The virus is not meters away. It is hops away.

Lex Fridman introduces Po-Shen Loh as a Carnegie Mellon mathematician, national coach of the USA International Mathematical Olympiad team, founder of the education project Expii, and founder of NOVID, an anonymous contact-tracing app. Fridman wants tools that coordinate large numbers of people by giving them freedom and knowledge rather than orders. The conversation starts in the air. Both men still find it astonishing that a metal box stays up in turbulence, that a forty-story elevator does not fail, that a city of towers does not fall.

Fridman used to build toothpick bridges in high school for the joy of seeing which structure held. That feeling scales: a suspension bridge spanning water is a reminder that each person holds only a little expertise and depends on a network of other people’s knowledge. He maps the same stack onto software, from bits to assembly to C++ to an operating system to Python to machine learning. Loh recognizes the impulse. In the late 1990s he wrote the shareware-style games *Alien Attack* 1 through 4 from scratch, first in Pascal and then in C++, because he wanted to see an idea executed all the way to something that worked.

There was no font library. Even the letters were pixels, drawn with assembly copied from a book that put a colored point on the screen. Fridman has played the games in a browser that emulates MS-DOS and runs the original exe. Programming contests were the opposite of a finished product: write a function that is extremely efficient, under time pressure, which rewards a clean idea more than fast typing. Loh still recommends them. When he later tried to help control COVID, a contest-trained back-of-the-envelope calculation showed that linear-time algorithms on a four-core cloud machine could serve hundreds of thousands or millions of people. Without that, the server bill would have killed the project before it grew.

What pulled Loh into mathematics was the same dare as the games: a challenge that looked impossible, but that extra work might crack, unlike getting people to Mars. Fridman notes that people say “math is hard” as a complaint, and that every serious sport, art, or science is hard if you want to do something special. Loh splits the hardness. Memory-hardness is remembering methods a teacher demonstrated and replaying them on an exam. Invention-hardness is facing a question you cannot start, then finding a path. The first is tiring. The second is the rewarding kind.

He does not want to tell anyone “here is how you do this.” He would rather say here is an interesting question, I know you do not quite know how, do you have any ideas. Watch-me-then-practice-it-twenty-times is the method he is arguing against. If more people learned the art of invention in pure thought, he thinks they might stop finding math tiresome and start finding it capable: I can look at something I do not know how to do and come up with how.

At Carnegie Mellon he walks in without a script. The class is improvisational comedy: students offer ideas, he yes-ands them, and together they build a proof the students helped create. Every term the same theorem comes out slightly different. During remote semester he put the lectures on YouTube. Online courses he designs cannot be live, so they open with a question that is slightly out of reach, then drip automatic hints so the student still has to invent. Exams are 90 percent of the grade, open notes, with every past exam available — and a guarantee that this year’s problems overlap neither those old exams nor anything he taught. The first test usually goes badly. By the third, students have learned how to invent inside the subject.

In March 2020, as the United States shut down, Loh’s group — he insists it is a team, and that user-experience designers were at the core from day one — asked what network theory plus smartphones plus anonymized health data could do. Months in, they stumbled onto a different way to control disease. For most of history the method was find the sick, find who they were around, and remove those people from society against their will. That feedback loop fights human incentives. You cannot steer eight billion people through things they individually do not want to do.

They flipped the product question. In a pandemic, what would you want? To live as much of your life as possible and not get sick. So build an app that helps you *avoid* getting sick, not one that removes you after you were near someone infectious. The obstacle is that COVID spreads about two days before symptoms. You cannot see who is sick. Network theory supplies the substitute metric. You caught it from someone. Tell everyone how many physical relationships separate them from the sickness. Distance in feet or seconds is the wrong unit. Degrees of separation, LinkedIn-style, is the trivial idea that unlocks a different behavioral loop.

To show why the loop strengthens as the disease gets worse, Loh asks you to imagine an airborne hemorrhagic fever. Who would install an app that said how many relationships away that disease had struck? A very large number of people. A friend pointed out the property he is proudest of: the deadlier and more transmissible the pathogen, the stronger the incentive to install, which is a good characteristic if you are trying to keep civilization running. Informal surveys said that as people saw the disease getting closer they would hide and cut contacts.

