Editing 2821: Path Minimization

The second path travels more obliquely from Cueball to the water and then at a sharper angle to the swimmer. This path would take the shortest amount of time, presuming that Cueball would move faster on land (covering more of the distance) and slower through the water (but less distance). The exact angles would depend on how much faster Cueball is on land than in the water (according to Snell's law).

The third path travels at a far more oblique angle to the water, such that the subsequent swimming path is entirely perpendicular to the shoreline, adding to the amount of time spent on land in order to minimize the time spent swimming. Depending on one's swimming ability versus running ability, this could be the safest path to take. It might also be more sensible to keep the target in clear sight for as long as possible, from the land, then aim exactly away from shore when both your head and theirs are barely at wave-height. But this is a completely different reason from the distance or time preferences.

The fourth path travels nearly parallel to the beach. In fact moving slightly ''away'' from the swimmer but towards an intermediate goal: an ice cream stand. After that, the path turns and aims straight towards the swimmer, as all the others eventually do (although it is not made clear at this point if Cueball will also be eating, or just carrying, an ice cream even whilst swimming).

The fifth and final path, barely visible directly above Cueball, is labeled as the path that ''maximizes'' time. This path, presumably, travels around the entire world, likely stopping for many, ''many'' rest breaks. It should be noted that, by the definition given, it is theoretically possible to stretch the maximum time taken out forever by simply walking away and never returning.

The title text adds the stipulation, to the ice-cream path, that you also carry an ice-cream to the target swimmer to 'justify' that choice of route. But how this squares with the reason to rendezvous with the swimmer, or the manner in which this would further complicate the swimming stage, goes unsaid. But it makes it clear that ''not'' doing this isn't considered socially permissible, whether or not he had stopped to eat an ice-cream of his own beforehand.

The comic pokes fun at two famous physical/mathematical problems that are usually stated as happening on a beach. The first is the Lifeguard problem, used by Richard Feynman in his book ''QED'' to illustrate {{w|Fermat's principle}}, or principle of least time, which states that the path taken by a light ray between two given points is the path that can be traveled in the least time. This is closely related to {{w|Stationary-action principle}} for mechanical systems. In Feynman's words:

The second problem referenced in this comic is the [https://gametheory101.com/courses/game-theory-101/hotellings-game-and-the-median-voter-theorem Beach Vendor Problem], which is stated as follow. Suppose that on a long beach there are two ice cream vendors. Customers are uniformly distributed on the beach and each person will go get the ice cream at the closest vendor. Each vendor wants to maximize the number of customers that buy at their place. To minimize the customer's walking time, the optimal configuration would be to have one vendor at 1/4 of the beach length and the other at 3/4, but {{w|Hotelling's law}} predicts that the two shops will converge to the middle of the beach, in an attempt to steal as many customers as possible from the competition. This is a case of {{w|Nash equilibrium}} that is also related to the {{w|Median voter theorem}}. If the number of vendors is larger than 2, the problem may become [https://gametheory101.com/tag/hotellings-game/ considerably more complicated].

:Path that minimizes time until you get ice cream [A line from beach cueball to an ice cream stand manned by Ponytail, bearing about 90, then angling toward ocean cueball, bearing about 190]