Addenda to recent articles 201910
Several people have written in with helpful remarks about recent posts:
Regarding online tracking of legislation:
 Ed Davies directed my attention to
www.legislation.gov.uk, an
official organ of the British government, which says:
The aim is to publish legislation on this site simultaneously, or at
least within 24 hours, of its publication in printed form.
M. Davies is impressed. So am I. Here is the European Union
(Withdrawal) Act
2018.
This then led me to Standardizing the World’s Legislative
Information — One hackathon at a
time
on the LII's VOXPOPULII blog.
(Reminder to readers: I do not normally read Twitter, and it is
not a reliable way to contact me.)
Regarding the mysteriously wide letter
‘O’ on the
Yeadon firehouse. I had
I had guessed that it was not in the same family as the others, perhaps
because the original one had been damaged. I asked Jonathan
Hoefler, a noted font expert;
he agreed.
But one reader, Steve Nicholson, pointed out that it is quite
common, in Art Deco fonts, for the ‘O’ to be circular even when that
makes it much wider than the other letters. He provided ten
examples, such as Haute
Corniche.
I suggested this to M. Hoefler, but he rejected the theory
decisively:
True; it's a Deco mannerism to have 'modulated capitals'… .
But this isn't a deco font, or a deco building, and in any case it
would have been HIGHLY unlikely for a municipal sign shop to spec
something like this for any purpose, let alone a firehouse. It's a
wrong sort O, probably installed from the outset.
(The letter spacing suggests that this is the original ‘O’.)
Several people wrote to me about the problem of taking half a pill
every day, in which I overlooked
that the solution was simply the harmonic numbers.
Robin Houston linked to this YouTube video, “the frog
problem”,
which has the same solution, and observed that the two problems are
isomorphic, proceeding essentially as Jonathan Dushoff does below.
Shreevatsa R. wrote a long blog article detailing their thoughts
about the solution.
I have not yet read the whole thing carefully but knowing
M. Shreevatsa, it is well worth reading. M. Shreevatsa concludes,
as I did, that a Markov chain approach is unlikely to be fruitful,
but then finds an interesting approach to the problem using
probability generating functions, and then another reformulating
it as a ballsinbins problem.
Jonathan Dushoff sent me a very clear and elegant solution and
kindly gave me permission to publish it here:
The first key to my solution is the fact that you can add expectations
even when variables are not independent.
In this case, that means that each time we break a pill we can
calculate the probability that the half pill we produce will "survive"
to be counted at the endpoint. That's the same as the expectation of
the number of halfpills that pill will contribute to the final total.
We can then just add these expectations to get the answer! A little
counterintuitive, but absolutely solid.
The next key is symmetry. If I break a half pill and there are !!j!! whole
pills left, the only question for that half pill is the relative
order in which I pick those !!j+1!! objects. In particular, any other half
pills that exist or might be generated can be ignored for the purpose
of this part of the question. By symmetry, any of these !!j+1!! objects is
equally likely to be last, so the survival probability is !!\frac1{j+1}!!.
If I start with !!n!! pills and break one, I have !!n1!! whole pills left, so
the
probability of that pill surviving is !!\frac1n!!. Going through to the end we
get the answer:
$$\frac1n + \frac1{n1} + \ldots + 1.$$
I have gotten feedback from several people about my Haskell type
constructor clutter, which
I will write up separately, probably, once I digest it.
Thanks to everyone who wrote in, even people I forgot to mention
above, and even to the Twitter person who didn't actually write in.
[Other articles in category /addenda]
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