It is easy to dissect a square into an odd number of congruent pieces such that the center of the square is completely inside one of the pieces. It is harder to do so with an even number of pieces.

(1) Find the smallest even number of pieces for which this is possible.

(2) Find the smallest even number of rectangular pieces for which this is possible.

I don't know if this is minimum but I have a 24 piece answer for part 2. This size 12 square is divided into 2 by 3 rectangles. The interior of the rectangle denoted by E's has the center of the main square.

AABBAABBAABB

AABBAABBAABB

AABBAABBAABB

BBAACCCDDDAA

BBAACCCDDDAA

BBAAEEECCCAA

AABBEEECCCBB

AABBCCCDDDBB

AABBCCCDDDBB

BBAABBAABBAA

BBAABBAABBAA

BBAABBAABBAA