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Ref: https://en.wikipedia.org/wiki/Squaring_the_square A perfect cubing of a cube is a partition of the cube into some finite number of smaller cubes that are pair-wise non-congruent. The above page ...

Definition: A partition of a planar figure into finitely many pieces that are all similar to itself and also mutually non-congruent may be called a self-similar perfect partition. A classical example ...

For which positive integers n is $n^2$ the sum of precisely n smaller positive squares? Of these n x n squares, which can be actually cut into n smaller squares?