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

Yes. The three negatives amount to the same as one negative (or as they say in math, 3 Opposites are the same as 1 Opposite). This means that the first two negatives (Kills Bill to Repeal the Law) cancel each other, leaving only the last negative (Against Smoking in Public Places). That is Congress is *against smoking* in public places, or, *for non-smoking* in public places.

Author Lewis Carrol 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At first sight, it would appear that inasmuch as the state of the bag after the white chip is withdrawn is identical to the initial state of the bag, the probability of drawing a white chip is just what it was initially, namely \(\frac{1}{2}\).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. Here is my explanation of why.

If a black chip is initially in the bag, and a white chip is added, the white chip withdrawn must be the added white chip, leaving a black chip in the bag (possibility 1).

If a white chip is initially in the bag, and another white chip is added, the white chip withdrawn may be *either* the initial white chip, leaving the added white chip in the bag (possibility 2), *or* the added white chip, leaving the initial white chip in the bag (possibility 3).

In two of the three possibilities, a white chip remains in the bag after the white chip is withdrawn. Therefore the probability of drawing a white chip the second time is \(2\text{ out of }3\).

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

If A is telling the truth, then B must be lying and C must be telling the truth. If C is telling the truth, A and B must lying, but this is inconsistent with the initial assumption that A is telling the truth. This implies that A is lying.

If A is lying, then B must be telling the truth and C must be lying. If C is lying, A or B )or both) must be telling the truth. This condition is satisfied since while A is lying, B is telling the truth.

Therefore one answer to the problem is that A lies, B tells the truth, and C lies.

Extending this line of reasoning to all the other possibilities, it can be shown that the above answer is unique.

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?

Lets say that Country A decides to grow only wheat, and that Country B decides to grow only corn.

In Country A, the \(300\text{ farmers}\) that previously grew corn will now be growing wheat, making a total of \(500\text{ farmers}\) growing wheat. Since a farmer in Country A grows 6 bushels of wheat a year, a total of (6)(500) or 3000 bushels of wheat will be grown in that country each year.

In Country B, the \(600\text{ farmers}\) that previously grew wheat will now be growing corn, making a total of 1000 farmers growing corn. Since a farmer in Country B grows \(3 \text{ bushels of corn}\) a year, a total of \((3)(1000)\) or \(3000\text{ bushels of corn}\) will be grown in that country each year.

Suppose that Country A trades \(1500\) of its \(3000\text{ bushels of wheat}\) to Country B in exchange for \(1500\) of its \(3000\text{ bushels of corn}\).

Country A now has: \(1500\text{ bushels of wheat}\) (whereas before the trade it had \(1200\)), and

\(1500\text{ bushels of corn}\) (whereas before the trade it had \(1200\))

Country B now has: \(1500\text{ bushels of wheat}\) (whereas before the trade it had \(1200\)), and

\(1500\text{ bushels of corn}\) (whereas before the trade it had \(1200\))

In this case, the trade was clearly beneficial (mathematically speaking) to both countries even though one of the countries was more efficient at growing both wheat and corn than the other..

John Schwarz