Find all possible positive integers $$n$$ such that $$n^4 - 4n^3 + 22n^2 - 36n + 18$$ is a perfect square.

PLACEHOLDER ANSWER - the answer is a set of integers. With $$m = n^2 - 2n + 9$$, the expression equals $$m^2 - 63$$. If it is $$k^2$$, then $$(m - k)(m + k) = 63$$, so $$m = 32$$, 12 or 8. Since $$m = (n - 1)^2 + 8$$, $$m = 8$$ gives $$n = 1$$ (value 1) and $$m = 12$$ gives $$n = 3$$ (value 81), while $$m = 32$$ has no integer $$n$$. So $$n = 1$$ and $$n = 3$$.