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Infinitely many.

For example, find the centre of the square: this is the point where the two diagonals cross. Next, take any one of the infinitely many points on a side of the square and draw a line joining that point to the centre and extend it to meet the opposite side of the square. Each such line will divide the square into two parts that are 3 sixths each.

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Q: How many ways are to shade 3 sixths of a square?
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