Call the possibly infectious people set A and everyone else set B. Old contact tracing only pushes on A. There is a symmetry: you can also let people in B, at the fringe of A, remove themselves from the interface. Nobody has to be told what to do. Loh analogizes it to a free market versus central planning. Line up incentives so that purely selfish behavior contributes to the global function. Fridman, who spent 2020 furious at every institution except the vaccine biologists, hears a design that uses information without limiting personal freedom.

NOVID was not allowed to demand personal information. Confidence that they could skip GPS is what let Loh run at the problem. Two phones with the app talk over Bluetooth, estimate how far apart they were, and that relative proximity is enough to build the physical network. Duration is estimated the way a movie is filmed: a snapshot every five to eight minutes, because a battery-honest app cannot stream distance. A ten-minute grocery brush with a stranger is statistically weak and, in their paradigm, irrelevant. They are not doing damage control on the past. They are predicting the future. Hours of repeated contact survive the sampling; a one-off stranger washes out.

The mobile work is not a standard UI talking to a web server. It hooks Bluetooth for a job phones were not really designed for, and iOS is famously hostile to background work. The UK ran into that wall. Loh’s team dug into operating-system docs and is proud that NOVID uses little battery, even compared with Apple’s own system. On the server side, contest people and olympiad people wrote C++. There are only seven to eight billion humans. If the algorithms are linear or nearly linear, a gigahertz core and a recalculation once an hour — 3,600 seconds — can keep up. Fridman objects that n-squared possible edges would explode. Loh’s answer is sparsity: the average person has at most on the order of a hundred strong contacts worth tracking, a bound he inflates to cover college social butterflies.

Each user sees how many direct connections they have, but not how those people connect to one another. They also see counts at distance two and three. In university pilots, that chart became an adoption loop: people who discovered they were indirectly tied to hundreds or thousands of others told friends to download. The trigger was about two or three direct connections per typical user. They were, in Loh’s phrase, making a viral app to fight a virus on the same network the virus uses.

Can this information keep a society open, or do you still need stay-at-home orders? Loh says they do not know yet, which is why they want epidemiologists, not a mathematician declaring victory. The hope is about the endgame: few cases, reopen, cases rise, lock down again, forever, unless vaccines end it. Even with vaccines, makers were already talking about next year’s dose. Places that cannot afford a $20-a-shot rollout still need other techniques. If some people must go out and others can hide, giving everyone a personal risk signal might pull aggregate R0 below one without forcing anyone.

Fridman wanted, from March 2020, not a GPS map of bodies but information about which activities were actually risky. Politicians spoke with certainty; scientists, he thinks, spoke with too much authority, and being wrong after claiming to be sure destroyed more trust than saying “I am not sure” would have. Other contact-tracing efforts, including Google and Apple’s, earned Loh’s respect. The value proposition was different. Install a standard app and it tells you *after* you were exposed, so you can protect other people. What does that do to your own probability of getting sick? Close to zero. Free-market capitalism, he notes, was not based on altruism. Mechanism design that lets everyone maximize their own situation can still help the whole.

When they roll NOVID out at a university, the first question is usually whether people know what it does. Most have read about the other apps and assume it will tell them to quarantine. “NOVID never wants to ask you to quarantine.” The principle is: if something is coming close, tell you, so you can protect yourself if you want. If you shut the door, you are shutting it from the inside. Loh did not come on the podcast to tell everyone to install it. It works best when a local health authority partners, because otherwise a high-school senior prank is to declare a massive outbreak during finals week. Verified positives get a short-lived passcode, similar in spirit to Google and Apple; without a code you can still self-report as unverified. Public-health researchers can reach the team through novid.org. The Philippines is not the United States; local experts have to own deployment.

