I came across a number of puzzle-type items this summer. Here are a few that I found particularly interesting. See me if you want my answers. (There is more than one way to reason through these problems.).

Headline: "Congress Kills Bill to Repeal the Law Against Smoking in Public Places". Does this mean Congress favors no-smoking in public places?

Author Lewis Carroll was a noted mathematician. The following is from his *Alice in Wonderland*.

A bag initially contains one chip, known to be either white or black. A white chip is then put in the bag, and a chip drawn out, which is seen to be white. What is now the probability of drawing a white chip from the bag?

(At first sight, it would appear that inasmuch as the state of the bag after the white chip is withdrawn is identical to the initial state of the bag, the probability of drawing a white chip is just what it was initially, namely \(\frac{1}{2}\). However, this is not true.)

A says B lies, B says C lies, C says A and B lie. Who lies and who tells the truth?

Here is an interesting application of mathematics to solve a problem in economics and world trade.

Consider two countries (call them Country A and Country B) both of which grow corn and wheat. Country A has \(200\text{ farmers}\) that grow a total of \(1200\text{ bushels of wheat}\) a year (\(6\text{ bushels per farmer}\)), and \(300\text{ farmers}\) that grow a total of \(1200\text{ bushels of corn}\) a year (\(4\text{ bushels per farmer}\)).

Country B is much less efficient than Country A, and has \(600\text{ farmers}\) that grow a total of \(1200\text{ bushels of wheat}\) a year (\(2\text{ bushels per farmer}\)), and \(400\text{ farmers}\) that grow a total of \(1200\text{ bushels of corn}\) a year (\(3\text{ bushels per farmer}\)).

Country A is clearly more efficient at growing both corn and wheat than Country B. The question is: Can these two countries reach a trade agreement that is beneficial to both sides?

John Schwarz