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Jim
I think that you are right about this problem being related to the traveling salesman problem--in fact I think this problem maps directly on to the traveling salesman problem, meaning it requires brute force approach...
EF
Better still, just shove everything into MySQL and then ask it to sort it. Hey, it might work! Of course, I doubt that's the sort of approach ITA is looking for.
I suspect that the Right Way to tackle the problem is to break the spelled-out numbers into the word-tokens that will comprise them (e.g. "eight", "forty", "million", etc). Those'll be easy to alphabetize -- you'll know what the first column of a properly-sorted list would look like.
If you can develop an algorithm that can fully explore the space underneath a given token (not simple, but not mind-bendingly difficult I don't think), then proceed through that space while counting letters (and ignoring order) you can eliminate chunks of the problem's space, token by token. Narrow it down to the left-most token that contains the 51 billionth letter -- the order of the elements belonging to the tokens before it doesn't matter, thanks to the commutative property. This lets you simply keep a count of total letters instead of keeping each word in memory. Once you hit the target letter, you'll know the left-most token in whose neighborhood the answer resides. You can repeat the process with the next-to-left-most token; then do it with the next-to-next-to-left-most token; etc. Eventually you'll have a set of possible answers that's small enough to sort easily.
In practice this would be pretty tedious to code (and I'd hate to risk missing a token while defining the first step), so I can't say I'm champing at the bit to try implementing it. I bet there's a more elegant solution.
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