Gravitational potential energy of a mass can be written as GPE = mass times acceleration of gravity time height change.Maybe the key words here are "height change". It doesn't matter which route they take. Not, at least, in terms of GPE, although the actual physical demands of a longer route would presumably (but maybe not necessarily) bigger than those of a shorter route.Since all three climbers start from the same altitude, and finish at the same altitude, they all have gained the same GPE. Therefore, NO climber has gained the greatest GPE.It's in the nature of the earth's gravitational FORCE field, that the idea of a POTENTIAL ENERGY can be "defined". The earth's attractive force is INVERSE SQUARE, and this is also known as a CONSERVATIVE field.It seems that ALL conservative fields have a potential associated with them. This applies also to the ELECTROSTATIC force field, and gives rise to the idea of VOLTAGE and/or POTENTIAL DIFFERENCE.
