Take a square and cut it as you wish into a finite number of pieces. Rearrange the pieces without rotating or flipping any of them to form an new square that is rotated 45-degrees compared to the original.

(In reply to One possible Solution (visual) by SteveH)

Bravo!

Whoops! Sorry, have to take my Bravo back.  You did do a good job of fooling me, though, but as Jer points out in a later post, your solution is flawed.  When I print the two squares on separate sheets, superimpose them, and hold them up to the light, the pieces are congruent, but the new figure is a rectangle, not a square. An interesting dissection without rotation or reflection, nevertheless.

Edited on October 20, 2004, 6:19 pm

Posted by Richard on 2004-10-20 00:12:02