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

(In reply to Solution by David Shin)

Nice solution.  However, constructing your point Q can be done more simply as follows.

Originally crease between the midpoints M' and N' of the sides (AD and BC) instead of between the midpoints M and N of the top and the bottom.

Then fold C upward toward M'N' in such a way that a crease can be made from B to a point Q' on CD such that C is at a point P' on M'N' and angle P'BQ' equals angle Q'BC.

Then BP'C is equilateral, and angle Q'BC is half of angle P'BC, or 30 degrees. Hence Q' is the same point as your Q.

 

 

Edited on September 15, 2004, 7:48 pm

Posted by Richard on 2004-09-15 19:29:42