If a cube of dimension 4 by 4 by 4 is cut into 1 by 1 by 1 after painting it by red color then which of the following is/are true.
I) Ratio of number of painted surfaces of resultant cubes to total number of surfaces is 1/4
II) Number of cubes having no faces pointed is equal to number of cubes having three faces painted.
III) Number of cubes having 2 faces painted is same as number of cubes having one face painted  Choose the correct choice from the following:
A) None  B) II and III  C) II only  D) I, II & III  E) I & III only
Long explanation, probably too detailed   Hypercube facts (n=3 for a cube):   2^n vertices 2n−1n edges 2+E−V faces   For a cube: 8 vertices, 12 edges and 6 faces   Let G=4be the size of the original cube.   I) Ratio of number of painted surfaces of resultant cubes to total number of surfaces is 1/4   There is 1 large cube with 6 sides of G2 painted faces.   There is G3x1 cubes with 6 sides of surface area 1.   1×6×G^2/G3×6×1=G^2/G^3=G−1=1/4   True   II) Number of cubes having no faces pointed is equal to number of cubes having three faces painted.   There are always 8 vertices on a cube, each vertex is the only 1x1 block which has 3 coloured faces.   Inside the cube, there is a secondary unpainted cube. Each side of this unpainted cube is equal to the size of the original cube minus two (on each axis, there will be two 1x1 cubes on the exterior of the larger cube). The size of this cube is (G−2)3   There are (G−2)3( cubes with 0 sides painted. =23=8   There are V=23 cubes with 3 sides painted =8   True   III) Number of cubes having 2 faces painted is same as number of cubes having one face painted   There are 12 edges on a cube. Each edge consists of G cubes, two of which are the vertices of the cube (and have 3 colours), the rest have two colours.   On each exterior face, there is a square of (G−2) cubes, surrounded by 4 vertex cubes and (G−2)×4 edge cubes.   Each square consists of (G−2)^2 coloured faces. There is 1 square per face.   There are (G−2)×E cubes with 2 sides painted. =2×12=24   There are (G−2)2×F cubes with 1 side painted. =22×6=24   True   ANSWER:I, II &amp; III
<result>D</result>
If a cube of dimension 4 by 4 by 4 is cut into 1 by 1 by 1 after painting it by red color then which of the following is/are true.
I) Ratio of number of painted surfaces of resultant cubes to total number of surfaces is 1/4
II) Number of cubes having no faces pointed is equal to number of cubes having three faces painted.
III) Number of cubes having 2 faces painted is same as number of cubes having one face painted