A quadrilateral is drawn on a sheet of transparent paper. What is the minimum number of times the sheet needs to be folded to ensure that it is a square?

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Prove that among all quadrilaterals, only a square has non-perpendicular axes of symmetry.

## Solution

One fold is not enough. When folding a quadrilateral \(ABCD\) along the diagonal \(AC\), where \(AB = AD\), \(BC = CD\), and \(AB \neq BC\), the quadrilateral maps onto itself, but it is not a square.

For a fold not along the diagonal, a counterexample can be an isosceles trapezoid.

Two folds are sufficient. Fold the sheet along the perpendicular bisector of a side of the quadrilateral, then along the diagonal. Among all quadrilaterals, only a square has non-perpendicular axes of symmetry.

## Answer

Two folds.