A rich merchant had collected many gold coins. He did not want anybody to know about them. One day, his wife asked, “How many gold coins do we have?”

After pausing a moment, he replied, “Well! If I divide the coins into two unequal numbers, then 32 times the difference between the two numbers equals the difference between the squares of the two numbers.”

The wife looked puzzled. Can you help the merchant’s wife by finding out how many gold coins they have?

The merchant has 32 gold coins.

To verify this, divide the 32 coins into two unequal numbers, say, 27 and 5. Then:

32 (27 – 5) = (27^{2}) – (5^{2}).

704 = 729 – 25

If the two unequal numbers are 22 and 10, then:

32 (22 – 10) = (22^{2}) – (10^{2}).

384 = 484 – 100

Alternatively, let’s say that the 2 numbers are x and y. We can then come up with the following equation:
32(x – y) = x^{2} – y^{2}

x^{2} – y^{2} can be expanded to: (x – y)(x + y)

So, now the equation will become: 32 = x + y

(x + y) will give the total number of gold coins.