The problem I’d set out last week: I have a teapot good for about three cups of tea. I want to put milk in the once, before the first cup. How much should I drink before topping up the cup, to have the most milk at the end?
I have expectations. Some of this I know from experience, doing other problems where things get replaced at random. Here, tea or milk particles get swallowed at random, and replaced with tea particles. Yes, ‘particle’ is a strange word to apply to “a small bit of tea”. But it’s not like I can call them tea molecules. “Particle” will do and stop seeming weird someday.
Random replacement problems tend to be exponential decays. That I know from experience doing problems like this. So if I get an answer that doesn’t look like an exponential decay I’ll doubt it. I might be right, but I’ll need more convincing.
I also get some insight from extreme cases. We can call them reductios. Here “reductio” as in the word we usually follow with “ad absurdum”. Make the case ridiculous and see if that offers insight. The first reductio is to suppose I drink the entire first cup down to the last particle, then pour new tea in. By the second cup, there’s no milk left. The second reductio is to suppose I drink not a bit of the first cup of milk-with-tea. Then I have the most milk preserved. It’s not a satisfying break. But it leads me to suppose the most milk makes it through to the end if I have a lot of small sips and replacements of tea. And to look skeptically if my work suggests otherwise.
So that’s what I expect. What actually happens? Here, I do a bit of reasoning. Suppose that I have a mug. It can hold up to 1 unit of tea-and-milk. And the teapot, which holds up to 2 more units of tea-and-milk. What units? For the mathematics, I don’t care.
I’m going to suppose that I start with some amount — call it — of milk. is some number between 0 and 1. I fill the cup up to full, that is, 1 unit of tea-and-milk. And I drink some amount of the mixture. Call the amount I drink . It, too, is between 0 and 1. After this, I refill the mug up to full, so, putting in units of tea. And I repeat this until I empty the teapot. So I can do this times.
I know you noticed that I’m short on tea here. The teapot should hold 3 units of tea. I’m only pouring out . I could be more precise by refilling the mug times. I’m also going to suppose that I refill the mug with amount of tea a whole number of times. This sounds necessarily true. But consider: what if I drank and re-filled three-quarters of a cup of tea each time? How much tea is poured that third time?
I make these simplifications for good reasons. They reduce the complexity of the calculations I do without, I trust, making the result misleading. I can justify it too. I don’t drink tea from a graduated cylinder. It’s a false precision to pretend I do. I drink (say) about half my cup and refill it. How much tea I get in the teapot is variable too. Also, I don’t want to do that much work for this problem.
In fact, I’m going to do most of the work of this problem with a single drawing of a square. Here it is.
So! I start out with units of tea in the mixture. After drinking units of milk-and-tea, what’s left is units of milk in the mixture.
How about the second refill? The process is the same as the first refill. But where, before, there had been units of milk in the tea, now there are only units in. So that horizontal strip is a little narrower is all. The same reasoning applies and so, after the second refill, there’s milk in the mixture.
If you nodded to that, you’d agree that after the third refill there’s . And are pretty sure what happens at the fourth and fifth and so on. If you didn’t nod to that, it’s all right. If you’re willing to take me on faith we can continue. If you’re not, that’s good too. Try doing a couple drawings yourself and you may convince yourself. If not, I don’t know. Maybe try, like, getting six white and 24 brown beads, stir them up, take out four at random. Replace all four with brown beads and count, and do that several times over. If you’re short on beads, cut up some paper into squares and write ‘B’ and ‘W’ on each square.
But anyone comfortable with algebra can see how to reduce this. The amount of milk remaining after j refills is going to be
How many refills does it take to run out of tea? That we knew from above: it’s refills. So my last full mug of tea will have left in it
units of milk.
Anyone who does differential equations recognizes this. It’s the discrete approximation of the exponential decay curve. Discrete, here, because we take out some finite but nonzero amount of milk-and-tea, , and replace it with the same amount of pure tea.
Now, again, I’ve seen this before so I know its conclusions. The most milk will make it to the end of is as small as possible. The best possible case would be if I drink and replace an infinitesimal bit of milk-and-tea each time. Then the last mug would end with of milk. That’s as in the base of the natural logarithm. Every mathematics problem has an somewhere in it and I’m not exaggerating much. All told this would be about 13 and a half percent of the original milk.
Drinking more realistic amounts, like, half the mug before refilling, makes the milk situation more dire. Replacing half the mug at a time means the last full mug has only one-sixteenth what I started with. Drinking a quarter of the mug and replacing it lets about one-tenth the original milk survive.
But all told the lesson is clear. If I want milk in the last mug, I should put some in each refill. Putting all the milk in at the start and letting it dissolve doesn’t work.