Editing 2821: Path Minimization

In this comic, it appears that Cueball, standing on shore, is observing a swimmer who is possibly (but not obviously) in distress. The comic illustrates five potential paths that can be taken to reach the swimmer, each with a different reason to make them viable, in the manner of demonstrating different optimal strategies that can be chosen.

The first path is a direct line from Cueball, straight to the swimmer, which allows for the minimum possible distance to be traveled, some on land and the remainder in the water.

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 (though currents may complicate this). 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 spend time eating his ice cream on the beach, or will attempt to carry and possibly eat an ice cream whilst swimming).

The fifth and final path, barely recognizable as a path, points off the top of the comic and reappears at the bottom. This path presumably travels around the entire world, likely stopping for many, ''many'' rest breaks. It is labeled as the path that ''maximizes'' time. 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.

You could also fulfill the criteria of reaching the target in finite, but arbitrarily long, time by following a {{w|random walk}}(+swim) or even follow a {{w|space-filling curve}} carefully chosen to be the maximally finite scenario. Or you could simply choose any path, and stop for an arbitrarily long time, or travel at a speed approaching zero. In the comic, however, a requirement for simplicity of path may dictate the use of something close to the opposing {{w|great-circle distance}}, or a variation that has a maximal swim-time even without ''undue'' time-wasting detours, and assume equal speeds of travel on all routes.

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, which Richard Feynman, in his book ''QED'', uses 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 follows. 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].

The title text adds to the ice-cream path the stipulation 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.