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Colouring and Cutting of Cubes

A cube is a three-dimensional solid. The three dimensions of the cube—length (l), breadth (b) and height (h)—are equal, and form the edges of the cube. Examples of a cube are ice cube sugar cube and dice.

Properties of A Cube

A cube is a three-dimensional solid having 8 corners, 12 edges, 4 diagonals and 6 faces. All the edges of a cube are equal, and all the faces of the cube are square in shape.
 
Description: 87408.png
 
In the adjacent figure,
  1. The eight corners are A, B, C, D, E, F, O and P.
  2. The twelve edges are OA, OC, OE, EB, AB, FA, FP, FC, CD, DP, DE and PB.
  3. The four diagonals (all equal) are OP, AD, BC and EF.
  4. The six faces (all squares) are FPBA, FCDP, CDEO, PDEB, AOEB and FCOA.
If a solid wooden cube is cut into smaller cubes of equal size, then the total number of smaller cubes is always equal to a cube of a natural number (i.e., 8, 27, 64, 125, etc.). If a cube is painted on all of its faces with a colour and thereafter cut into smaller cubes of equal size, then the number of smaller cubes so obtained, along with the number of their faces painted, can be calculated as follows:
  1. Number of smaller cubes with 3 adjacent faces painted = 8
  2. Number of smaller cubes with 2 adjacent faces painted = (n – 2) × 12
  3. Number of smaller cubes with only 1 face painted = (n – 2)2 × 6
  4. Number of smaller cubes with no face painted = (n – 2)3
where Description: 87435.png
 
If a cube is painted on all of its faces and thereafter cut into smaller cubes, we find that
 

Description: 87445.png 

  1. The smaller cubes with 3 adjacent faces painted are present on the corners of the big cube.
  2. The smaller cubes with 2 adjacent faces painted are present along the edges of the big cube.
  3. The smaller cubes with only 1 face painted are present on the faces of the big cube.
  4. The smaller cubes with no face painted are present inside the big cube.
The smaller cubes have paints only on their adjacent faces. To be more specific, no smaller cube can have paint on its opposite faces. Further, the maximum number of faces a smaller cube can have paint is three only.





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