# The Universe of Discourse

Mon, 01 May 2006

Once again I am going to write up the interesting Google queries that my blog has attracted this month.

Probably the one I found most interesting was:

           1 if n + 1 are put inside n boxes, then at least one box
will contain more than one ball. prove this principle by
induction.

But I found this so interesting that I wrote a 1,000 word article about it, which is not finished. Briefly: I believe that nearly all questions of the form "solve the problem using/without using technique X" are pedagogically bogus and represent a failure of instructor or curriculum. Well, it will have to wait for another time.

Another mathematical question that came up was:

           1 a collection of 2 billion points is completely enclosed
by a circle.  does there exist a straight line having
exactly 1 billion of these points on each side

This one is rather interesting. The basic idea is that you take a line, any line, and put it way off to one side of the points; all the points are now on one side of the line. Then you move the line smoothly across the points to the other side. As you do this, the number of points on one side decreases and the number of points on the other side increases until all the points are on the other side. Unless something funny happens in the middle, then somewhere along the way, half the points will be on one side and half on the other. For concreteness, let's say that the line is moving from left to right, and that the points start out to the right of the line.

What might happen in the middle is that you might have one billion minus n points on the left, and then suddenly the line intersects more than n points at once, so that the number of points on the left jumps up by a whole bunch, skipping right past one billion, instead of ticking up by one at a time. So what we really need is to ensure that this never happens. But that's no trouble. Taking the points two at a time, we can find the slope of the line that will pass through the two points. There are at most 499,999,999,500,000,000 such slopes. If we pick a line that has a slope different from one of these, then no matter where we put it, it cannot possibly intersect more than one of the points. Then as we slide the line from one side to the other, as above, the count of the number of points on the left never goes up by more than 1 at a time, and we win.

Another math query that did not come from Google is:

        Why can't there be a Heron's formula for an arbitrary quadrilateral

Heron's formula, you will recall, gives the area of a triangle in terms of the lengths of its sides. The following example shows that there can be no such formula for the area of a quadrilateral:

All three quadrilaterals have exactly the same edge lengths, but their areas are different, so there couldn't possibly be a formula to tell you the area from just the edge lengths.

The following query is a little puzzling:

           1 undecidable problems not in np

It's puzzling because no undecidable problem is in NP. NP is the class of problems for which proposed solutions can be checked in polynomial time. Problems in NP can therefore be solved by the simple algorithm of: generate everything that could possibly be a solution, and check each one to see if it is a solution. Undecidable problems, on the other hand, cannot be solved at all, by any method. So if you want an example of an undecidable problem that is not in NP, you start by choosing an undecidable problem, and then you are done.

It might be that the querent was looking for decidable problems that are not in NP. Here the answer is more interesting. There are many possibilities, but surprisingly few known examples. The problem of determining whether there is any string that is matched by a given regex is known to require exponential time, if regular expressions are extended with a {2} notation so that a{2} is synonymous with aa. Normally, if someone asks you if there is any string that matches a regex, you can answer just by presenting such a string, and then the querent can check the answer by running the regex engine and checking (in polynomial time) that the string you have provided does indeed match.

But for regexes with the {2} notation, the string you would provide might have to be gigantic, so gigantic that it could not be checked efficiently. This is because one can build a relatively short regex that matches only enormous strings: a{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2} is only 61 characters long, and it does indeed match one string, but the string it matches is 1,048,576 characters long.

Many problems involving finding the good moves in certain games are known to be decidable and believed to not be in NP, but it isn't known for sure.

Many problems that involve counting things are known to be decidable but are believed to not be in NP. For example, consider the NP-complete problem discussed here, called X3C. X3C is in NP. If Sesame Workshop presents you with a list of episodes and a list of approved topics for dividing the episodes into groups of 3, and you come up with a purported solution, Sesame Workshop can quickly check whether you did it right, whether each segment of Elmo's World is on exactly one video, and whether all the videos are on the approved list.

