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I think Pear begins to display the quality of intuition even before she is 7. It starts in Chapter 6 - when Pearl seems to only recognize her mother because of the scarlet letter. As a baby and a small child it is referenced that she is frequently touching it and responding to it, almost as if she knows and understands its power over her mother.
As she gets older, she starts to show signs of understanding things about people in the rest of the colony. She is unafraid to chase the children to attempt to throw rocks at her and her mother as they walk through town. When asked at the Governor's house (loosely quoting here) "Where did you come from/Who made you?" (this said to see if Hester was raising her child in proper catechism), Pearl responds with something to the effect of: "I was not born at all. I was plucked from the rose bush that grows beside the prison door." What a response from a 7 year old.
It is also right about this time that Pearl begins to repeatedly question her mother about the scarlet letter. And this question always comes coupled with another, (again, loosely quoting) "Why do you [Hester] wear the scarlet letter on your breast" and "Why does the minister hold his hand over his heart?" She also asks why the minister is always seen with Chillingworth.
Rather than coming across as innocent questions, these seem meant to be rhetorical. It is as if only Pearl recognizes the answers the lie in these questions, and by asking them, she is attempting to reveal the answers, rather than her curiosity.