Scribble, scribble, scribble

notes

As I have done for several previous concerts, I've put together some rambling notes on the text for my choir's next concert, in the hope that they'll be useful to other choir members. I'm putting a link here in case anyone else is interested.

cake

My all-purpose chocolate-and-nut cake recipe (which I'm posting because someone asked me for it). It's a slight variation on the one at the end of Nigel Slater's Appetite, which in turn is a slight variation on someone else's.

The recipe is pretty much infinitely malleable; I've never made it quite the same way twice. It's pretty hard to make it not work. The numbers here are what I did the last time I made it.

Toast 200–250g of hazelnuts and almonds (whatever proportion you prefer) lightly; maybe 8 minutes in the oven at 180°C. Let them cool, then chop them up into whatever size bits you prefer along with 250g good dark chocolate.

Soften 250g butter (I just put it in the microwave at very low power for a few minutes) and beat it up with 250g of sugar (some combination of demerara and light or dark brown soft; I usually use more of the former) until pale and fluffy. I'm usually pretty lax about getting it really fluffy, and it works fine.

Add 4 large eggs one by one, beating each one in.

Add 3–4tbsp amaretto and stir in.

Measure out 170g plain flour and 100g ground almonds; add 2tsp (rounded) of baking powder. Sift them into the mixture if you can be bothered; I usually just weigh them out and dump them in. Fold them in.

Stir in most of the chocolate and nuts; leave out 2–3tbsp.

Put it into a lined cake tin of suitable size (say 23cm), preferably one of those with a spring-clip side and a removable base.

Sprinkle the rest of the chocolate and nuts on top, along with another 2tsp demerara sugar.

Bake at 170°C for 70 minutes.

You may need to adjust the cooking temperature and time for your oven. Mine is fan-assisted and runs about 10°C hot, and I more often overdo than underdo this cake.

I usually make this with Green & Black's chocolate: some combination of their 85% dark and their "Maya Gold" (55% or thereabouts; quite orangey).

Slater's recipe has coffee where I have amaretto; his source has milk; I'm sure you could use brandy or Grand Marnier or even water.

cars

Proposition: Cars and similar vehicles should be required by law to have at the rear a prominently-visible digital readout of their current speed.

Advantage 1: it would be that bit easier to match the speed of the car in front, which is generally a Good Thing for efficient traffic flow. (Actually you might want a low-pass-filtered version of the speed of the car two in front, or something; anyway, whatever it is you want, the extra information will do no harm.)

Advantage 2: it would act as a mild deterrent to speeding.

Advantage 3: it would make it slightly easier for police to catch people speeding.

(What's that you say? Sometimes driving faster than the limit is a perfectly reasonable thing to do, and sometimes the police prioritize fundraising over safety when it comes to catching speeders? Quite possibly. In which case, the thing to do is to fix the laws or the speed limits, not to rely on the fact that sometimes it's inconvenient to tell whether someone's speeding or to prove they are.)

It might be difficult to do this visibly enough to be useful without obscuring the view through the rear window; that might be sufficient reason not to do it.

cost

How much does a book cost?

The room I’m in right now contains approximately 1800 books. The shelving on which those books (along with not very much else) are stored cost, I think, about £1500 or so. (You can get bookshelves that cost a lot less, if you don’t mind them sagging and/or looking unsightly. You can get bookshelves that cost a lot more, if you want beauty as well as quality.) So, crudely: shelving a book costs £1.

That room, though. It didn’t come for free, and if we had drastically more books we’d need a bigger house to put them in. (I’m fairly sure that every time we’ve been through the soul-destroying business of buying a house we’ve rejected some houses on the grounds that they didn’t have enough spare wall-space for bookshelves, so this isn’t a purely theoretical concern.) A quick look at house prices in our area suggests that one decent-sized room typically adds something of the order of £40,000 to the price of a house. (Less for smaller houses, more for larger houses; larger houses have larger rooms.) So, crudely: housing a book costs £20.

I’m not sure I wanted to know that.

Of course there’s lots wrong with that analysis. For instance, this room contains not only books but also computers, desk space, etc., and we can fit quite a lot more books into our house before it’s so full we have to buy a new one. Even so, I think it would be quite difficult to justify an estimate of the overhead cost per book that’s below, let’s say, £5. But do I think of that in second-hand bookshops? Why, no, I do not.

There is presumably a broader lesson here.

machine

I just got Knuth’s newly published book of algorithm papers. There are many nice things in it; here’s one. (It’s not actually Knuth’s; it’s due to Larry Stockmeyer.)

Suppose you have a machine whose basic operations are integer addition, subtraction and multiplication, and greater-than-or-equal testing. Theorem: you can’t determine whether a number n is odd faster than some constant times log(n).

Proof: suppose you can; then, in particular, for large enough k your program should be able to determine for any number up to (let’s say) 2^10k whether the number is odd or even, and do something different in each case, using at most k steps. I’ll show that it can’t.

Well, what can the program have done within k steps? It’s done at most k comparisons so there are at most 2^k code paths. Each comparison is testing the sign of some polynomial in n; what polynomial depends on the results of previous comparisons. In any case, the path the program has taken up to this point can depend only on the values of at most 2^k polynomials in n. Oh, and they all have degree at most 2^k.

Well, all these polynomials collectively have at most 2^2k roots. We can therefore find an integer m no bigger than 2^2k+1 such that none of those roots lies between m and m+1 inclusive. (Each root excludes at most two choices of m.) But then our program must produce the same results for m and m+1 since all the comparisons it does produce the same answer for both; since m and m+1 have different parity, it therefore can’t be doing the right thing for both of them.

Knuth’s book contains many clever algorithms. From the cursory reading I’ve given it so far, its contents don’t seem to be “papers on the design of algorithms” any more than the works of Shakespeare constitute “papers on the writing of plays”. So the title’s a bit misleading, which is a pity (it would be very interesting indeed to read Knuth’s thoughts on how to design algorithms) but not a surprise. The book is of course very good anyway.