I’ve been taking milk in my tea lately. I have a teapot good for about three cups of tea. So that’s got me thinking about how to keep the most milk in the last of my tea. You may ask why I don’t just get some more milk when I refill the cup. I answer that if I were willing to work that hard I wouldn’t be a mathematician.

It’s easy to spot the lowest amount of milk I could have. If I drank the whole of the first cup, there’d be only whatever milk was stuck by surface tension to the cup for the second. And so even less than that for the third. But if I drank half a cup, poured more tea in, drank half again, poured more in … without doing the calculation, that’s surely more milk for the last full cup.

So what’s the strategy for the most milk I could get in the final cup? And how much is in there?

I haven’t done the calculations yet. Wanted to put the problem out and see if my intuition about this matches anyone else’s, and how close that might be to right. Or at least calculated. I suspect it’s one of a particular kind of problem, though.

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## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.
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My guess would be that the best strategy is to get the tea in as quickly as possible- literally refilling with each sip- but I’m afraid that’s instinct and not calculation.

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It’s exactly my instinct too, though! This sort of gradual replacement problem is almost always an exponential decay. And then consider the extreme cases. Drink all the tea and refill the cup and no milk remains. Drink zero tea molecules and replace zero tea molecules and the milk never decreases. So, tiniest possible change is likely the way to go. But that can be made more rigorous.

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Have you tried showing the door both tea and notea at the same time? Of course, this only works when your inprobability drive is on.

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I have not. What I have done is the next-best thing: get out my iPad Pro and drawn a silly little picture! You’ll get to see that soon.

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