# Grandpa William and the crack of the bag

- Grandpa William, you are an old man and yet you have managed to make a fortune in the stock market. How did he survive the crack of 1929?

- I sold all my gold mine shares a few weeks before the crack. One week I sold a quarter of the shares, the following week another quarter, the third week another quarter and the fourth week I got rid of all the remaining shares for sixteen dollars.

The product of the sale price of the first week for that of the last week was equal to the square of the price of the second week. The money I got for the sale of the second week was equal to the average of the first and third. The last one, was greater than double the first and every week I got an even number of dollars.

What was the price at which he sold the shares the third week?

#### Solution

We will call p1, p2, p3 and p4 at the respective prices. They tell us that p4 = 16.

The statement says that p4> 2 * p1. On the other hand, we know that p1 * p4 is equal to the square of p2. p1 can only be worth 2, 4 or 6, because it must be even and if it is worth 8 or more the condition p4> 2 * p1 is not met. Therefore, p1 * p4 can take the values ​​32, 64 or 96. Of these three values ​​only 64 is a square, so p2 = 8 and p1 = 4. Since p2 is the average of p1 and p3, then p3 = 12