Category Archives: Mathematics

The Earth and the Rope Riddle

  By Robert Watson | 21 December, 2013 - 17:01 | ,
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Here’s a riddle I read on What If? (xkcd). I did a little experiment with it. But first, I’ll let you try to answer it for yourself.

Suppose you tied a rope around the Earth and tightened it so that every inch was hugging the surface. Now suppose you want to raise the rope one meter off the ground at every point. How much extra length would you need to add to the rope?

A. 6.28cm;  B. 6.28m;  C. 6.28km;  D. 734km

The multiple choices aren’t part of the original riddle, but I added them for my own purposes.

I tend to test my lower-level classes more frequently (so that the stakes on each test are lower). During the Fall semester, I have classes on three different schedules. Around Thanksgiving all the schedules line up. It’s a madhouse of grading that is actually harder to work with than finals week. The circumstances did make for a nice opportunity, though.

I gave the same multiple choice riddle on all five tests. It was a bonus, and the results were a little surprising. If you haven’t answered it for yourself, give it a try before reading on.

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The answer is B, which may come as a surprise that so little rope would be needed. But by raising the rope 1m, you’re increasing the diameter of the rope circle by 2m. This means you’re increasing the circumference by about 6.28m (2 pi). The initial size of the circle doesn’t matter; we’re only discussing how much we’re changing its size.

So how did my classes do? Here’s what I taught last semester, excluding the online class that didn’t receive the question.

1. Intermediate Algebra. This course covers the prerequisites to College Algebra. I had expected most of the students would guess. The question is a bit beyond that level of mathematics. As such, about 25% of the students got the question right – which is what you’d expect if everyone guessed. The selections were also chosen more uniformly.

The surprising result to me is only 13 of 27 students attempted it, although I emphasized there was no penalty for getting it wrong. I think by the end of the hour, the majority of the class simply forgot about it. It was written on the board, not the exam itself.

2. College Algebra. About a quarter of the students got this question right, but fewer people selected A. More thought seemed to go into the guesses – but they were still just that – guesses.

3. Precalculus. I was a little disappointed that only a quarter of the students got this right. We had just finished the trigonometry unit. More people worked out some math this time, but there was still a tendency to guess too high. Almost nobody selected A.

4. Calculus I. This is a 200-level class. About a third of the class got the question right – some improvement! Just like in Precalculus, almost nobody selected A.

5. Calculus III. Now we’re in the 300-level range. 80% of the class got it right. This was also the only class in which everyone attempted the problem.

I spent some time trying to figure out why only one class did significantly better than 25%, and I could only come up with one explanation. The first three classes are 100-level, and tend to emphasize mathematical skills more than anything else. They’re prerequisites. I emphasize how to “think like a mathematician,” but you do need to know how to do some math beforehand. After all, you can’t teach someone to write a novel until they have a basic grasp of spelling and grammar. You can’t think like a mathematician until you have a basic grasp of fundamental mathematics.

Calculus III was populated by math and math-ed majors – mostly students who will be teaching mathematics themselves after they graduate. Calculus I was a mix, mostly consisting of freshmen students who had Precalculus or Calculus AB in high school the year before. High schools tend to not emphasize critical thinking as heavily as in college. The primary difficulty in this question is resisting the temptation to guess too high.

How Many Novels Are There?

  By Robert Watson | 15 November, 2013 - 02:32 | ,
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We are now halfway into November, and if you are behind on your Nanowrimo novel, you may be looking into how to cheat catch up. You’ve probably heard that a thousand monkeys at a thousand typewriters will eventually turn out the works of Shakespeare.

Is it true? Technically, yes. Likely is a different question. The odds are small.

Very small.

Let’s suppose we want our monkeys to type out a particular 50,000 word novel. Hitchhiker’s Guide to the Galaxy is a good example, as it’s roughly around that mark.

There are 26 letters and seven basic punctuation marks (, . ? ‘ ” ! space) that are frequently used. There are others, but these are enough to make my point.

So the chances of a monkey hitting a particular character is 1 in 33.

