The cards played are not random. It makes sense in the game. So when someone play a card you instantly analyse why they played exactly this card and also what follow up cards would make sense to play. And if a different card is played you also notice this. So good players are able to use all this information, their analysis and short term memory, to replay all the cards that were played in their heads. It all makes sense and tells a full story to them.
Usually, they don’t remember all the cards, and they don’t need to. This is heavily game dependent, but for instance blackjack is a game famous for counting cards. However, blackjack players do not actually memorize the cards ; they assign a value to each card, and only keep track of the sum of the values for the cards which have already passed. There are more complicated systems of course, but the idea there is to only remember a “summary” of the cards. In other games, you may only be interested about specific cards ; say, you have the king of spades and what matters is whether or not the ace of spades have been played, and maybe how many trump cards are left. Or maybe only face cards store points, so you focus on those. Or maybe you want to target a specific suit, in which case you will focus on remembering those. Sometimes cards are played in combinations, which also helps remembering them.
The bottom line is: they rarely need to actually remember every card.
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