Puzzle:
A solid, four-inch cube of wood is coated with blue paint on all six sides. Then the cube is cut into smaller one-inch cubes. These new one-inch cubes will have either three blue sides, two blue sides, one blue side, or no blue sides. How many of each will there be?
Puzzle Solution:
Subtract the outside squares, which will all have some paint on them
16 top
16 bottom
8 more left
8 more right
4 more front
4 more back
= 56
So, only 8 will have no blue side.
The only cubes that will have blue on only one side will be the four center squares of each side of the 64-cube — so, 4 x 6 sides = 24 cubes
The only cubes that will have blue on three sides are the corner pieces — there are 8 corners, so 8 cubes.
The cubes with two sides blue are the edge cubes between the corners — two on each side of the top and bottom, so 2 x 4 sides x 2 (top and bottom) = 16, + the side edge/non-corner pieces, which will be another 2 x 4 = 8
So
No blue side = 8
1 side = 24
2 sides = 24
3 sides = 8
Total = 64 cubes
I f a cube is painted blue and cut into 64 pieces .
Approach this question as ,,
64 pieces = 4 *4* 4 so take n=4
just use this formula
1. no sided painted =(n-2)^3
2. 1 sided painted =6 * (n-2)^2
3. 2 sided painted = 12 *(n-2 )
4. 3 sided painted = always 8 (a cube has 8 corners)