Well, when I left you last Saturday morning, I was on my way to join some friends for a look inside the Lawrence Livermore National Laboratory.

“Credit: Lawrence Livermore National Laboratory.”

It went well… very well, in fact.

We probably got there at 10 a.m. First we went to something called HEAF: high explosives applications facility. As you would expect, it’s got lots of chambers where things can go boom! in relative safety. What I particularly enjoyed was a study of why the Lusitania sank in 18 minutes back in World War I. Two explosions rocked the ship… what caused the second one? And was it the second one that did most of the damage?

The Lab investigated explosions of aluminum powder, coal dust, and gun cotton – all of which were somewhere on the ship – and did a simulation of a boiler explosion. We got to see high-speed photography of three explosions. Aluminum powder would have been violent, but also searingly bright – and eyewitness accounts did not correspond to that. I think they decided that coal dust would not have been a very powerful explosion, and that the gun cotton wouldn’t have gone off soon enough. (I don’t really remember the details.)

Ultimately, the Lab decided that the second explosion was a ship’s boiler… and that the damage from the torpedo itself was enough to take the ship down that quickly. Wikipedia has an interesting article; the very end of it discusses the alternatives.

Then we went to the Center for Accelerated Mass Spectroscopy – radio-isotope dating. Saw a few video news clips and so on – in addition to getting nice description of how it all worked. Although the Lab was not one of the places that dated the Shroud of Turin to ~ 1300 A.D. that, after all, may be the most famous example of radiocarbon dating.

By then it was about 1 p.m. and time for lunch. Afterwards… well, we wanted to see the NIF – National Ignition Facility – which will attempt to use lasers to achieve nuclear fusion. But reservations had been required, and we had been too late to get on the list. Bummer. Nevertheless, our inviter – you had to be invited by a lab employee – thought that we should check and make sure. The worst that happens is they say, “No.” We diffidently approached the entrance to the NIF, and said, “I know that reservations were required –” and we were interrupted and told, “Go right in.”

So we did.

What can I say? It’s huge. Part of it reminded me of a scene out of Forbidden Planet, when we first see the huge Krell (Crell?) underground structure. And I hear we’ll all get to see some of it in the next Star Trek movie.

It was most impressive. I’m very glad I saw it. Here’s a general link, and here’s a link to pictures – which aren’t the same as being there.

As it happened, just a week before this expedition, my electronic copy of IEEE Spectrum had contained a less-than-flattering review of the NIF.

It may indeed be a white elephant, but it’s one hell of an elephant.

Mathematically, I stand about where I did last week. The next technical post will either be related to control theory, but also filed under “calculus”: the second-order linear differential equation with constant coefficients – or about number theoretic functions such as Euler’s . Oh, maybe I’ll do the Bode plots for a second-order system before I do the differential equation….

But there’s one more thing I want to add today. I had thought of mentioning it last week, but I didn’t have time before heading out to the Lab. This is another of those things a math geek should know. (This was my latest post about such things; I had a summary of the geek posts here.)

I may be endangering this blog, but I’ll risk it.

More than 20 years ago, in 1990 I think, Marilyn vos Savant was asked a question about a TV game show. I think it was “Let’s Make A Deal”, and the host was Monty Hall. (This is called the Monty Hall problem or paradox.) She answered the question, and got a whole lot of flak. Wikipedia, as often, has an informative article. It says that about a thousand PhDs were among the outraged people who yelled at her. (I remember that my math journals talked about it for a few months, both the problem itself and the shameful response of many professionals.)

It may be that after more than 20 years, the answer is no longer controversial; on the other hand, after 20 years, we’ve got a new generation of academics who might not get it. I may get yelled at, too.

Anyway, she was right, and a lot of professionals had egg on their faces for calling her ignorant and stupid.

A couple of weeks ago a friend sent me a short proof. Then I was reminded of an even shorter one. Here we go.

The rules seem easy, but the Wiki article points out that one could be very pedantic about them. Since the point is that the answer seems counter-intuitive, the rules just have to be what they have to be.

There are three closed doors. There’s a valuable prize (“car”) behind one, and junk (“goat”) behind the other two. Suppose I’m the contestant; I pick a door. The host always opens one of the other two doors – and it is always a door with a goat behind it. Now there are two closed doors, one with a car, one with a goat.

The host asks me if I want to change my choice, to choose the other door. And the question, of course, is: should I switch?

The answer is yes. Two times out of three the car is behind the other door. You might think the odds should be 50-50, but they’re not.

The relatively short proof goes like this. Suppose the car is behind door 2. If I pick door 1, the host must open door 3 to show a goat. If I switch – to door 2, by necessity – I win. Similarly, if I initially pick door 3, the host must open door 1, and if I switch, I win. Only if I had picked door 2, and the host opens either 1 or 3, then if I switch, I lose.

I win two out of three.

(If it bothers you that the host could pick door 1 or 3, enlarge the universe to six equally-probable choices: I choose each of the three doors two out of six times. Then I win 4 out of six. If it bothers you that I assumed the car was behind door 2, then enlarge your universe to 18 equally-likely possibilities. But really, three cases suffice.)

But there’s a much simpler way to see this. The Wiki article has it, but I’m sure I saw this a long time ago and forgot it. Shame on me.

I win if I switch if and only if my initial choice was wrong. And my initial choice is wrong two out of three times.

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