The FARMER and THE HORSE DRING AT THE RIVER : a folklor problem.
A farmer together with his horse (at point A) must go home (at point B). Points A and B are on the same side of a river (a line D) . The horse must be led to the river to drink ( anywhere on the river is possible). PROBLEM : Find X on D that minimize their walk back home = d(A,X)+d(X,B)
SOLUTION : Ask the same problem in 3D space : D is still a line and A,B are still points.
a) Special cases: If A (resp B) is on the river the solution is trivially X=A (resp.X=B).
b) Fact : (A,D) (resp (B,D)) defines a half plane Pa (resp. Pb)
Now think of Pa,Pb as the two sides of a book partly open ( spine touching the table) where D is the spine of the book , and A,B are on left and right page of it.
Open the book flat on the table : clearly drawing a straight line between A and B is minimal and the intersection of the line with D (the spine) is the point X wanted.
There is a special case where the book is closed at start ( Pa=Pb i.e. zero partly open) yet you may still open it flat , this is the original problem.
REMARK : If not careful you may produce a wrong demonstration when missing that Pa=Pb is possible and need to look at the half plane as a double one folded on itself.
**QUESTION** : Find a 4-dimensional version of this?
A small remark for the general question: the more specific the problem is the more data you have so the more options (combinatorialy) for proof you have. Yert the specific problem might not be 'representive' enough of the general one, in which case generalising might well be harder.