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.

It is easiest to describe this by starting with a unit square tilted 45 degrees. Cut the square in half with a horizontal line from left vertex to right, and bisect the lower triangle with a vertical line through the bottom vertex, resulting in two smaller triangles. Move the lower left triangle so that it mates with the upper right edge of the big triangle, and move the lower right triangle so that it mates with the upper left edge of the big triangle. We now have an untilted rectangle measuring sqrt2 units long and half as high.

Pick the point on the top edge of the rectangle that is one unit from the left edge. Bisect the rectangle with a line passing through this point and the lower left corner of the rectangle. This bisection results in a trapezoid to the right of the line (sort of a ramp) and a triangle to the left of the line. Slide the triangle up the ramp until its right vertex is directly above the right edge of the trapezoid.

Extend a line through the left edge of the triangle in order to cut the triangular tip off the trapezoid. Move this triangle to fill the triangular void in the upper right corner of the figure. We now have an untilted rectangle that is one unit wide, and since no area was gained or lost in this process, the rectangle must be one unit tall and is therefor a square.

Thanks to Brian for the hint.

 

Edited because I can never get the darn sqrt sign to work :P

Edited on October 21, 2004, 9:42 pm

Posted by Bryan on 2004-10-21 21:40:44