Square craze

Geometry Level 2

There is a square with side \(a\). Inside of this square, there are four squares on each corner. These small squares have side \(\frac a4\). There are two more small squares inside the big square, and they also have side \(\frac a4\). These two small squares touch each other by their vertices. These two squares are in the middle of the big square, and each touch one of the squares that are on the corners, creating a "chain", in such way that the diagonal \(D\) of the big square can be expressed as \(4d\), where \(d\) is the diagonal of the small squares. After all, the big square has now a space that is not filled with small squares.

Given that the apothem of the small squares is 1 cm, calculate the area of this space. Give your answer in \( \text{cm}^2\).

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