There are an infinite number of \(1\times 1, 2\times 2,\) and \(3\times 3\) squares available. If we want to completely cover a \(23\times 23\) square with these elements, what is the fewest \(1\times 1 \) squares that can possibly be used?

\(\)

**Details and Assumptions:**

- The squares are not allowed to overlap or stick out of the \(23\times 23\) board.
- We get to choose the placement of the squares.

This problem is from the Hungarian KöMal (High School Mathematical Pages).

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