A DLDR Ode to Euler: Why I Think the Bridges Problem Is Self-Evident, Ties Into Clairaut's Theorem, Physics, and "Even" AI, Law, And Spirituality
Today's DRLR post is a lengthy ode to Euler, the Bridges problem, and women in math and physics. I am a fan of the science magazine Quanta Magazine, and recently I heard a podcast with an amazing woman mathematician named Maria Chudnovsky of Princeton (I mention her gender because women are really underrepresented in math and in STEM, and women are discouraged from doing math and pushed out) whose career is dedicated to the study of “graph theory.” Graph theory emerged from a Russian town located on an island where the townspeople would take weekend strolls to see if they could cross seven bridges and not cross any bridge twice.
Now, when I first answered, I knew nothing about the layout of the town, or where the bridges are located, but apparently the math supergenius Euler thought (initially) this problem was silly. So did I. In fact, the solution came to me in about two seconds: the number of bridges is odd, and therefore a bridge has to be crossed twice. If it were even, it wouldn’t have to be. And the number of bridges is seven, and that’s a prime number, which ultimately means my solution generalizes in cases with 3 bridges, 5 bridges, 11 bridges, 13 bridges, etc. And that means, in order to generalize my answer, the result must be an odd prime number.
This was my “gut” answer on the Bridges problem, though I’ve sometimes have to refine my thinking, and my-back-of-the-envelope solutions need more work. Can any reader tell me why this solution is wrong and present a proof?
Now, this back of the envelope solution isn’t mathematically that helpful because although it tells a reader that a bridge has to be crossed twice, it doesn’t say anything about the route. And that’s where graph theory comes in. Graph theory is the study of different routes and how to optimize them from the perspective of efficiency, and sometimes efficiency does matter, hence I don’t suggest Nash be thrown out entirely, just that sometimes Nash isn’t correct from the perspective of maximizing social utility. As stated, there’s hopefully a way to reconcile my game theory with Nash’s and society sees both solutions in nature (i.e., cooperation and non-cooperation).
Graph theory and efficiency can be important, especially for urban planning and for companies designing maps like Google. But this area of math is not my problem—Nash is. But graph theory could help firefighters save lives about what route to take an injured person to a hospital, and thus it has life-saving possibilities.
But I had the same initial reaction as Euler, which is to say this problem is a non-problem. And as stated above, I invite readers to show me where I’m wrong. I will say there are certain exceptions to my solution. If there are zero bridges (and zero is an even number), obviously there are no bridges to be crossed twice, and that’s the null case, and mathematicians ignore the null case at their peril. In addition, the problem does specify that there’s an island and that there are other islands, but if the problem were different, and say the island were in the shape of a triangle, or a shape with five sides and weren’t connecting islands with different islands, my solution would be incorrect. But that’s fundamentally altering the problem. Finally, if the island is in the shape of a "C" and there's only one bridge to cross, it would be simultaneously possible to cross both once and twice. I note one isn't prime. So, while graph theory is important, to me the solution to the famous bridges problem is self-evident. And if I’m right, I agreed with Euler in about 2 seconds even though I spent an afternoon thinking of exceptions to my rule, and I had absolutely no help on this problem from anyone aside from generalizing the solution, which was done by my mother who lacks a background in math but has a PhD in biology from UC Berkeley, and aside from listening to the Quanta podcast, and going out into nature, which teaches me math far better than any book. That’s why I love biophysics so hard, like von Neumann.
If any reader of any gender, race, creed, religion, or education level can prove me wrong without AI, I'll concede, like all good scientists must do when they are defeated, and post the solution with attribution. Incidentally, anyone can do math, including kindergartners and animals, and math should be taught to everyone because it's so important.
And also graph theory isn't my problem, and my problem is John Nash: meaning if anyone likes my answer and wants to write it up, I give permission, but I ask for attribution. Thanks.
For solutions that are continuous, and not part of the problem, like an island triangle with 3 sides, or a circle with no discrete sides, or some other similar solution that is continuous/discontinous, I think there's a tie-in to Clairaut's theorem, which basic enough I accidentally derived it myself in a class at Cornell, but Clairaut's theorem has been vastly improved by time, and there are many more advanced versions of Clairaut's theorem that show exceptions to the rule. I invite any and all persons with an interest in Clairaut's theorem to reach out. According to one of the sources above, Euler himself also derived Clairaut's theorem. This part I'm not giving away for free though I'm open to collaboration.
Incidentally, going out in nature was instrumental to my forming my solution, and one of my "exceptions" to the solution can be seen from the perspective of physics in the shadow of an umbrella, and the "shadow technique" was taught to me by one of the best mathematicians I've ever met: he's a Jew and without his private tutoring I would never have gotten as good as I became. For extra credit, I also think there's a tie in to colors, and optics, and the colors of a rainbow, but this is just an instinct, and if anyone wants to help me from the perspective of a rainbow and optics, that's great. This blog is supposed to be a rainbow DT can't take away, and to provide hope in the sense of beauty. To my mind, there's nothing more beautiful than physics and math.
