubes painted on (one face + two faces + three faces + no face).
So, the total number of cubes painted on only one of their faces = 64 - 8 - 8 - 24 = 24

Example 5:

In this question, three views of a cube are given. If the same cube is rotated in a particular way, it will give rise to different views. Four such views are given in the options. However, out of the four options given, one of the options does not conform to the original cube. Mark that option as your answer. (The letters used are only to mark the different faces of the cube.)
From the given figure, it can be inferred that the four faces adjacent to face A are—B, C, D and E.
Hence, face ‘F’ cannot be adjacent to face ‘A’.
Hence, option (c) is wrong.

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