Let's quote the current author's notes:
One thing I did notice was that many readers (a) neglected simple solutions in favor of complex ones, (b) neglected obvious solutions in favor of nonobvious ones, and (c) suggested that the correct hints had been put there for deliberately deceptive purposes.
General announcement: I do not lie to my readers. Almost everything in HPMOR is generated by the underlying facts of the story. Sometimes it is generated by humor – I can’t realistically claim that comic timing that precise would occur in a purely natural magical universe. But nothing is there to deliberately fool the readers.
Methods of Rationality is a rationalist story. Your job is to outwit the universe, not the author. If it taught the lesson that the simple solution is always wrong because it is “too obvious”, it would be teaching rather the wrong moral. There are some cases where people have scored additional points by successful literary analysis, e.g. Checkov’s Gun principles. But the author is not your enemy, and the facts aren’t lies.
Now, yes, it is possible that Eliezer Yudkowsky's Author Note on this very chapter is a lie, and he will suddenly reveal a whole series of barriers to paying the debt that will shut off everything from arbitrage to time turners to Dumbledore using the Philosopher's Stone in the manner allowed in canon, without having given us any previous hints as to what they are. But I think Eliezer Yudkowsky is not lying, and that at least one of the many simple solutions proposed (or another simple solution) will work.
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The foreign exchange(i.e., currency) market is both very liquid and very volatile. With advance knowledge of major changes from a Time Turner, it's easy to make very fast exponential growth of your seed money - even a 1% change per day, in any direction, multiplies your money by a factor of 12 every year(or x3 million before he graduates Hogwarts). Of course, anyone with that sort of record would get investigated hard and fast, but a more cautious approach can still result in absurd growth.
And there is leverage, so you can invest 100 $, but get profits or loses like you invested 10000 $. So if theres 100x leverage, and 1% profit to make on currencies each day, you can double your money every day.
So to turn 100 galeons to 100000 galeons Harry would need 10 transactions with 1% profit each.
http://www.investopedia.com/ask/answers/06/forexleverage.asp#axzz1qS7tVBFp