You have 10 pieces of string, each with two ends. You randomly pick two ends of string (possibly from the same string, possibly from different ones) and tie them together, creating either a longer piece of string or a loop. You keep doing this until you run out of free ends.

What is the expected number of loops you end up with?

Things to note are the following:

- Once a loop is made from a string, it is removed from further consideration.
- Picking two ends from the same string immediately makes a loop.
- Picking two ends from different strings makes a longer string.

So in the end, no matter which two ends are picked, we have one fewer open string than we started with.

Let \(f(n)\) be the expected number of loops with \(n\) open strings. We know \(f(1)=1\). With \(n\) strings, the probability of picking two ends from the same string is \(n/\binom{2n}{2}\). So:

That is, \(f(n) = \sum_{i=1}^n 1/(2i-1)\), \(f(10) = 1841/863\).

For large \(n\), this sum does not converge, in fact it is obvious that \(f(n)\) grows like \(\log n\).

At first, I thought it was the CD version that was being shut down, but no, it’s the online version; apparently the former, along with many Microsoft Home products (some were classics), had long been discontinued. Incidentally, I’ve used the CD product, but never the online product — I’ve been aware of it because it comes up in searches, but since it’s just the CD version put online, I’m not surprised it is meeting the same fate. It just goes to show that whatever process is driving traditional publishing into the ground is rather far along.

I, for one, still remember paper encyclopedias. For that matter, I still remember when libraries used card catalogues (you pull them out of a small drawer to find the Dewey decimal), but these became extinct at about the same time as the 5.25″ floppy disk. As for encyclopedias, they sat as multivolume collections in the reference section — maybe they still do? Haven’t been to a public library in a long time…

The first CD encyclopedia I remember was Grolier’s. Its selling point was some animations in articles. For a time these encyclopedias were useful for school projects, but by high school they seemed pretty useless — the articles just had too low a signal-to-noise ratio. Maybe they did not provide enough depth, or the short list of references were not adequate, or there was too much fluff that simple queries could not be answered in a well matched way. Often the writers were totally full of themselves, too (reminds me of about.com). The end result was these references could neither be used directly (plagiarism aside), nor were the raw data in them easily extractable. I think that’s one reason why I stopped using them, whatever the media encyclopedias came in. The other reason was that such generalist information was not difficult to find on the internet, even without a Wikipedia.

So while the comparison to Wikipedia is appealing as a foil, these products really failed on their own merits: they were generally inadequate and inferior products and they were not even free for being so.* The economic realities of that are only catching up now. And if newspapers follow them there, it would be because newspapers have long become wire service repeaters, not because of the existence of Google News. Interestingly, I haven’t had the interest to subscribe to these newspapers for a long time, either.

* Inferior compared to what, you say. Isn’t it the existence of a “better” alternative that lies at the crux of the matter? Actually, no. The inferiority is measured from the amount of nagging feeling of not having learned much. The reality is, without an alternative, one would just know less and unless effort be expended, be resigned to that… (Economically, of course the existence of an alternative matters, but that’s a separate issue.)

However, there is a way to figure this matter of structure from first principles (and perhaps generate a more unique style as a result), albeit with the tradeoff that you cannot be quick, you must be careful.

The first principle for aesthetics is that the character must stand … this is something my old man told me, actually, so I didn’t figure this out myself, but it is very true. If you hold up the piece of paper and look at the strokes as struts of a building, it must look like the character is architecturally sound, i.e. reasonably symmetric if need be, balanced in weight so will not tip over, is not poorly supported with too small a bottom and too big a top, etc. This isn’t too difficult if the character is mechanically drawn, but the trick is to do it even with asymmetric calligraphic strokes and multi-part characters with asymmetric radicals and caps.

The second principle for aesthetics is about spacing, and this is much like optimal typography and typesetting. The strokes should be spread out evenly so that where they appear parallel, they appear to have nearly identical spacing as other such spaces. Otherwise there will be ugly bunching and voids. This is very difficult because the strokes are written in order so there is a pre-commitment issue. Once you commit to a particular stroke, it also commits the spacing requirements for the rest of the character. So one slightly off stroke and you are screwed. This is more a problem for large writing, since bigger mistakes are possible.

Then is the issue of multiple character layout. This wouldn’t be so much of an issue if all characters were the same shape and complexity, but they are not. Some are extremely sparse, and some are very dense. Some are tall and some are fat. They all have to be laid out on paper to look like they take up the same space and also evenly spaced from each other. There is also the compromise of making inter-stroke space appear similar in multiple characters. So one needs to deal with some visual artifacts and vision tricks. As a result, the characters will not all be the same size and will not be spaced evenly, so this is a very tricky thing to get right. You can have perfectly written individual characters but still a terrible collection.

And finally here is a side point: people say Simplified characters are uglier than Traditional characters for calligraphy. In fact this cannot be true. What happens is Simplified characters are sparser and sparser characters writ large are the most difficult to get correctly (not to mention there are no classic master’s character books to trace in Simplified). They are ugly only because (or to the extent that) they are not written well. The bastion of poor practioners (like me) is in small dense characters that distract from scrutiny and generally look pretty good no matter how you write them.

For that matter, what is the fear of a central bank? In some countries, banks are like utilities and post offices — public services provided by the government. The First and Second National Banks were killed because people did not trust the government with their money. Well, it seems like big commercial banks can be trusted even less. Also, it’s not like anything would function with just community banks. It’s not a country of farmers any more…

I mean, capital allocation decisions can still be locally made and subject to market forces, as they should, but banking infrastructural issues like described here (and regulatory ones, some say) should have no reason not to be national, am I wrong?