## Math Magic: To Predict Unknown

April 1, 2014 Leave a comment

One of the math magic is to predict unknown facts. Please find below several situations where math gives unexpected and useful answers.

**1. Code Testers**

John detected 2 errors and Mary – 3 errors in a code. There is one error in common. How many errors are still undetected?

It looks like a joke. However there is a mathematical solution of the problem that shows that the number of undetected errors is N. I am sure that you can easily to find the number N.

99.9% fail or refuse to solve it.

**2. Lake Width**

How estimate the width of a lake without crossing it? You just walk and make some measures at a lakeside.

For example in the situation shown at the picture the width of the lake is 200 meters.

**3. How many fish are there in the lake?**

Yesterday, I caught 30 fish of a certain size in the lake.

I marked and released them without any harm.

Today I also caught and released 80 fish of the same size and noticed that there were 6 marked fish in the second catch.

How many fish of the same size are there in the lake?

**4. Seller’s decision**

I sell my car. People come to my garage one at a time and make bids to buy it. I make an immediate decision whether to accept or reject an offer after receiving it. I decide to reject the first N offers, mark the highest price P, and accept the first offer that is greater than P.

What number N do you recommend me if I expect that 100 people can make a bid?

If N is small, I can accept a small amount of money.

If N is large, I can reject the best offer.

This is the famous problem of the optimal stopping theory (Secretary Problem).

**5. Winning Strategy**

In a game, Anna and Bill take 1, 2, or 3 coins on each turn. The player to take the last coin from the pile wins. If Anna goes first and there are 40 coins on the table, how many coins should she take to guarantee that she would win?

The truth is “She always loses if Bill knows the winning strategy.” Do you know Bill’s strategy?