In today’s New York Times is a math riddle from Charles Darwin.
If I have a cup of brandy and a cup of water. I take a teaspoon of water and add it to the brandy. Then I take a teaspoon of the mixture and add it to the water cup, does the water have more brandy in it than the brandy cup has water in it?
No, both have the same amount of impurities as the other.
OK, so what is so tough about this? Isn’t the answer obvious?
This is usually attributed to Lewis Carroll.
I don’t think the answer is obvious. Well, actually, I think the answer is intuitive, but once you start thinking about it, you doubt your intuition, and after you’ve puzzled over it, you decide that your original intuition was correct.
[spoiler]You have two containers with one unit of brandy and water respectively. You take a quarter-unit of water and put it in the brandy.
Now container 1 has 3/4 units of 100% water, and container 2 has 5/4 units of 80% brandy.
Here’s where you start to doubt your initial intuition: will a quarter unit of 80% brandy, added to 3/4 units of 100% water create 1 unit of 80% water? The answer is not immediately apparent.
Puzzling it out, you realize that, yes, it will: that quarter unit of 80% brandy is 1/20 unit water and 4/20 units water, added to 15/20 units water, making 16/20 units of water, or 80%. The containers match.[/spoiler]
You don’t even have to do any math.
Consider that the cup of water is now “contaminated” with some quantity X of brandy. The same volume of water X displaced by the brandy is now over in the cup of brandy.
Yes - as long as the levels remain at their original points after any amount of swapping, it can only be the case that each contains an equal volume of the other as contaminant.
In reality, however, it might not be quite so simple - add 100ml of alcohol to 100ml of water and after mixing, the volume is less than 200ml - because of the way the molecules pack together in mixture.
Not sure whether it can be assumed that a tablespoon of water in a tumbler of brandy will enjoy the same packing advantage as a tablespoon of brandy in a tumbler of water - therefore, any operation after the first one might be transferring a different molar amount than expected.
Lewis Carroll it was. I read some sort of book review on the plane, and also a review of a Darwin biography.
No, it’s not obvious. You take pure water and add it to the brandy. Then you take diluted brandy and add it to the water. And there’s the thing, it’s diluted. Instinctively you might think that you are putting more water into the brandy, and less brandy into the water.
Yep. I first encountered this problem in 8th grade algebra when we studied mixtures and rate/time/distance and I forget what the third class of problems was that makes up the triumvirate.
I was suprised to find out that the algebra showed that each glass has the same amount. Because I was thinking pure versus diluted like you mention.
Once you’ve seen the math behind it, it does become apparent, but intuitively it is not so.
But wait, there is another facet. Remember, one cup of alcohol mixed with one cup of water does not yield two cups of liquid. I think that might play a factor here, because if you put one teaspoon of water in the brandy, you won’t have one cup plus one teaspoon in volume, so when you remove the teaspoon of liquid to return it to the first glass, you will leave less than a cup of liquid behind.
Even as I am an AA-er, I find it infinitely more entertaining
to prefer drinking the brandy, and to hell with Babe Ruth.
Of course you actually are doing that. It’s just that you’re also putting water back in the water cup.
(Unless you meant ‘putting’ in the sense of “the state of things when you’re done mixing back and forth”.)
I’ll admit, I had to work through tedious symbol-shuffling to override the intuition preparing me for otherwise in order to obtain the right answer, but this is a much nicer way to see it.