In this sentence you are referring to:
Juguemos un rato a las cartas.
Let's play cards for a bit.
The subject of the sentence is not "las cartas"; The subject of "juguemos" is "nosotros". It is a bit difficult to see it as in English the translation is not literal, but if we analyse the English sentence we can see that "Let's play cards" means "Let US play cards"; it is referring to us.
In Spanish as you know we need to agree subject with verb: (Nosotros) juguemos...
I hope this clarified it.
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