Will do math for chocolate

I don’t usually pay too much attention to the conversations in my office that are in German. My office mates both have many people coming in to talk to them and are both German, so speak in German with their German visitors. This particular conversation caught my attention primarily because it repeatedly used the work ‘Schokolade” one of the few German words I have no trouble with. Also there were four people involved. Eventually one of my office mates said, “Ask Dan, he is very interested in chocolate!”

So the conversation was about a simple demography puzzle, and my boss had offered a chocolate bar to whoever solved it first.
The problem is this: 100% of a population were alive at the beginning of a year. 60% were alive at the end of the year. Assuming that the mortality rate is constant throughout the year, what percentage of individuals were alive at the exact middle of the year?

Note that a constant mortality rate does not mean that the same absolute number of individuals die each day, but rather than the same proportion of those starting the day alive end the day dead.
Motivated by chocolate, and using one simple bit of algebra, I secured the Intense Orange Lindt dark chocolate bar.

So I can’t hang a chocolate bar in the comments section, but I can promise kudos if you can tell me what portion of individuals were alive after six months, and how you got that answer.

Why I can love chocolate

The conversation usually goes something like this:

Me- I'm allergic to caffeine. It makes me get a terrible headache, then get really sleepy and sleep for 12 hours and then I still have a terrible headache.

You- Um, but I see you eat chocolate all the time. You're always talking about chocolate and writing about chocolate. You are a total chocolate adict.

Me- Yeah! I love chocolate. It's my main drug.

You- Um, but chocolate has caffeine.

Me- Well, it has a tiny bit, but mostly it has other closely related chemicals.

I've had this conversation with enough dozens of people that I figured I should look up what was in chocolate. And it turns out I actually did know what I was talking about (for once). According to this article in the journal European Food Research and Technology, cacao beans have three main kinds of very similar chemicals in the group called methylxanthines. These are theobromine (named for the Cacao tree, Theobroma cacao), caffeine, and theophylline. Raw fermented beans straight off the cacao tree have lots of theobromine, very little caffeine and almost no theophylline. In the various preparation steps between then and when I actually eat it, much of the caffeine is lost. The concentration of these various chemicals depends a lot on the strain of cacao, the growing conditions and the processing, but most chocolate has 20 to 100 times as much theobromine as caffeine. A cup of hot cocoa has about half as much caffeine as a cup of decaf coffee.

The fact that chocolate doesn't make me have a terrible headache and put me to sleep is likely (likely meaning I am speculating) either because there is too little caffeine in it to matter or because it has so much theobromine. Theobromine could be counteracting the caffeine, or it could be competitively excluding the caffeine from the neuroreceptors it normally binds to. Basically this means that theobromine and caffeine are so similar that they stick to the same spots on my neurons, and if the theobromine gets there first, the caffeine may not be able to stick, and therefore not affect me. But the moral of the story is I can eat chocolate without worrying about the caffeine making me sick.

Elastic replacement

I've been thinking on an off for several years about the concept of 'elastic replacement.' That is term I made up to mean when the rate at which new things arise increases in response to the effort to remove old things. The example that got me started thinking about this was a political one. For the early years of the US occupation of Iraq, the Bush administration frequently cited statistics on how many terrorists had been killed or captured as a result of US action in Iraq. What they never seemed to consider was that US action in Iraq might also be increasing the replacement of terrorists by making it easier for terrorists to find and recruit young people who might have otherwise been non-violent. The size of any population is determined by not only the number leaving, but also the number entering, and by measuring only one portion of our effect, the US government necessarily got a biased picture of our effect. The CIA has confirmed that the number of active terrorist in Iraq rapidly increased rather than decreasing through the first years of the Iraq war.

This general idea carries over into all sorts of arenas. When a giant old tree falls, it makes space for hundreds of seedlings. When I comb my cat to remove loose hairs, I loosen lots more hairs and possibly even stimulate increased hair growth. WWII caused far fewer deaths of US soldiers than the ensuing baby-boom caused births of US babies. Eating all the chocolate so that my wife won't be tempted by it inspires her to go out and buy more chocolate. Responding to all the emails I should have responded to a long time ago brings in even more emails that need to be responded to.

Elastic replacement seems to be an extremely common class of unintended consequences. Humans don't seem to be good at considering it, but we should try.