Wine & Tea & Coins

Wine & Tea

We start with a wine and tea problem that came from a 1997 interview with the Russian mathematician Vladimir Arnol’d.

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 (wine) in the cup and some foreign substance (tea) in the barrel.

Which is larger at the end of your manipulations: the quantity of wine in the cup or the quantity of tea in the barrel?

The objective is not only to solve the problem, but to come up with an elegant and insightful diagram that makes your solution clear.

We let the students work on this problem for a while, but then we switch over to the coins problem which seems completely different.

Coins

I take a pile of coins, give them to a student, put on a blind-fold, and ask the student dump them randomly on the table. Then I ask them to tell me how many heads there are. They tell me that there are 3 heads.

I then declare that, without taking off the blindfold, I will make two piles of coins with the same number of heads in each pile. I add that if I wished I could turn some of the coins over.

That doesn’t at first seem possible, but that’s exactly what I do. When I have finished, I ask them to count the number of heads in each pile. Indeed I have been true to my word: there are 2 heads in each pile.

As I have said, the two problems seem completely different but they are in many ways the same problem in disguise. The two types of coins are, indeed, the wine and the tea. This isomorphism is what is illustrated so well with the diagrams.