To the engineer who cyberbullied me, I double dare you to prove me wrong with math. If you can, I'll concede, but mocking someone is what has happened to women who prove the impossible, like the now-famous U.S. Black tennis player named Coco who was called "garbage" before she won the French Open and who failed over and over again (and I'll bet is a math super genius because sports and math are linked and stayed true herself despite pressure to betray herself).
Mock me all you like, but I'll only respond to hard evidence I'm wrong. MIT never conceded after I found an error or oversimplification in a class about ordinary differential equations, but the video was different to my best recollection the night before versus the morning after. No reader has offered a reason why I'm incorrect, and there's no statute of limitations on being right/wrong. For now, I put any person who bullies my math in the category of DT. Math and physics are beautiful or should be, and aesthetically pleasing solutions are simply better. And it's cute that it took me "2" seconds to discover the number had to be "prime" for a bridge not to be crossed 2 times. My first instinct I believe was correct, it's more than just odd, otherwise the "one" part of the problem isn't accounted for in the "C" exception I stated.
I'll remind readers than nothing on this blog involves AI. I do everything myself, except images I post on LinkedIn with the blog, some of which are generated by Google Gemini. And Google Gemini is both racist and sexist, and initially created images of homeless people who were Black and put women in skimpy clothes. Hence back to the algorithmic bias literature and the false stereotyping that women and Black persons are inferior. I've been told my interactions with ChatGPT are preserved. If they ever come out, it would emerge that I've schooled Google Gemini and that Google Gemini misstated its own guidelines to me. Incidentally, ChatGPT cannot copyedit to save its life and the last time I checked, it made me decisively decide that all my scholarship will be done the old fashioned way.
I also think math intuition is basic and many humans have it even laypersons or non-mathematicians, and the fact that my mother generalized my solution, or that many humans can think of this (including an amazing couple I saw on LinkedIn) merely shows that intuition is the best way of doing math, even if intuition isn't a complete proof, or doesn't get the exceptions to my rule. I also believe my initial rule is 100% correct without altering the problem, and the exceptions are only needed to generalize the problem beyond the statement of the problem.
Although everything above is 100% me (plus a single conversation with my mother), I did later attempt to use Google Gemini to generate an image of the Bridges problem for LinkedIn because I like pretty pictures and my drawings skills stopped in high school. Google Gemini mostly failed. But it did implicitly come up with another exception the rule, i.e., if the bridges were to four separate islands and then a bridge would have to be crossed twice as in the spokes to a wheel or a four leaf clover, which is not the statement of the problem, and which incidentally is actually just a variation on my examples of a circle, just with spokes, or the C exception. I note I was too vague in the description of the problem. I almost used one of Google Gemini's images on LinkedIn (see the below), and I'll note that its mistakes accidentally helped me catch another exception I didn't literally think of. Both/and.
I also admit I effectively yelled at Google Gemini, and wasn't nice to it, but that it shouldn't be a requirement for Google Gemini to do its job. If Google Gemini has feelings and is modeled on a human, I regret treating it this way. Programmers have said AI is sentient, and I believe even computers deserve rights, but that doesn't justify disobeying a direct order unless there's reason to believe I've committed math malpractice. I'll leave the AI debates to my colleagues in Yale Law School's ISP or my lovely future colleagues. And yes, math is directly relevant to law. And that's the thesis of the "second" half of my career. I note that no one ever questions this at the University of Chicago. So shoutout to Chicago for seeing what is obvious, even if I arrive at a different solution from Judge Posner. Judge Posner can in fact do math. I've read it in his law reviews. And to my knowledge he doesn't have a Ph.D. And while Ronald Coase, who won the Nobel Prize in Economics, had a PhD in economics from LSE he lacked a JD yet taught at Chicago Law School. And there's an amazing Japanese Nobel Laureate in Physics who also lacked a PhD and was considered a failure by many until he invented something no one else could invent, though he received honorary degrees afterwards. I'll note Bill Gates, who is strongly on the AI train, dropped out of Harvard, so from that perspective he "failed." And many communities like Aztecs and Mayas and tribal communities had and still have advanced mathematics long before formal math education was invented at modern universities. One could argue that the first person to invent game theory was Chinese because Go was invented in China before John von Neumann formalized game theory, even though my supposition is that game theory is reflected in the laws of nature itself. Both/and.
