If Harry finished the potion, how could Dumbledore get past the black flames?

To get through to the final chamber containing the Philosopher's Stone Harry and Hermione have to defeat Snape's logic challenge. When Hermione discovers the correct potion they find that there is only a small quantity remaining.

"Got it," she said. "The smallest bottle will get us through the black fire - towards the Stone."
Harry looked at the tiny bottle.
"There's only enough there for one of us," he said. "That's hardly one swallow."
_{(Philosopher's Stone, Chapter 16, Through the Trapdoor).}

I've always assumed that there wasn't much potion left because Quirrell had already consumed part of it to gain access himself. Yet Dumbledore later went through the flames to rescue Harry from Quirrell. He did this despite it being made perfectly clear that Harry had finished the potion.

"Here I come," he said and he drained the little bottle in one gulp.
_{(Philosopher's Stone, Chapter 16, Through the Trapdoor).}

How then did Dumbledore get through the challenge? Did the bottle refill itself every time the potion was drunk (but contained a small quantity of potion so as to only allow one person entry)? Or did Dumbledore have some kind of 'cheat code' that allowed him to bypass having to drink the potion at all?

Answer

I would imagine that the potions refill. If they do not refill then when Harry and Hermione arrived in the potions room they would have seen that one bottle was half empty and known that that was the correct one, just like in the room with the keys:

Not for nothing, though, was Harry the youngest Seeker in a century. He had a knack for spotting things other people didn’t. After a minute’s weaving about through the whirl of rainbow feathers, he noticed a large silver key that had a bent wing, as if it had already been caught and stuffed roughly into the keyhole.

We do not find that they noticed anything different about the correct bottle, and in fact Harry was still skeptical that it was the correct one:

“You drink first,” said Harry. “You are sure which is which, aren’t you?”

It would thus seem that they were unable to determine the correct bottle by noticing one that had already been used; therefore, it stands to reason that the bottle refilled itself.

Of course one could argue that the bottle was in fact half empty but Harry and Hermione did not think of using such evidence. However, this is probably unlikely as this is precisely where we find out that Hermione's logical reasoning exceeds the average wizard:

“Brilliant,” said Hermione. “This isn’t magic — it’s logic — a puzzle. A lot of the greatest wizards haven’t got an ounce of logic, they’d be stuck in here forever.”

“But so will we, won’t we?”

“Of course not,” said Hermione.