ASAP MATH problem

Question

A cube (made from 27 smaller cubes)( 3x3x3 ) was dipped into pink paint.

 

how many small cubes will have 3 pink faces? 

how many small cubes will have 2 pink faces?

how many small cubes will have a pink face?

how many small cubes will have no pink faces?

 

how could u predict the above for any size cube? 4x4x4? pxpxp?

 

Answer 1

A cube of edge length s has:

 

  ALL CUBES                 Cube with edge s                     Cube with edge 3        Cube with edge 4

  8 corners           --      8 pink on 3 sides                               8                                8

  12 edges           --      12*(s-2) pink on two sides                12                              24

   6 sides             --     6*(s-2)² pink on one side                     6                              24

  (s-2)³ internal    --     (s-2)³ no pink                                     1                                8

                                 ------------------------------               ------                          ------

                                       s³                                               27                              64

 

Note that 8 + 12(s-2) + 6(s-2)² + (s-2)³

           = 8 + 12s - 24 + (6s² - 24s +24) + (s³ - 6s² + 12s -8)

           = s³      [accounts for all smaller cubes]

Answer 2

8 (the corner pieces) will have 3 pink faces.

 

6 (the center piece on each side) will have 1 pink face

 

1 (the internal cube) will have no pink faces

 

12 (THE REMAINDER) WILL HAVE 2 PINK FACES