Ruler And Compass
1. Given two points, draw the straight line passing through both of them.
2. Given two (non-parallel) straight lines, find the point where they intersect.
3. Given two points, draw the circle with one as centre and passing through the other.
4. Given two circles, or a circle and a straight line, find the point or points (if any) where they intersect.
Some problems can be solved using only these operations and some can't. For instance, you can do these:
a1. Given a triangle, find a square of equal area.
b1. Construct a regular polygon with 102 sides.
but you can't do these:
a2. Given a circle, find a square of equal area.
b2. Construct a regular polygon with 100 sides.
How to distinguish the possible problems from the impossible? That's a task for the (at first glance quite unrelated) field of Galois theory...