I have a cross in the form of one square with four identical squares surrounding it
Cut it into the fewest number of pieces possible such that, when rearranged the pieces form two smaller crosses of identical size.
I've gotten it to 9 pieces.
The trick was to realize that to get two similarly shape crosses of half the size (area), then the linear dimension shrinks to sqrt(2)*original. Then start cutting on the diagonals.
|\ / |
|/_ \ /\|
I'll leave the assembly of the pieces as an exercise for the reader