What Loh finds beautiful in mathematics is a change of perspective that makes an unsolvable-looking problem crystallize. NOVID was that move: quantify proximity by relationship hops instead of feet, and the rest follows. Fridman uses the same graph picture for civilization, Twitter bubbles, and recommender systems, then steers to the cleanest version of a perspective-shift: the International Mathematical Olympiad. It is the most prestigious pre-college math contest in the world. Questions are not “the answer is 42.” They are essays: explain why this strange fact, which you have never seen, is true. Two days, four and a half hours and three problems each day. Solve any one of the six and you are a genius with an honorable mention.

Each country sends six students. Country rankings are unofficial sums. Fridman, who grew up in the former Soviet Union, says the IMO there was a source of national pride on the order of wrestling or hockey, trained as seriously as gymnastics, and that beating Russia or China is brutally hard. The United States does not wear it that way. Loh thinks the Soviet investment was rational: many of those high-school medalists later did serious research. Getting good at splitting a mess into a simple crystal is a tool for real problems. Fridman adds that the culture treated invention as something anyone might do, not a gifted-and-talented track, at the cost of enormous pressure.

Loh does not think you need authoritarianism to send the message that being a thinker is a good thing; Americans already train that hard for football. When the Mathematical Association of America asked him in 2013 to be head coach, he told the executive director they might do worse, because he would not define success as winning. He gets a summer camp with about sixty strong minds. He wanted to maximize how many of them he would read about in the *New York Times* in twenty years. That is why some of them later worked on COVID: in twenty or thirty years he will not have the energy, and another catastrophe will need people who step up.

Each problem is worth seven points, six problems, a maximum of 42, with partial credit for true facts along the way. Scoring is a negotiation. The host country brings about fifty judges. A delegation sits across from two or three hosts for half an hour over a rubric. The US team, Loh says, does not bargain upward. Sometimes they ask for *less*. Scripts arrive in every language; Mongolian papers are explained by Mongolian coaches to people who cannot read them. It is a jury whose members sometimes do not share a language. The Americans write in English, so everyone can see, which makes cheating hard. They hope others are equally clean, and they care more that every stage of the pipeline teaches a general skill.

Loh quantifies hardness as the number of leaps of insight — changes of perspective — along the path. Each reframing is a large fan-out in the search space. The number of leaps is the exponent; the base is how many possibilities you might try. A three-insight problem is not three times a one-insight problem, because after the first leap you still do not know you are on the right track. Those multi-leap questions are the hard ones on the olympiad.

In college he was bad at reading math books: a long page of true sentences after which you have no idea what you read, or how anyone thought of it. His fix was to look at the theorem statement, look at the length of the proof, close the book, and try to reprove it. Length is roughly linear in the number of insights, because each insight takes space. A short proof means one insight, so you can hunt it. On a contest he glances at whether a problem is number one, two, or three and estimates the leap count the same way. The rule is approximate, and it works best inside one book on one subject. A two-page proof still means: this will be insane.

Fridman brings up Andrew Wiles spending seven years on Fermat’s Last Theorem, living with ideas that feel like leaps and might be dead ends. That, Loh says, is what attracts professional researchers: the pain of fighting, and the feeling when the insights finally string together. Short-term, both men enjoy the rush of an insight that *might* be the one, even knowing the base rate is a dead end. If insights are still arriving, Loh has not hit a wall yet. From his PhD advisor Benny Sudakov, who grew up in Soviet Georgia, he learned to choose problems that might matter if you succeed — the reason they dug into COVID. From Jacob Fox at Stanford, a graduate-school friend, he learned to ask after a proof: what did we invent that we could use somewhere else?

Asked whether mathematics is discovered or invented, Loh comes down on discovered. He is drawn to statements that look simple and are brutally hard, Fermat’s Last Theorem being the type specimen: why should a child-readable claim resist centuries? If another intelligent species compared notes, he suspects they would also have noticed that pi matters. You can say humans like circles because of how we perceive; stars and planets like circles too. A circle is a point and a constant distance. It does not get simpler. Fridman wonders how much work it would take to align two minds, not just notation, and mentions that Stephen Wolfram and his son helped on the language in the film *Arrival*. Visual proofs, back-and-forth, “does this concept make sense to you,” might be the only conversation possible.

