The FARMER and THE HORSE DRING AT THE RIVER : a folklore 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 to deal with, so combinatorialy it means more options for proof hence larger space for search. Yet a specific problem might not be 'representative' enough of the general one which might well be harder to solve.