NOTE: Spoilers below!

Difficulty: Easy.

Idea: Exhaustive search, but reducing the search space to something manageable. I attempted to do this with pencil + paper, but could not constrain the space enough.

First, we find strict upper and lower bounds, the roots of 1020304050607080900 and 1929394959697989990. That still leaves a search space (upper - lower) of about 375M numbers. We want to constrain that space further.

Now, notice that the target, 1_2_3_4_5_6_7_8_9_0, ends in 0, which means 2 and 5 are prime factors of it. But the target is a square, so all of its factors' exponents must be even. Therefore the target is of the form 1_2_3_4_5_6_7_8_900.

Next, notice that the only numbers whose square ends in 900 must end in 30 or 70.

With that in mind, we can reduce the initial search space by a factor of 100 (we only need to check 2 numbers out of every 100), leaving a manageable space of ~4M, which we just search over with a script.

--

This solution runs in 6.5 seconds and spits out the answer of 1389019170. The answer is near the upper bound of our search. If we run the search starting from the upper bound instead of the lower one, the script runs in 60ms.