In 1798, we're told, Napolean Bonaparte took an army to Egypt. What we are not told, however, is the incident at 'Plato's Plateau', apparently constructed by Alexandrians to honour Plato's solids.

Nineteen of the enterprising invaders happened upon a large tetrahedral pyramid in the desert, with three of its four points on the ground. Sitting at the top was an old man. Three soldiers, wanting to capture the edifice, called upon the old man to surrender. They then positioned themselves so that one soldier was at each of the three lower vertices, and the old man was at the top. If each of the three soldiers, and the old man, picked a random adjacent vertex of the pyramid, and traveled, by climbing or walking, to it, then what is the probability that no on met someone else on the way? What is the probability that the old man was caught?

There were nineteen soldiers so that this same scene could be performed at the hexahedral, octahedral, dodecahedral and icosahedral edifices on Plato's Plateau, with one old man at each. What are the corresponding answers at each of these?

Unfortunately, Napoleon's soldiers were unable to capture any old men (chances?) and in frustration destryoed Plato's Plateau with cannon fire.