A solid, n-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 type 1-inch cube will there be?
Check if you were right - full answer with solution below.
Solution: the
This puzzle can be solved logically or mathematically. After cutting, the total no of cubes will be: n*n*n = n^3. Then, the Number of each type of cube can be found by using the following formulas:
1 sided painted =6 * (n-2)^2
2 sided painted = 12 *(n-2 )
3 sided painted = Always 8 (a cube has 8 corner)
No sided painted =(n-2)^3
Examples:
Question 1:
A solid, 4-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.then How many of each 1-inch cube will there be?
Solution:
given n=4 then No sided painted = (4-2)^3 =>8
1 sided painted = 6 * (n-2)^2 =>24
2 sided painted = 12 *(n-2 ) => 24
3 sided painted = Always 8 (a cube has 8 corner)
Question 2:
A solid, 8-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.then How many of each 1-inch cube will there be?
Solution:
given n=8 then No sided painted = (8-2)^3 =>216
1 sided painted = 6 * (n-2)^2 => 216
2 sided painted = three12 *(n-2 )
=> 72 3 sided painted =
Always 8
(a cube has 8 corner)
Question 3:
when 64 inch cube cut into smaller 4 inch cube then
n=given_size/cutting_size
for given question n=64/4 =16 then No sided painted =
(16-2)^3
=>2744 1 sided painted =
6 * (16-2)^2
=>1176 2 sided painted =
12 *(n-2 )
=> 168 3 sided painted =
Always 8
(a cube has 8 corner)