3. Maga Magò has a deck of 52 cards, stacked with the back facing up. Magò separates the seven cards at the top of the stack, flips them, and places them at the bottom of the stack. Now all the cards are stacked again, but not all of them still have the back facing up: the seven at the bottom are turned the other way. Magò repeats the previous operation until all the cards have the back facing up again. In total, how many sets of seven cards has Magò flipped?
Solution: We color the cards of the deck blue and red, in alternating groups of three and four, respectively. The top three are blue, the next four are red, the next three are blue, and so on. We observe that when Magician Magò turns a small deck, this alternating group coloring is maintained. If we focus on the blue cards alone, we see that there are eight groups of three cards, and with each move, the top group is placed at the bottom of the sequence, inverted. Therefore, it takes eight moves to invert all the groups, and another eight moves to return them to their initial orientation. Consequently, the blue cards return to all facing downwards every 16 moves. Similarly, this happens for the red cards, which form seven groups of four cards, so they return to their initial state every 14 moves. All the cards will then be facing downwards again for the first time after $112=\operatorname{lcm}(16,14)$ moves.