# Cluemeister's Corner

#### Results for September

Can you feel the changes of autumn in the air? Our first puzzle has come and gone! In September, the Cluemeister asked readers to name the *second* number in alphabetical order, assuming that "eight" is the first. He specified a long list of conditions to adhere to, representing what he considers the most reasonable way to approach this otherwise surprisingly vague question.

Only four answers came in during September, so the Cluemeister extended the deadline to October 5th. This gesture brought the number of entrants to nine, which is acceptable for the first month of a feature that so far has been advertised only on Facebook and perhaps by word of mouth! The Cluemeister also wishes to apologize for including the wrong e-mail address - a (dot) was left out - and hopes that not too many people had their submissions bounce before it was fixed.

So, what is the answer to the September question? By the Cluemeister's best reckoning, the second number in alphabetical order, when spelled out in English, is:

#### 8 8,000,000,000/8,808,808,881

No, seriously! Not a joke. How did the Cluemeister arrive at such an ornate figure? Read on.

One reader guessed that "fifty" comes after "eight" in alphabetical order. If one would rather not leave the e's, though, "eleven" is available. But why strike out across new morphemic ground when can instead duplicate the success of the number one entry? Several readers made "eighteen" their guess.

One reader guessed "eight hundred," which leads us to Rule 2 and perhaps the most debatable of the Cluemeister's clarifications: which comes first when alphabetizing, a space or a letter? Dictionaries all list e.g. "be" before "before", or "star" before "stark", but that only proves that an imaginary "end-of-word" character comes before any other. Dictionaries also tend to list "coaster" before "coast guard" and the like, suggesting that they adhere to the advice of several style guides in that non-alphanumeric characters are ignored when alphabetizing. Under this rule, "eight hundred" comes before "eighteen". On the other hand, the computer age has popularized the inclusion of all characters when alphabetizing, probably because it's so easy to program. You just define a space as coming before any letter, as it does in ASCII, and let the program do its thing. This mentality is reflected in this list of the first thousand natural numbers in alphabetical order, which lists "eight hundred" well before "eighteen".

The Cluemeister decided to be anal inclusive and include spaces and hyphens as part of the alphabetization process. However, in this case, it doesn't even matter, as the number "eight billion" beats both other entries regardless.

All English-language numbers begin with a determiner drawn from the numbers "zero" through "nineteen" or "twenty", "thirty", or the like - sometimes after the word "negative", but clearly we don't need to worry about negative numbers. So clearly, the second number in order will begin with an "eight". But is there any follow-up quantity that comes before "billion"? Maybe "bajillion", perhaps? Well, a bit of research confirms that there is no such number. When one chooses to spell out very large numbers, one uses a system of Latin prefixes along with "-illion" that ends, typically, at the equivalent of 999-illion: "novenonagintanongentillion". There is some disagreement about which exact spellings to use along the way (e.g. "noven" vs. "novem"), but none of these prefixes comes before "billion" in alphabetical order. One reader suggested "centrillion", by which we assume he means "centillion", or 10 to the 33rd power, but this, of course, begins with a "c" and thus falls short.

When we look beyond 10^3003-1, however, things get weird. (This assumes the American system of counting large numbers. Using the European system makes no difference for our puzzle; it only changes the magnitude of any potential answer.) Many scholars suggest that we draw the line there and call enough enough - what need is there to spell out numbers that huge? But since when did *need* have anything to do with it? One source suggests a series of second-order prefixes beginning with consecutive letters of the alphabet, but this is silly - it just postpones the problem. (This does mean that "zero" is not necessarily the last number in alphabetical order - it may come before 'zilli-" something, if you choose this system.) If we want to name every number conceivable, no matter how large, we need a system that stacks prefixes.

The Conway-Wechsler system referred to in Rule 9 seems to be the best such system. It takes the power of 10 in question, divides it by three and subtracts one, as is standard. Then it breaks results greater than 999 into groups of three digits and treats each group separately, appending all the relevant prefixes together. A group of three zeroes is given the nifty prefix "nilli-". This means that 10^3003 is called "one millinillion", while 10^6003 is called...wait for it..."one billinillion"! And moreover, 8x10^6009 is "eight billibillion", which comes alphabetically before "eight billion"!

Should this, then, be our final answer? No, for we can extend the process indefinitely. 8x10^6006009 is "eight billibillibillion", and 8x10^6006006009 is "eight billibillibillibillion", and so on. What this means is that if we adopt the full Conway Wechsler system, there *is* no answer to our question! Any proposed number to come second in alphabetical order can be met by a challenger starting with "eight billbillibilli..." that includes more "billi-"s than the reigning champion. However, it would be a shame not to reach an actual answer, and moreover this system is not widely accepted, so the Cluemeister decided to put the kibosh on such insanely high numbers in Rule 9.

Where does that leave us? Back at "eight billion", which several readers suggested and which is, indeed, the second number in alphabetical order...if we restrict ourselves to the integers!

But why do that? Fractions are numbers too, aren't they? Rule 1 banned irrational numbers from consideration for being unspellable, but fractions can easily be spelled. You use "and" to divide the whole component from the fractional component, as described in Rule 3. 8 1/2, then, is written "eight and one half". And, wouldn't you know it, "and" comes before "billion" in alphabetical order!

