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An infinite number of ways. Cut along a line from anywhere on a side to the centre of the square. Make three more cuts, at 90, 180 and 270 degrees to the first at the centre. Each point on a side of the square will give rise to a different set of four identical pieces of the square. And there are an infinite number of points on the side of the square. So an infinite number of answers to the question.

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Q: How many ways can you divide a square into four identical pieces?
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