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Mathematical circles are deeply about persevering through difficult problems that are often unlike any the student has seen before (Common Core Math Standard #1). Circle problems and session seldom start in any familiar place, and may seem to make little sense at first—but are highly attractive mysteries. Students must learn the standard “tools of the trade” of problem solving, such as working backwards, finding invariants, drawing pictures, making tables, the extreme principal, making a simpler problem/scale down, wishful thinking, and looking for symmetry. Further, since the leader will almost never reveal the answers, the students must learn to explain clearly their solutions and approaches—the beginnings of mathematical proof. Computing some quantity or following some rote algorithm will never be enough.