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?

### Solution:

This puzzle can be solved logically or mathematically.

After cutting,

total no of cubes will be: n*n*n = n^3

Then,

Number of each type of cube can be found out 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 = **12 *(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)

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