Given a square piece of paper, show how by creasing and folding only, a square of half the area of the original can be obtained.

Call the side length of the square s. Then we wish to construct a square of area 1/2s^2. This corresponds to a square of side length ��(1/2)s. Note then that a right triangle with legs 1/2*s will have hypotenuse of length ��(1/2)s. This is the distance on the lines connecting the midpoints of adjacent sides of the original square. Thus, a square with these lines as sides satisfies the condition.

Construct this by folding in half twice and creasing (be it corner to corner or side to side as you wish, so long as you fold it in two different ways). The intersection of these creases is the center of the square. Fold each corner in to the center, creating the desired square. This is intuitively seen to be correct by noting that the entire square has exactly 2 layers of paper at all points.

Posted by Timothy on 2004-04-09 23:07:16