A very rich lady goes with a bag of coins (not more than 400) to the city. In the city she meets four vagabonds. She gives the first vagabond 4 coins and a quarter of the number of coins which is left in the bag. She applies the same procedure to the second, third and fourth vagabond (so division by 4 worked always). With how many coins did she leave home?

The lady went to the city with 244 coins in her bag.

There are many ways to solve this problem. But there is at least one simple way to solve it. Suppose that the lady starts with 40 coins. The first vagabond gets 4 and 1/4 * 36 = 9, in total 13 coins. You would have found this number 13 immediately if the lady had brought 12 coins in addition (52) and then divided by four! This implies that if the lady brings 12 coins in addition, she had be able to divide 4 times by 4 without a remainder. So she should have brought 4 * 4 * 4 * 4 = 256 coins. But since we "borrowed" 12 coins, in reality the lady brought 256 - 12 = 244 coins to the city. (the other possibilities are 500, 756, ....)