## 45. Thirty-two Cards

Alfred, Bruno, Christoph, and Damien are playing cards using a deck of 32 cards. Damien dealt them all to the players, and when it was pointed out that the number of cards each player had was different, he said: "If you want us all to have the same number of cards, do as I say. You, Alfred, divide half of your cards equally between Bruno and Christoph. Then let Bruno divide half of his cards equally between Christoph and Alfred, and Christoph - half of his cards between Alfred and Bruno."

How many cards did Damien deal to each of the players initially?
45. In the end, each player ends up with eight cards. Christoph had previously divided[^31]half of his cards equally between Alfred and Bruno; thus, before this, Christoph had $2 \cdot 8=16$ cards, and Alfred and Bruno each had 4 cards. But before this, Bruno had divided half of his cards between Christoph and Alfred, and before this, Bruno had $2 \cdot 4=8$ cards, Alfred had 2 cards, and Christoph had 14 cards. This was the distribution of cards immediately after Alfred had divided half of his cards between Christoph and Bruno; therefore, initially, Alfred had $2 \cdot 2=4$ cards, Bruno had 7 cards, Christoph had 13 cards, and Damien (who did not participate in any of the card redistribution procedures) had 8 cards.