What about decimals? We could choose to spell numbers as decimals rather than as fractions. The Cluemeister didn't bother proscribing this because it couldn't affect the answer: anything beginning "eight point..." or "eight billion point..." will come after "eight billion". The same is true of imaginary numbers: any whole number of "i" will always come later in the alphabet than its real equivalent.

So that means (barring complex numbers, which we consider later) our answer must be a fraction beginning with "eight and". What, then, is our numerator? Clearly it, too, must begin with "eight".

One reader submitted the oral guess of "eight and eight eighteenths." Not too shabby, but it ignored a clause of Rule 8, which states that fractions must be in simplest form in order to prevent ambiguity. 8 8/18 reduces to 8 4/9, but since "eight and four ninths" was the best answer the Cluemeister had at the time, he was prepared to declare this reader the winner regardless! Since then, however, another reader submitted a better answer: "eight and eight elevenths", followed by the improvement "eight and eight eighty-fifths." Neither of these beats the unreduced "eight and eight eighteenths", but unlike that answer, these are irreducible.

However, we can do still better. Why follow one "eight" with another when you can follow it with "billion"? "Eight and eight billion" can be followed by another "eight", at which point we must decide whether to extend the numerator with "million", "thousand" or "hundred", cut the numerator off at 8,000,000,008, cut it off at 8,000,000,000 and make the last "eight" part of the denominator, or even throw an "i" on the end and make it a complex number. (It's debatable whether you could write the plus sign in "a+bi" as "and" rather than "plus", but as it turns out, it makes no difference.) Reflection shows that the third of these options is the way to go, as adding another "billion" beats all alternatives. Thus far, we have "eight and eight billion eight billion".

Our denominator needs to be larger than our numerator, of course (Rule 8), and must not share any common factors with eight billion. Tacking on another "eight", is, naturally, the way to proceed. Now we have to choose between "million", "thousand" and "hundred", so of course we choose "hundred".

At this point, the Cluemeister originally envisioned "8 8,000,000,000/8,000,000,8??", and proceeded from there. But on third or fourth thought, he realized that the "hundred" used here could be measuring thousands or even millions. This allows us to add another "eight" and arrive at "eight and eight billion eight billion eight hundred eight million", as "million" comes before "thousand". We proceed similarly to get "eight and eight billion eight billion eight hundred eight million eight hundred eight thousand eight hundred". Notice how unwieldy the number is becoming, which is why the Cluemeister was reluctant to ban commas in Rule 6, though it seems to be a necessity for maintaining sanity.

We now need to cap off our denominator. We can't use "eighths", though, or "eightieths", or even "eighteenths", because all of these produce even denominators and thus reducible fractions. The only way to use the string "eight" is in the number "eighty-" followed by some pluralized ordinal number after the hyphen. And not just any pluralized ordinal number. An *odd* pluralized ordinal number!

That leaves us with "firsts", "thirds", "fifths", "sevenths", and "ninths". Of these, "fifths" is first in alphabetical order, but poses a problem of its own: 8,000,000,000/8,808,808,885 reduces to 1,600,000,000/1,761,761,777. We go, then, to the next choice on the list, "firsts", which does in fact produce an irreducible fraction. That means that we are finally done, and the second number in alphabetical order is:

**Eight and eight billion eight billion eight hundred eight million eight hundred eight thousand eight hundred eighty-firsts.**

Getting tricky and sticking an "i" onto the denominator at some point isn't valid: complex numbers, written properly, have real denominators. Note also that the number "eight billiard", or 8,000,000,000,000,000 in the European system, beats "eight billion", as one reader observed. The Cluemeister did, however, specify in Rule 7 that this word isn't allowed - this number is more commonly known as "eight thousand billion", or "eight quadrillion" in the U.S.

A Czech friend informs the Cluemeister that this puzzle would have been much easier if posed in the Czech language, where the first number alphabetically is "čtvrtý" (one fourth), not requiring a determiner, and the second number is "čtyři" (four). But he apparently overlooked "čtrnáct" (fourteen)! So that opens up a whole can of worms - Fourteen quadrillion? Fourteen quattuordecillion?) ...Which suggests that not even the Czechs have it easy on this one. (Who'd have guessed?)

The only two entrants to submit numbers beginning with "eight and" were both Andys, fittingly enough. Did their names serve a clue for how to sneak in under the word "billion"? In any case, **Andy Pierce**, who drove a hard bargain with the Cluemeister by demanding to know whether anyone had found the best solution yet, is the September winner, with the entry "eight and eight eighty-fifths".

Will this victory actually mean something come Minicon? A few Dealer Dollars, perhaps, or some prize from the Medallion Hunt? Too soon to say, but the Cluemeister predicts "probably". Congratulations, Mr. Pierce!

The October puzzle will be up in the next few days. And don't worry if you're not a math person - it'll be about presidents!

Hoping to see you soon, The Cluemeister.

*Follow-ups or questions? Write the Cluemeister at thorin (dot) tatge (at) pobox (dot) com.
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Click here to see the September puzzle.*