François Chollet’s IQ-style tasks, Fridman says, are easy for humans and hard for AI for the same reason olympiad problems are: you have to notice a symmetry or invent a pattern you have not been shown. What is the intelligence that does that? Loh looks at the raw search space and calls it huge. Strong solvers have a chess-player’s heuristic for not analyzing a line, and if you ask they will say something that sounds right, which is probably only part of the function their brain is computing. Ask Garry Kasparov why a position is good and you get words; the words are not the whole evaluation. He is not sure a computer built from scratch would grow the same tendencies the topology of neurons grows. He is also not enough of an expert, he says, to pretend otherwise.

To show students where ideas come from, he live-solves middle-school contest problems on YouTube, seeing them for the first time. The point is to make the invention audible: why this move is intuitive, why the wrong ones are wrong. He does get stuck. Live chat is for that. Students suggest tries; some suggestions have more power than others; if he is stuck he goes fishing in the comments, and he is willing to be on camera as the coach who cannot do this one.

Loh thinks every student can pick up this skill. Middle school is the right moment, because fractions, multiplication, and the area of a triangle are enough to start noticing the coincidences that excited the Greeks — when the Pythagorean theorem was a discovery, not homework. Geometry is his favorite on-ramp: visual, like computer vision, and the earliest school subject that invites proof. Combinatorics showing up inside geometry is what excites him most. Fridman fell in love with geometry too and regrets how briefly it appears before vanishing until much later courses.

The way to let someone feel the power to invent is a question they do not know how to do, that they can think about, and then solve, so they have the experience of moving from cannot to can. For adults who gave up on math, he recommends actual middle-school contest problems at the state level. They are hard enough that a professional who does not do this for a living will struggle. Search MATHCOUNTS, mathleague.org, and the MAA’s AMC 8, 10, and 12 for old questions. His own Daily Challenge with Po-Shen Loh is paid — it funds the rest of his work — and aimed at middle-school contest thinking: an interesting question, a minute of thought, a hint, another minute, a bigger hint, then several ways to finish, with an extra concept smuggled in. High-schoolers who skipped contests in middle school still use it to learn how to think without the kid in the front row answering in two seconds.

Lifelong, he asks what he can do now that he could not a year or a month or a week ago, in math or anything else. Fridman has been using Anki and setting aside ten to thirty minutes a day to think through a non-work question from common sense, no looking things up — gun control, for example — to build what he calls the thinking muscle. Loh’s practical version, if you have children, is to work state contest problems in order to teach your elementary or middle-school kid. That is how his wife, who never did the contests, taught herself. If you want to learn something, teach it. He also posted his Carnegie Mellon combinatorics class on YouTube for free; you do not need calculus to follow it. Combinatorics, for him, is the finite and discrete: graphs where you have a node or you do not, reasoning that is iterative, algorithmic, inductive. He landed there because Benny Sudakov looked like he did interesting work and cared about students, even though Loh’s prior training was continuous analysis.

Computational complexity is almost question one when something has to serve a large population. That was the first filter on disease control: one server or no project. Loh then pulls a paper on voting trees. Two candidates are easy. Many candidates are a mess, which is why voting theory exists. The theoretical toy is a system of head-to-head runoffs wired like a circuit, inputs as candidates, every pairwise preference known. A nice outcome would be that whoever wins is preferred to a lot of people.

A clean fact: among 1,024 candidates, someone always beats at least 512 of the others. The proof is an averaging trick. If every match awarded a point to both players, everyone would have the same score, equal to the number of other people. Award a point only to the winner and you have half as many points to go around, so the average is half the field, so someone is at least average. Expected value in a different costume. It is obvious after the explanation and not obvious as a claim about a circuit.

The default circuit is a balanced binary tree: 1 plays 2, 3 plays 4, winners play, up through 1,024. That circuit can be outsmarted. A candidate who beats only the ten people on their path to the top, and loses to everyone else, can still be crowned. You cannot trust that wiring for every election. Loh and coauthors improved the guarantee from about log base two of n — the length of the path — to the square root of n. Ideal would be n/2; square root is still a large jump at a thousand candidates.

