Friday, September 20, 2013


The puzzles in my last post led to great answers in the comments, and also some great thinking in email threads with friends. A variety of puzzle answers and open questions are below.

First, however, for further reading on the topic of numbers, or even just a gift idea, you might check out The Book of Numbers by John Conway and Richard Guy. I haven't read it (I saw the citation while browsing here), but Conway is a famous mathematician with a flair for games and what he writes is usually well worth reading. He spoke to my college Linear Algebra class one morning, and it was an amusing experience.

Without further ado:

1) Amy subtracted a three-syllable number from a three-syllable number and obtained a thirty-seven syllable number. What could her numbers have been?

For this puzzle, a reader posted the following answer (which was my answer too):

    12,000,000,000,000 - 23 = 11,999,999,999,977.

To devise this puzzle, I first thought about the difference "1 trillion minus one," which would have led to a puzzle of the form [3] - [1] = [30]. But then it seemed more intriguing for the first two numbers to have the same (small) number of syllables---a decision that also allowed me to pump up the syllable count of the difference in two ways:  choosing the subtrahend 23 (which leads to a 77 in the difference) and upping the minuend to 12 trillion (which leads to an 11 in the difference). That resulted in a puzzle of the form [3] - [3] = [37].

We can find some amusing new kinds of answers if we relax the whole-number constraint, an idea suggested by the first comment on the original post. Then we can find cases like this one:

    (0.000000000003) - (8 + i) = -7.999999997 - i.

In words,

    "three trillionths minus eight plus i equals negative seven and nine hundred ninety-nine billion, nine hundred ninety-nine million, nine hundred ninety-nine thousand, nine hundred ninety-seven trillionths minus i"

or in terms of syllable counts, [3] - [3] = [42].

2) What is the smallest number you can find with nineteen syllables?

A reader posted this answer, which I believe is optimal:


He also observed that Spanish beats English in this context, because one can get to 19 syllables with a smaller number. The smallest I have found is


I don't know if this is optimal for Spanish. Nor do I know whether another language would do even better. (German might be worth a look, but Esperanto doesn't look promising.) For those interested in pursuing the Spanish question, this tool is helpful.

Relaxing the whole number constraint once again, note that the words "decillionths" and "one hundred" both have three syllables. So if we just delete the 1 from 177,777 and append "decillionths," then we find a very small, 19-syllable (rational) number:

    77,777 decillionths = 0.000000000000000000000000000077777.

3) What is the largest number you can find for which the number of syllables equals the sum of the digits?

One reader found several examples for this one. First,


or "seven hundred twenty-one decillion, etc. etc." (Number of syllables = 108 = sum of digits.)

Second, working again in the decillions, he found the improved answer


(number of syllables = 101 = sum of digits).

This was my own answer as well. I suspect it is optimal relative to the decillions-and-below nomenclature that I linked to in the original post. Nomenclature for higher numbers can be found here and here.

My own initial strategy for this puzzle was just to futz around until I found a 36-digit example that began with 999. Then I said, Can I add another 9? If I do, then my partial digit sum will be 4*9 = 36, while my partial syllable count will be 12 ("nine hundred ninety-nine decillion, nine hundred..."). That puts the digit sum 24 points ahead of the syllable count already. If I want to recover parity, then the digits I add from here on out will have to give me a lot of net syllables. Adding groups of 111 does so, because each 111 adds eight or nine to the syllable count, but only adds three to the digit sum. (Each group of 111 buys me five or six points net.) In fact, I can add enough 111's at the back of the number to buy a lot more 9's at the front. Finally, when the 9's and the 1's met up in the middle, I knitted them together with a 2, the digit that made the number of syllables and the number of digits equal.

Applying this proto-algorithm to numbers in the trillions and below leads to


which I suspect is optimal for 15 digits.

The reader's third number, a clever tongue-in-cheek suggestion, refers to a naming convention that I didn't even know. It is

   999,999,992,111,111,111,111,111,111,111,111,111 googolplexian 6.

One more note on puzzle #3: in the email thread, an interesting question was raised as to whether there is a largest whole number for which the syllable count equals the digit sum. Unfortunately, none of the conventions I've seen so far for assigning words to numbers has been spelled out to infinity, so the question would have to be made precise in some fashion. For example, consider any naming convention for which there exists a value of k such that all period names from the kth onward have greater than 27 syllables. In such a naming system, is the digit sum eventually doomed to fall behind the syllable count forever? If so, then relative to that system, there will exist a largest whole number for which the syllable count equals the digit sum.  (Assuming there is at least one such number in the naming system!)

4) What is the largest prime number you can find for which the number of syllables equals the sum of the digits?

Here a reader found


and my mathematical computing platform helped me in finding a pretty big example. But before I give it, let's first note the smallest prime with this property, namely


This is the 27th prime number, which makes puzzle #4 a little more difficult than I would have liked, because you have to go a fair ways up through the primes before the first example can be found. (Puzzle #3 was more accessible, because both 1 and 10 are solutions to it.)

Working in the decillions, the largest prime I could find with syllable count equal to digit sum was


(Number of syllables = 101 = sum of digits.)

I don't think there are any larger examples in the decillions. I actually found this number by permuting digits of the answer to puzzle #3 and just hoping that the result would turn out to be a prime number with a syllable count of 101. (Scientific, huh?)

Finally, in case interesting, here is a scatter plot that shows, in a randomly sampled way, the relationship between the number of digits in a number (horizontal axis) and the number of syllables in the number (vertical axis).

To make the plot, I randomly selected ~200,000 exponents uniformly in the range from 0 to 36; raised 10 to the power of each; and rounded the result. For each such integer, I created an (x, y) data point by counting the number of digits and the number of syllables. Unfortunately, because this data set is large and lives on a lattice, a simple scatter plot is deceptive: some data markers would represent hundreds of data points, while other data markers might represent only one data point---and there would be no way to see the difference. I usually deal with this problem by adding a little random jitter to the data when I plot it. But here that didn't suffice. So instead I gave each data marker a color commensurate with the number of data points that live on the lattice point it occupies. The resulting graph better reveals the "main sequence" within the data (darker blue dots).

I'll end this post with a similar plot for the digit sum vs. the number of syllables - n.b., this one uses the same general technique but a different color function.

Thanks everyone for your creative answers and ideas!

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