Gopal用立方体拼成符合某种规律的几何体。前四个几何体如下所示。

1. Identify the pattern for the number of painted cubes

Looking at the 'Number of painted cubes' column:
Solid 1: $1 = 1^2$
Solid 2: $4 = 2^2$
Solid 3: $9 = 3^2$
Solid 4: $16 = 4^2$
Therefore, for Solid 5, the number of painted cubes is $5^2 = 25$.

2. Calculate the number of unpainted cubes for Solid 5

The total number of cubes for Solid 5 is given as $125$.
The number of unpainted cubes is the total number of cubes minus the number of painted cubes.
Unpainted cubes $= 125 - 25 = 100$.

1. Identify the pattern for the number of painted cubes for Solid $n$

From the table, the number of painted cubes for Solid 1, 2, 3, and 4 are $1, 4, 9,$ and $16$ respectively.
These are perfect squares: $1^2, 2^2, 3^2, 4^2$.
Thus, the number of painted cubes for Solid $n$ is $n^2$.

2. Find the solid number that has 196 painted cubes

Let $n^2 = 196$.
$n = \sqrt{196} = 14$
Therefore, it is Solid 14.

1. Find the solid number $n$ from the total number of cubes.

From the pattern, the total number of cubes for Solid $n$ is $n^3$.
$n^3 = 5832$
$n = \sqrt[3]{5832} = 18$

2. Calculate the number of unpainted cubes using the established pattern.

From the table, the number of painted cubes is $n^2$.
Number of painted cubes $= 18^2 = 324$
Number of unpainted cubes $= \text{Total cubes} - \text{Painted cubes}$
$= 5832 - 324 = 5508$