But consider the related problem in which Sesame Workshop comes to you with the same episodes and the same list of approved combinations, and asks, not for a distribution of episodes onto videotapes, but a count of the number of possible distributions. You might come back with an answer, say 23,487, but Sesame Workshop has no way to check that this is in fact the right number. Nobody has been able to think of a way that they might do this, anyhow.

Such a problem is clearly decidable: enumerate all possible distributions of episodes onto videos, check each one to see if it satisfies Sesame Workshop's criteria, and increment a counter if so. It is clearly at least as hard as the NP-complete problem of determining whether there is any legal distribution, because if you can count the number of legal distributions, there is a legal distribution if and only if the count of legal distributions is not 0. But it may well be outside of NP, because seems quite plausible that there is no quick way to verify a purported count of solutions, short of generating, checking, and recounting all possible distributions.

This is on my mind anyway, because this month I got email from two separate people both asking me for examples of problems that were outside of NP but still decidable, one on March 18:

I was hoping to get some information on a problem that is considered np-hard but not in NP (other than halting problem).

And then one on March 31:

I was visiting your website in search of problems that are NP-Hard but not NP-Complete which are decision problems, but not the halting problem.
It turned out that they were both in the same class at Brock University in Canada.

Here's one that was surprising:

           1 wife site:plover.com

My first thought was that this might have been posted by my wife, looking to see if I was talking about her on my blog. But that is really not her style. She would be much more likely just to ask me rather than to do something sneaky. That is why I married her, instead of a sneaky person. And indeed, the query came from Australia. I still wonder what it was about, though.

           1 mathematical solution for eliminating debt

Something like this has come up month after month:

        [18/Jan/2006:09:10:46 -0500] eliminate debt using linear math
[08/Feb/2006:13:48:13 -0500] linear math system eliminate debt
[08/Feb/2006:13:53:28 -0500] linear math system eliminate debt
[25/Feb/2006:15:12:01 -0500] how to get out of debt using linear math
[23/Apr/2006:10:32:18 -0400] Mathematical solution for eliminating debt
[23/Apr/2006:10:33:43 -0400] Mathematical solution for eliminating debt

At first I assumed that it was the same person. But analysis of the logs suggests it's not. I tried the query myself and found that many community colleges and continuing education programs offer courses on using linear math to eliminate debt. I don't know what it's about. I don't even know what "linear" means in this context. I am unlikely to shell out \$90 to learn the big secret, so a secret it will have to remain.

           1 which 2 fraction did archimedes add together to write 3/4

I don't know what this was about, but it reminded me of Egyptian fractions. Apparently, the Egyptians had no general notation for fractions. They did, however, have notations for numbers of the form 1/n, and they could write sums of these. They also had a special notation for 2/3. So they could in fact write all fractions, although it wasn't always easy.

There are several algorithms for writing any fraction as a sum of fractions of the form 1/n. The greedy algorithm suffices. Say you want to write 2/5. This is bigger than 1/3 and smaller than 1/2, so write 2/5 = 1/3 + x. x is now 1/15 and we are done. Had x had a numerator larger than 1, we would have repeated the process.

The greedy algorithm produces an Egyptian fraction representation of any number, but does not always do so conveniently. For example, consider 19/20. The greedy algorithm finds 19/20 = 1/2 + 1/3 + 1/9 + 1/180. But 19/20 = 1/2 + 1/4 + 1/5, which is much more convenient. So the problem of finding good representations for various numbers was of some interest, and the Ahmes papyrus, one of the very oldest mathematical manuscripts known, devotes a large amount of space to the representations of various numbers of the form 2/n.

           1 why does pi appear in both circle area and circumference?

This is not a coincidence. I have a mostly-written article about this; I will post it here when I finish it.

           1 the only two things in our universe starting with m & e

I hope the high school science teacher who asked this idiotic question burns in hell.

It does, however, remind me of the opening episode of The Cyberiad, by Stanislaw Lem, in which Trurl the inventor builds a machine that can manufacture anything that begins with the letter N. Trurl orders his machine to manufacture "needles, then nankeens and negligees, which it did, then nail the lot to narghiles filled with nepenthe and numerous other narcotics." The story goes on from there, with the machine refusing to manufacture natrium:

"Never heard of it," said the machine.

