Cluemeister's Corner

The deadline for this puzzle has passed. You can find the answer here!

Corners come in all shapes and sizes

This is a new feature for Minicon 46! Each month, the Cluemeister behind the Minicon Medallion Hunt will post a new puzzle of one sort or another on this page. You can send your answers, comments and speculations to thorin (dot) tatge (at) pobox (dot) com, and the correct answers, together with those who found them, will be posted at the start of the following month, together with the next puzzle.

In addition, all the puzzles will touch on themes or include ingredients relevant to clues in the Sixth Annual Minicon Medallion Hunt, to be held at Minicon 46. Those who follow along or participate here will therefore have an edge come Minicon - and while this Corner may not be giving out prizes, the Hunt will be!

Now that the Corner's been broken in, let's set out our first quandary...a geeky combination of mathematics, logic, and grammar!

September Puzzle: Alphabetical Numbers

Way back on 8/8/88, when the Cluemeister was 8, he read in the paper that the first number in alphabetical order, when spelled out in English, is..."eight"! This seems true enough, and one can find the fact cited in many a tome of numerical curiosities. But the follow-up question that recently wracked the Cluemeister's brain - the question for this month - is:

What number is -second- in alphabetical order when spelled out in English?

This question turns out to be amazingly tricky! In order to answer it, we need to make a lot of assumptions about how numbers are spelled out and alphabetized. Before proceeding, think about the question and use your own conventions to come up with an answer.

Got one?

The Cluemeister wonders whether anyone in the world would come up with the same answer as him in this position. Well, perhaps you would if you were following the same rules! There's plenty of room for argument, but here are the assumptions the Cluemeister considers most reasonable:

1. A person might argue that "eight" doesn't come first at all, but rather that that honor goes to "e", Euler's constant! Other transcendental numbers, however - and for that matter, irrational numbers in general - have no clear uniform way to spell them out, as they can be defined in various ways. Therefore, we will assume that they cannot be spelled out, and that we are therefore concerned only with rational numbers.
2. Spaces come before letters in alphabetical order. Hyphens, as in "forty-two", come between spaces and letters.
3. While one may choose informally to say something like "four hundred and three", the word "and" is technically reserved for the division between the whole part of a mixed number and the fractional part, as in "two and one half".
4. There is wide disagreement about whether and when to hyphenate the parts of spelled-out fractions. Would one write, "Thirteen-thousand seven-million-thirty-firsts"? "Thirteen-thousand seven million thirty-firsts"? We adopt what appears to be the most common rule - that fractions are not hyphenated (beyond ordinarily hyphenated components they may contain) when used as nouns, and therefore that this number would be "Thirteen thousand seven million thirty-firsts." It's less clear how to treat fractions used as adjectives, such as "a two-thirds majority", but we can assume that for purposes of this question all numbers are treated as nouns.
5. No number contains the word "a". Thus, "a billion" does not come before "eight". Nor can a word like "billion" stand alone. Rather, it should be written as "one billion". This also rules out numbers like "Five and seven thousandths", which should technically be "Five and seven one thousandths".
6. Does one use commas to separate the parts of large numbers, as used in the American style to separate groups of three digits, when large numbers (like 56,829) are written as numerals? Should one include the comma in "fifty-six thousand, eight hundred twenty-nine"? There is no consensus. However, it would look strange to write "four thousand, ten" rather than "four thousand ten". One could try to construct a set of rules for special cases in which commas are not needed, but the Cluemeister considers it better and simpler to omit commas when spelling out numbers.
7. There's no reason to allow more than one way to spell the same number, if one is going to the trouble of alphabetizing them. For consistency, therefore, unofficial number nicknames like "dozen", "score", and "googol" are not allowed. This also bars the obscure European form "milliard" and its relatives.
8. For the same reason, improper fractions are not allowed, and all fractions must be in simplest form. This also prevents the potential ambiguity of numbers like "three hundred thousand eight hundred sixty-fourths", because both 300,000/864 and 300,800/64 are improper fractions.
9. There is no solid consensus on how to write out extremely large numbers, if one is going to be silly and eschew scientific notation. We choose to follow the Conway-Wechsler System, as set forth at http://mrob.com/pub/math/largenum.html#conway-wechsler, for numbers up to (but not including) ten to the three thousand third power, as it appears to be the most logical choice. For larger numbers still, the Conway-Wechsler system is more tenuous and would make our question unanswerable(!), so we restrict ourselves to numbers (and numerators and denominators) below the lofty figure of 10^3003.

The Cluemeister pondered all this for a while and came up with an answer. Then he thought some more and found a better one, and finally one still better. Using the above rules, can you match or beat his final answer?

The deadline for this puzzle has passed. You can find the answer here!