Cube Cutting

Concepts -

Cube has 6 faces, 8 vetices and 12 edges Number of cubes of size yyy formed after cutting a large cube of size xxx

= (x/y)^3^ If unit cubes than number of unit cubes formed after cutting a cube of size x = x^3^

Shortcut Formulae

For a cube of side nnn painted on all sides which is uniformly cut into smaller cubes of dimension 111,
- Number of cubes with 0 side painted= (n-2) ^3
- Number of cubes with 1 sides painted =6(n - 2) ^2
- Number of cubes with 2 sides painted= 12(n-2)
- Number of cubes with 3 sidess painted= 8(always)
For a cuboid of dimension abc painted on all sides which is cut into smaller cubes of dimension 111,
- Number of cubes with 0 side painted= (a-2) (b-2) (c-2)
- Number of cubes with 1 sides painted =2 [(a-2) (b-2) + (b-2)(c-2) + (a-2)(c-2) ]
- Number of cubes with 2 sides painted= 4(a+b+c -6)
- Number of cubes with 3 sidess painted= 8