Formulas For Cube

Important Formulas for Cube

1. For a Cube of side n x n x n painted on all sides which is uniformly cut into smaller cubes of dimension 1 x 1 x 1,

• Number of cubes with 0 side painted = (n-2) ^{3}
• Number of cubes with 2 sides painted = 12(n-2)
• Number of cubes with 1 sides painted = 6(n – 2) ^{2}
• Number of cubes with 3 sidess painted = 8(always)

2. Rotation Formulas:

Single Rotations:

• Clockwise Rotation by 90 degrees (x-axis):
  (x, y, z) -> (x, -z, y)
• Counterclockwise Rotation by 90 degrees
  (x-axis): 
  (x, y, z) -> (x, z, -y)
• Clockwise Rotation by 90 degrees (y-axis): 
  (x, y, z) -> (z, y, -x)
• Counterclockwise Rotation by 90 degrees (y-axis):   
  (x, y, z) -> (-z, y, x)
• Clockwise Rotation by 90 degrees (z-axis):   
  (x, y, z) -> (-y, x, z)
• Counterclockwise Rotation by 90 degrees (z-axis): 
   (x, y, z) -> (y, -x, z)

Combined Rotations:

• Clockwise Rotation by 180 degrees (x-axis):
  (x, y, z) -> (x, -y, -z)
• Clockwise Rotation by 180 degrees (y-axis):
  (x, y, z) -> (-x, y, -z)
• Clockwise Rotation by 180 degrees (z-axis):
  (x, y, z) -> (-x, -y, z)

3. Hidden Face Formula:
If you need to determine the face opposite a given face (when the cube is unfolded), you can use this formula:

Opposite Face of F = F ± 3

Where F is the face number (1 for front, 2 for right, 3 for back, etc.). The ± sign indicates that you can add or subtract 3 to get the opposite face.

4. Percentage Formula:
When calculating percentages related to cubes, use the formula:

Percentage =\frac{Part}{Whole}\times100

Where “Part” is the number of units in question (like the number of cells with a certain color or pattern), and “Whole” is the total number of units in the cube (total number of cells).

5. Counting Formula:
For counting specific attributes within a cube (like the number of cells with a particular color or pattern):

Count = Number of Units with Attribute