This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
Construct a square with only a compass and a straight edge.
Draw a circle (C1 at P1). Now draw a diameter D1 (intersects at P2 and
P3). Set the compass larger than before. From each of points P2 and
P3 draw another larger circle (C2 and C3). Where these two circles
cross, draw a line (D2). This line should go through the center of
circle C1 at a right angle to the original diameter line. This line
should cross circle C1 at P4 and P5. Points P2, P4, P3, P5 form a
For extra credit:
Reset the compass to its original size. From P2 and P4 draw a circle
(C4 and C5). These circles intersect at P6 and P1. Connect P6, P2,
P1, P4 for a square of the same size as the original compass setting.