One widget inside the construction: candidate 7 plays A, and separately B, C, D; those winners play upward. Every bottom match is 7 against someone. If 7 emerges at the top, 7 actually beat that whole group. If anyone else, call them F, is on top, F beat 7 somewhere on the way. Either way the champion’s defeated set has a lower bound. Asked whether this improves real elections, Loh will not sell it as a reform. Ranked-choice student government at his college made sense to him; getting a better system adopted is mechanism design, tricking human nature into accepting it, which is another fight.

A summer at Microsoft Research produced a paper on stochastic coalescence. Suppose a crowd each knows how many hours they slept, and you want the total. Asking one by one is linear. If people can add, they can build the same kind of tree from the bottom: 1 and 2 add, 2 disappears, 1 keeps the sum; 3 and 4 likewise; then 1 talks to 3. In parallel the wall-clock time is the number of layers, ten steps for 1,024 people — logarithmic. The cheat is labeling who is person 1 and person 2 in downtown. That labeling is itself linear.

An elegant distributed proposal, which Loh came in to analyze, starts with everyone as their own representative, pointer on themselves. Each round, people who still hold a pointer flip a coin: hunt for someone to absorb them, or wait. Hunters pick a random person and walk the delegation chain, with path compression, until they find an agent. Agents have limited bandwidth and can accept only one request; extras are rejected. Next round, only remaining agents flip. The catch is that a hunter hits a lump with probability proportional to the lump’s size, not uniformly among lumps.

That bias can create a bottleneck: one giant lump and about square-root-of-n singletons. The giant can swallow only one per round, so time degrades to about √n. Joint work showed the imbalance really does arise, so purely logarithmic time fails. A tiny patch repairs it: when an agent is propositioned by several lumps, accept the smallest, not a random one. That is enough to even the sizes out. Fridman hears a constitution. Instead of a tree of height one where everyone votes for president, you might delegate to someone you actually know on an issue, they delegate further, and a representative network grows from the bottom. Graph theory’s exciting theorems are mostly twentieth century; he thinks the applications are twenty-first, the same way NOVID distributes information about a disease.

Does P equal NP? Loh does not know. Around Carnegie Mellon’s theorists he has learned what he does not know well enough to answer. Fridman cites Scott Aaronson’s roughly 3 percent chance that P equals NP, and the way a specific number sticks in the head the way 42 does — high enough to matter, low enough to feel like a long shot, in a species that barely distinguishes 1 percent from 50. Humans also fail to intuit exponential growth, which is part of why pandemics surprise us. For books and stories, Loh does not offer a syllabus. He was shaped by campaigns, fictional and not: *The Lord of the Rings*, and a lot of World War II history, the idea that someone crazy enough can push through adversity after adversity, sometimes for years, starting from a single person.

Advice to young people comes in two pieces. Learn to invent, not to mimic. A lot of schooling is watch someone do X, then repeat X with different inputs. Loh never liked following directions exactly; if you said homework must be in pencil he would try a pen. Second, from Sudakov: make sure the thing is big enough. Once you can invent, do not settle for comfortable. Aim at something hard that might be a multi-year campaign and might matter. He does not score himself on succeed-or-fail. If NOVID failed entirely he would not feel terrible — it is a very hard problem — but he would regret not jumping in. Give yourself the shot at unbounded opportunities.

Meaning, for him, is a score: after he is no longer here, for how many person-years did what he did still matter, whether or not it is attributed to him. That is why we still talk about Newton. He later renormalized. He found a simpler way to solve quadratic equations than the textbooks use; if it enters the curriculum the chance of a still-easier replacement is small, so the score may already be unbounded. A few people using an idea for a lifetime, or trillions using it for fifteen minutes among the stars, are different shapes of the same ambition. He wants work that outlasts him. Humans are meat vehicles that carry ideas for a few years, borrow a pile of baby ideas, and occasionally invent a new one that then has a life of its own.