To solve the equation \( X - 3 = (3 - x)^2 \), we can start by rewriting it in a standard form.

First, we expand the right side:

\[
(3 - x)^2 = 9 - 6x + x^2
\]

So the equation becomes:

\[
x - 3 = 9 - 6x + x^2
\]

Next, we will move all terms to one side of the equation:

\[
x - 3 - 9 + 6x - x^2 = 0
\]

Combining like terms, we have:

\[
-x^2 + 7x - 12 = 0
\]

To make it easier to work with, we can multiply through by -1:

\[
x^2 - 7x + 12 = 0
\]

Now we can factor this quadratic equation:

\[
(x - 3)(x - 4) = 0
\]

Setting each factor to zero gives us the solutions:

\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]

Thus, the solutions to the equation \( x - 3 = (3 - x)^2 \) are:

\[
\boxed{3} \quad \text{and} \quad \boxed{4}
\]

Finally, we can verify the solutions by substituting them back into the original equation.

For \( x = 3 \):

\[
3 - 3 = (3 - 3)^2 \quad \Rightarrow \quad 0 = 0 \quad \text{(true)}
\]

For \( x = 4 \):

\[
4 - 3 = (3 - 4)^2 \quad \Rightarrow \quad 1 = 1 \quad \text{(true)}
\]

Both solutions are valid. Therefore, the final solutions are:

\[
\boxed{3} \quad \text{and} \quad \boxed{4}
\]