Editing 2694: Königsberg

{{comic
| number    = 2694
| date      = November 4, 2022
| title     = Königsberg
| image     = konigsberg_2x.png
| imagesize = 448x343px
| noexpand  = true
| titletext = At first I thought I would need some gold or something to pay him, but then I realized that it was the 18th century and I could just bring a roll of aluminum foil.
}}

==Explanation==

[[File:Konigsberg_bridges.png|frame|right|{{w|Königsberg}}, Prussia in Euler's time, showing the Pregel river and its seven bridges. Two of the original seven bridges no longer exist,[https://goo.gl/maps/ChdBoeQMr3AQPi446] although there are three new bridges. The Baltic port city is now Kaliningrad, a Russian exclave.]]

This comic is about the {{w|Seven Bridges of Königsberg}}, a seminal {{w|graph theory}} problem solved by the famous mathematician {{w|Leonhard Euler}}.[https://www.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem] The problem was whether a path through the city crossing each of the seven bridges just once exists, without crossing the river forks any other way. In 1736, Euler proved that no such path exists. This result is considered to be the first theorem of graph theory and the first proof in the theory of networks[http://www-personal.umich.edu/~mejn/courses/2004/cscs535/review.pdf] — a subject now generally regarded as a branch of {{w|combinatorics}} — and presaged the development of {{w|topology}}. Combinatorial problems of other types had been considered since antiquity. {{w|Graph (discrete mathematics)|Graphs}} are a data structure common in many algorithmic problems in computer science.

[[Cueball]] attempts to cheat on the final exam in his algorithms class by traveling back in time to commission the construction of an eighth bridge before Euler could learn of the problem, allowing a trivial solution that would remove the rationale for further analysis. He hopes that this would alter his present-day timeline in such a way that the test becomes easier because graph theory might never have been developed. The use of the word "tried" implies failure, which is probably a good thing since his success would create a {{w|Temporal_paradox#Grandfather_paradox|paradox}}. [[:Category:Time travel|Time travel]] is a recurring topic on xkcd and examples where attempts to change the past fails has also been used before like in [[1063: Kill Hitler]].

With the addition of the eighth bridge, it becomes possible to cross each bridge exactly once, starting at the north bank and ending on the larger eastern island, or vice-versa. However, there is still no way to traverse each bridge exactly once and return to the starting point, because the altered graph would have an {{w|Eulerian trail|Euler trail}} but not an Euler cycle. Thus the problem might still have been interesting to Euler.{{Citation needed}} (Adding a ninth bridge connecting the north bank to the east island would render the problem completely trivial.) We can't say whether Euler or others would have developed graph theory anyway, or whether Cueball's exam would have been any easier or more difficult.

An alternative modification allowing an easy solution is to remove bridges. During World War II, two bridges to the central island connecting it to the north and south banks were destroyed by bombing, so today there is an Eulerian trail across the five remaining bridges.

The title text alludes to the fact that ordinary {{w|aluminum foil}}, which was not commercially available until 1911, would have been a tremendously valuable curiosity in the 18th century. Aluminum was a highly priced metal before the 1880s when inexpensive methods were developed to refine it. The {{w|Washington Monument#Aluminum_apex|Washington Monument}} was constructed with a tip made of pure aluminum due to its value and conductive capacity (this turned out to be a bad idea, because it attracted lightning, which melted some of the aluminum). Aluminum had not been extracted in its pure form at the time of Euler, and was known only in compounds such as {{w|alum}}, so the metal would have been unique and exotic. The value of aluminum and the use of it as the tip of the Washington Monument was also mentioned in [[1608: Hoverboard]] where a heist to steal the tip is [https://www.explainxkcd.com/wiki/images/6/6f/1608_0995x1083y_Tip_of_Washington_monument.png depicted].

==Transcript==
:[Cueball, standing next to two men wearing wigs, pointing with a pointer at a map showing the seven bridges problem, with an extra bridge added in dashed lines]
:Cueball: Lord Mayor of Königsberg, I will reward you handsomely if you construct this bridge before my friend Leonhard arrives.

:[Caption below the panel:]
:I tried to use a time machine to cheat on my algorithms final by preventing graph theory from being invented.

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Comics featuring real people]]
[[Category:Time travel]]
[[Category:Math]]
[[Category:Programming]]