"What? But it's only sodium. You know, the metal, the element..."

"Sodium starts with an s, and I work only in n."

"But in Latin it's natrium."

"Look, old boy," said the machine, "if I could do everything starting with n in every possible language, I'd be a Machine That Could Do Everything in the Whole Alphabet, since any item you care to mention undoubtedly starts with n in one foreign language or another. It's not that easy. I can't go beyond what you programmed. So no Sodium."

I have a minor reading disorder: I hardly ever think books are funny, except when they are read out loud. When I tell people this, they always start incredulously enumerating funny books: "You didn't think that the Hitchhiker's Guide books were funny?" No, I didn't. I thought Fear and Loathing in Las Vegas was the dumbest thing I'd ever read, until I heard James Woodyatt reading it aloud in the back of the car on the way to a party in Las Vegas (New Mexico, not Nevada) and then I laughed my head off. I did not think Evelyn Waugh's humorous novel Scoop was funny. I did not think Daniel Pinkwater's books were funny. I do not think Christopher Moore's books are funny. Louise Erdrich's books are sometimes funny, but I only know this because Lorrie and I read them aloud to each other. Had I read them myself, I would have thought them unrelievedly depressing.

I think The Cyberiad is uproarious. It is easily the funniest book I have ever read. I laugh out out every time I read it.

The most amazing thing about it is that it was originally written in Polish. The translator, Michael Kandel, is a genius. I would not have dared to translate a Polish story about a machine that can make anything that begins with n. It is an obvious death trap. Eventually, the story is going to call for the machine to manufacture something like napad rabunkowy ("stickup") that begins with n in Polish but not in English. Perhaps the translator will get lucky and find a synonym that begins with n in English, but sooner or later his luck will run out.

I once met M. Kandel, and I asked him if it had been n in the original, or if it had been some other letter that was easy in Polish but impossible in English, like z, or even ł. No, he said it had indeed been n. He said that the only real difficulty he had was that Trurl asks the machine to make nauka ("science") and he had to settle for "nature".

(Incidentally, I put the word napad rabunkowy into the dictionary lookup at poltran.com and it told me angrily "PLEASE TYPE POLISH WORDS USING CAPITAL LETTERS ONLY!" Yeah, that's a good advertisement for your software.)

 Order Consciousness Explained from Powell's
In addition to being funny, The Cyberiad is philosophically serious. In the final chapter, Trurl is contracted by a wicked, violent despot to manufacture a new kingdom to replace the one that overthrew and exiled him. Trurl does so, and then he and his friend Klapaucius debate the ethics of this matter. Trurl argues that the miniature subjects in the replacement kingdom are not actually suffering under the oppressive rule of the despot, because they are only mechanical, and programmed to give the appearance of suffering and misery. Klapaucius persuades him otherwise, and me also. Perhaps someday I will write a blog article comparing Klapaucius's arguments with the similar arguments against zombies in Dennett's book Consciousness Explained.

By the way, narghiles are like hookahs, and nankeens are trousers made of a certain kind of cotton cloth, also called nankeen.

The most intriguing query I got was:

           1 consciousness torus photon core

Wow, isn't that something? I have no idea what this person was looking for. So I did the search myself, to see what would come up. I found a lot of amazingly nutty theories about physics. As I threatened a while back, I may do an article about crackpotism; if so, I will certainly make this query again, to gather material.

My favorite result from this query is unfortunately offline, available only from the Google cache:

The photon is actually composed of two tetrahedrons that are joined together, as we see in figure 4.6, and they then pass together through a cube that is only big enough to measure one of them at a time.
Wow! The photon is actually composed of two tetrahedrons! Who knew? But before you get too excited about this, I should point out that the sentence preceding this one asserted that the volume of a regular tetrahedron is one-third the volume of the smallest sphere containing it. (Exercise: without knowing very much about the volumes of tetrahedra and their circumscribed spheres, how can you quickly guess that this is false?)

Also, I got 26 queries from people looking for the Indiana Pacers' cheerleaders, and 31 queries from people looking for examples of NP-complete problems.

This is article #100 on my blog.