According to the multiplication principle, the chances of a monkey hitting a particular 2-character combination is $latex \left(\frac{1}{33}\right)^2 &bg=e6eaea&s=0$. The chances of a monkey hitting a particular 3-character combination is $latex \left(\frac{1}{33}\right)^3 &bg=e6eaea&s=0$, a 4 character combination is $latex \left(\frac{1}{33}\right)^4 &bg=e6eaea&s=0$, and so on.

If we assume an average of 3 letters per word, then the chances of our monkeys banging out Hitchhikers Guide to the Galaxy is roughly $latex \left(\frac{1}{33}\right)^{150000} &bg=e6eaea&s=0$

Note that there are roughly $latex 10^{80} &bg=e6eaea&s=0$  atoms in the universe, and the universe is about 13.8 billion years old. To put the odds that our monkeys will produce Hitchhikers Guide to the Galaxy into perspective, if every atom in the universe was a monkey, and they had been typing 150,000 characters every nanosecond since the big bang, it still ain’t gonna happen.

This is probably the least interesting problem, though. If you want to win Nanowrimo with monkeys, you don’t need to write a specific 50,000 word novel. You just need to write any 50,000 word novel. And this is where the numbers really get tricky.

The fundamental question becomes what constitutes a novel?

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There was an interesting discussion over at what-if.xkcd about how many unique English tweets are possible. The bottom line is that there are roughly $latex 2^{140 \times 1.1} \approx 2 \times 10^{46} &bg=e6eaea&s=0$ possible tweets. The details are in the linked post and its own references, but the gist is this:

In a 1950 article, Claude Shannon determined that the typical written English sentence contains roughly 1.1 bits of information per letter. If you have some text with n bits of information, there are $latex 2^{n} &bg=e6eaea&s=0$ ways it can be interpreted. So a written English sentence containing t letters, or about 1.1t bits, can be interpreted $latex 2^{1.1t} &bg=e6eaea&s=0$ ways.

This certainly cuts down the number of novels. There are roughly $latex 2^{165000} &bg=e6eaea&s=0$ possible 50,000 word “novels.”

Does that make much of a difference?

No. The probability that any of the 150,000-character strings constitutes a 50,000-word “novel” has 178,000 zeroes after the decimal.

And keep in mind that I’m using a very loose definition of the word “novel”. A 50,000-word “novel” in this context just needs to be a collection of sentences that are readable individually, but not necessarily together. Here’s an excerpt of my fan-fic sequel to Atlanta Nights:

“Jim, do I have a dingo on my back?”
The sun set three fortnights ago on Afdw-IX.

But it’s not hopeless for my team of hypothetical monkeys yet. Perhaps out there is another segment of our universe, not unlike our own, except that not only did my team of monkeys actually write a 50,000-word novel, but also this blog post. And I have a pet duck. Just because, I suppose. The other me has a reason.

I bet its name is Fred.

Revision Sunday – Mara of the Ori

  By Robert Watson | 3 November, 2013 - 03:28 | , ,
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Largely due to my other commitments, the monthly riddles are now on hiatus. But I’ve also been mulling over the Mara of the Ori riddles (the two that have been posted, and the unpublished ones).

Originally I just wrote them for fun. I think, though, there may be more potential in them – if I can get some consistency in the level of the mathematics – as well as changing that level itself.

The two published riddles have a very high level of math. The second one, which is my favorite of the two, was based off of a problem I was given in a course I took in functional analysis. For those who haven’t heard of it – it’s a 6000-level math course (graduate level) at the University of Tulsa.

I think Mara of the Ori would do better to serve as a means to share a wonderful subject with people who otherwise view it as a dry, soulless exercise they must perform to earn a degree.

This means revising the mathematical principles that support the riddles. The tales themselves would remain largely unchanged – except for the ends, of course.

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Consider this an impending reboot, I suppose, of the first two riddles. It’s something I’m a bit hesitant to do on this blog. I generally consider sites like DeviantART my repository for drafts, and the blog for final copies. But I think the project has the potential to mature, and I’d hate to hold it back for the purposes of a rule that exists only in my mind.

In the meantime, I should post a discussion of the answers to the original versions, since I did receive quite a few answers (and I thank those of you who were supportive!)

However, for now, I have the second draft of An Ember in the Wind to wrestle with.

justmara