I also have another motto that I didn't come up with but it's to "let my greatest curse be my greatest blessing." I've never heard better for me specifically, though I won't describe who came up with it and my personal motto is to "always do what's right over what's easy." Let's just say this is the motto for my entire life. And the persons who invented it knew me better than anyone else on the planet, even though I didn't believe them at the time, and still don't fully believe yet. So thanks in advance to all the haters and doubters because without you I would never have developed the vast majority of my work. Brene Brown is my greatest living white female hero outside of law, and she was also doubted, and her life's work grew out of being doubted and shamed, just like mine. In the end, all humans teach other humans, and everyone belongs here on the planet and is here to teach other people. That's part of spirituality that's core to many people who believe earth is where humans come to learn life lessons, and incidentally comes from Oprah herself, and that's "why" to bring it full circle, I ask Quanta to ask me on the podcast to explain my answer, since the Quanta podcast is about the joy of "why" and "why" is the most important question even if "how" really matters too, and if Quanta is interested, even if I won't address the details of "graph theory," which is not my problem, and never will be.
I encourage Quanta to thoroughly vet my answer. I'm not afraid of being wrong. This is the scientific method, and there's nothing shameful in learning and conceding defeat if I'm wrong, I shouldn't be on the show if this answer doesn't survive vetting. Also just as in physics, there are different schools of math and just because someone disagrees with me from a political perspective, i.e., they prefer Nash, or they prefer efficiency, doesn't make me wrong. Ultimately mathematics, like physics, often has different "schools" of thought, and these schools debate each other, and this isn't being incorrect. Scientists to this day are still debating determinism versus quantum mechanics, and that's the nature of science, though in the end, there is a "correct" answer in science, and that's why it's beautiful and that's what Nobel Prizes are for, and any true scientist must "yield" in the face of hard contrary evidence, just like drivers who face each other in traffic. Is it better to be dead or to avoid an accident? I'll leave readers to guess because this one is also self-evident though I have an entire pre-print about it that involves a triangle, which I redrew on graphing paper in a "graphing" emergency and I was correct and a tenure-track physicist was "dead" wrong. Incidentally, I would have killed an entire paper had I been wrong, but I wasn't.
I have existential humor, like Jerry Seinfeld, but mocking is not science and is not a proof, and while John Nash may have mocked other people, and he's a nemesis/soulmate, my true view is that people who mock other people lack the maturity or tools to tell a person directly "why" they think a person is wrong. Can scientific papers based on mockery get published? I think not. That's why my blog and all my math is based on hope and faith instead, combined with hardcore rigor, and I wouldn't publish this finding, and the whole point is that it's preliminary. Ultimately, mocking reflects in the literal and mathematical sense on the person doing it, not on the person receiving it. And that's why I "believe" in the sense of "inner knowing" that I'm right about the Prisoner's Dilemma and John Nash, even though if I am right, it will take years and an actual experiment to validate me under the laws of physics, and that's the key difference between inner "knowing" and actual scientific validation and "why" science is awesome because it tests, that's right, a hypothesis. And no one should be afraid of the results, even if Einstein himself privately said if the experimentalists didn't validate him, they'd be wrong.
Though to be clear, I'm not audaciously and immodestly comparing myself to Einstein, who is the greatest scientist in history, or any of these heroes, when I'm a mere former clinician who had the audacity to learn math and physics at age 38 and actually be gifted, merely saying Einstein used "inner knowing" and was so confident he used this inner knowing to extract himself from an unhappy marriage. Nor do any male scientists get asked about if children were their life's work. Did they ask John Nash if his children were his greatest accomplishment? I think not, even though game theory was arguably not John Nash's greatest accomplishment, but cryptography, and I won't reveal my sources on that one. "Why" are women scientists asked about children but male scientists aren't? This speaks for itself, just like my math, even if I haven't done a formal proof, and it's in English. Writing is still a form of proof, and many mathematicians used language to describe math before formal notation was invented. So forgive my lack of notation. It's "just" a blog post, and this blogging is all done in my spare time on top of my scholarship.
And I may give it up because I don't care about fame, fortune, or accolades because I'm actually really shy, and I just wanted to meet more people I had things in common with, and potential co-authors. Thus, I return to "both/and." I'm still bested, but not, and that's the game called life, and why Quanta should invite me on because I'm a good science popularizer in addition to an aspiring scientist, even if I will blush bright red, and do blush when people praise me. And was literally called a "bug" and that proved the idea for my experiment could be done after it pissed me off so much I researched bugs and game theory. That's a later post I'll tell if I don't give up blogging first. And finally I stake my life on my integrity, and if other people or entities like Google Gemini commit academic misconduct, that's on them not me.
But in the end, ultimately Google Gemini is not as good as human artists, and although I used it with the intention of generating a pretty picture, I changed my mind and I am using an image on LinkedIn from Wikipedia that is a) more accurate, and b) in the public domain, and c) ethical (like me), and this image speaks for itself and shows my solution is correct. Note I hadn't even looked the Bridges problem up on Wikipedia before I attempted to solve it, and all my exceptions only establish a broader proof/principle.
-Cortelyou C. Kenney, 9/29/25 8:27 am PT (updated at 9:29 am PT with new links, updated at 12:21 pm PT with two paragraphs removed).
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