The Universe of Discourse

Weird constants in math problems

Michael Lugo recently considered a problem involving the allocation of swimmers to swim lanes at random, ending with:

If we compute this for large !!n!! we get !!f(n) \sim 0.4323n!!, which agrees with the Monte Carlo simulations… The constant !!0.4323!! is $$\frac{(1-e^{-2})}2.$$

I love when stuff like this happens. The computer is great at doing a quick random simulation and getting you some weird number, and you have no idea what it really means. But mathematical technique can unmask the weird number and learn its true identity. (“It was Old Man Haskins all along!”)

A couple of years back Math Stack Exchange had Expected Number and Size of Contiguously Filled Bins, and although it wasn't exactly what was asked, I ended up looking into this question: We take !!n!! balls and throw them at random into !!n!! bins that are lined up in a row. A maximal contiguous sequence of all-empty or all-nonempty bins is called a “cluster”. For example, here we have 13 balls that I placed randomly into 13 bins:

In this example, there are 8 clusters, of sizes 1, 1, 1, 1, 4, 1, 3, 1. Is this typical? What's the expected cluster size?

It's easy to use Monte Carlo methods and find that when !!n!! is large, the average cluster size is approximately !!2.15013!!. Do you recognize this number? I didn't.

But it's not hard to do the calculation analytically and discover that that the reason it's approximately !!2.15013!! is that the actual answer is $$\frac1{2(e^{-1} - e^{-2})}$$ which is approximately !!2.15013!!.

Math is awesome and wonderful.

(Incidentally, I tried the Inverse Symbolic Calculator just now, but it was no help. It's also not in Plouffe's Miscellaneous Mathematical Constants)

[ Addendum 20200707: WolframAlpha does correctly identify the !!2.15013!! constant. ]

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Useful and informative article about privately funded border wall

The Philadelphia Inquirer's daily email newsletter referred me to this excellent article, by Jeremy Schwartz and Perla Trevizo.

Wow!” I said. “This is way better than the Inquirer's usual reporting. I wonder what that means?” Turns out it meant that the Inquirer was not responsible for the article. But thanks for the pointer, Inquirer folks!

The article is full of legal, political, and engineering details about why it's harder to build a border wall than I would have expected. I learned a lot! I had known about the issue that most of the land is privately owned. But I hadn't considered that there are international water-use treaties that come into play if the wall is built too close to the Rio Grande, or that the wall would be on the river's floodplain. (Or that the Rio Grande even had a floodplain.)

He built a privately funded border wall. It's already at risk of falling down if not fixed, courtesy of The Texas Tribune and ProPublica.

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