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Seems to me the water runs over the brim of the pitcher. Here's why:
If even one more drop of water goes into the pitcher, it spills. But when the ice is cold, some of it pokes up above the surface of the water (and hence it also goes above the brim) because ice floats in water. Therefore, when the ice is completely melted, that water that was frozen and in a position above the brim initially, is now in the liquid part of the water below the brim. Thus, the water spills because it can't hold that extra amount.
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