Is it possible to colour the faces of $27$ unit cubes and arrange them to form $3\times3\times3$ cube with all exterior faces red,then white,then blue Is it possible to colour the faces of $27$ unit cubes so that one can arrange them to form $3\times 3\times 3$ cube with all exterior faces red, then rearrange to form a cube with all exterior faces white, then blue?
There have be at least $24$ or $23$ cubes with all three of their faces in one colour... Thereafter trying various possibilities my guess seems to be no,  but I'm not sure and don't know how to prove it.
 A: This is possible. 
Let $(x,y,z)$ represent a cube with $x$ red faces, $y$ white faces, and $z$ blue faces. Then one way to paint the small cubes is as follows:


*

*Make one each of $(3,3,0)$, $(3,0,3)$, and $(0,3,3)$.

*Make three each of $(1,2,3)$, $(1,3,2)$, $(2,1,3)$, $(2,3,1)$, $(3,1,2)$, $(3,2,1)$.

*Make six of $(2,2,2)$.


You can check that for each color, there is $1$ cube with no faces of that color, $6$ cubes with one face of that color, $12$ cubes with two faces of that color, and $8$ cubes with three faces of that color.
A: *

*Start with 27 unpainted cubes arranged in a larger 3x3x3 cube.  

*Paint the outside faces white.  

*Remove the 3x3x1 bottom layer and place it on the top. Remove the 1x3x3 left slice and place it on the right hand side. Remove the 3x1x3 front slice and place it against the back side. The preceding rearrangement puts unpainted faces on the outside. Paint the outside faces blue.  

*Repeat the rearrangement to put all the remaining unpainted parts on the outside, and paint it red.


The result is that each of the cutting planes have one colour, and  three parallel cutting planes (where the two outside faces are considered as one single cutting plane) have three different colours. You can cyclically rearrange the slices to put any of the three colours on any of the opposing pairs of faces.
