A spoonful of cream is taken from a cup of cream and put into a cup of coffee. The coffee is then stirred. Then a spoonful of this mixture is put into the cup of cream.

Is there now more cream in the coffee cup or more coffee in the cup of cream?

 The following problem is from An Interview with Vladimir Arnol'd, Notices of the AMS 44:4, April 1997.

    "You take a spoon of wine from a barrel of wine, and you put it into your cup of tea. Then you return a spoon of the (nonuniform) mixture of tea from your cup to the barrel. Now you have some foreign substance (tea) in the barrel. Which is larger: the quantity of wine in the cup or the quantity of tea in the barrel at the end of your manipulations?"

    The big fog factor here is the barrel, which is largely irrelevant. What is relevant is the spoon. A spoonful of wine goes into the cup of tea, and the same spoon is employed a second time to remove a spoonful of the wine-and-tea mixture that the cup then contains. Some fraction w of the second spoonful is wine and the remaining fraction t=1-w is tea. Since the amount of wine originally added to the tea was exactly one whole spoonful, the amount of wine left behind in the cup after the second spoonful is removed must be the fraction 1-w of a spoonful, since the fraction w of a spoonful has been removed. This fraction of a spoonful of wine left behind in the cup is the very same fraction of a spoonful as that of the tea that has been removed. The amount of tea removed equals the amount of wine that remains. If you then put the fraction 1-w of a spoonful of tea (and the wine that accompanies it) in the wine barrel, you will without any good reason mess up the wine in the barrel, and you probably ought to be shot! But if you do this, or at least pretend to do this, the quantity of tea in the barrel will then equal the quantity of wine in the cup, and hence neither is the larger quantity.

Edited on January 10, 2006, 5:06 pm

Posted by Richard on 2005-05-03 04:28:37