A rectangle.
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cut in half then cut both halves in half again
No. There is no way you can cut one rectangle in half and only get one square. The other half would also have to be a square and this will only happen if the proportions are 1:2 and you cut the long sides.There are infinitely many ways to cut a rectangle in half none of which result in new rectangles. Only if the rectangle has proportions 1:2 can cutting it in half make a square, in fact the result could be two squares, not one.
Cut the exponent in half.
I think no because a cube is a perfect 3D square and if you cut it it becomes a rectangle type thing
/ \ - l and then do the same thing just with diffrent style lines