"Is this the Coldport Lost Property Office? I think I left my puzzle-book on the train this morning, and I was hoping someone had handed it in." The speaker was a tall gentleman with a six-sided flat cap.
"It is," said Sally, who was (wo-)manning the desk that morning. She sighed slightly. "However, the door to the Lost Property is locked with a code, and I haven't figured out what the code is yet."
"Figured it out? You mean... it's a puzzle?" A gleam appeared in the gentleman's eye. "Tell me about it, please!"
"The code entry is seven digits," said Sally. "But the door is faulty, so the digit $1$ is always there first, and when you press a digit on the code pad, the pad automatically immediately enters another digit. It starts at $1$ and increments each time."
"So if you press $6$," said the gentleman, "the pad entry reads $162$?"
Sally thought about this for a moment. "Right," she said.
"And if you pressed six again, it would read $16263$?"
"Exactly. And all I've been told so far is that the code to be entered is the unique perfect square." She sighed again. "I don't see how that helps me."
The gentleman grinned. "Oh, but that should be sufficient," he said. "One moment...." He pulled out a pad of paper and started scribbling on it. It look slightly longer than a single moment, but ten minutes later he suggested a seven-digit code to Sally, and to her slight surprise, the door unlocked. From the smell that gusted out as she opened it, it had been locked for quite some time.
What is the seven digit code to unlock the door? I'm adding the no-computers tag to this as a brute-force search is quite easy, but using a calculator to find squares or square-roots is permitted. (A good solution will ideally describe how to go about solving it without a computer :) )