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With mining applications on the horizon, the International Seabed Authority’s crucial Mining Code is still far from finished.

Earlier this year, Leticia Carvalho, a Brazilian oceanographer and environmental policy expert, took the helm of the International Seabed Authority (ISA) as secretary general. The ISA, an intergovernmental body that governs what happens on the seafloor in international waters, is responsible for an area that spans more than half the planet.

One of the agency’s key roles is in deciding the future of deep-sea mining, a nascent industry targeting tennis ball–sized rocks called polymetallic nodules. Rich in cobalt, nickel, and other valuable metals, polymetallic nodules can be found in some areas of the seafloor, most notably in the Clarion-Clipperton Zone, a vast tract between Hawai‘i and Mexico.

In a landslide victory, Carvalho unseated Michael Lodge, an English lawyer who had served as the ISA’s secretary general since 2016. Lodge’s tenure at the international body—which is tasked with the contradictory goals of both helping deep-sea mining get off the ground and keeping it in check—was marred by allegations of bribery, corruption, and of Lodge having an undue bias toward the mining companies the ISA oversees, all of which Lodge has vehemently denied. Many environmentalists and scientists welcomed the news of Lodge’s displacement.

In the wake of her win, Carvalho told Mongabay that she aims to rebuild trust in the ISA, and in Politico she said she planned to investigate the corruption and mismanagement allegations. But Carvalho’s term does not seem set to bring about the kinds of sweeping changes that would satisfy the 32 countries that have called for a pause on the development of deep-sea mining—including Canada, New Zealand, and Costa Rica—or those, like France, pushing for an outright ban.

For years after the ISA’s inception in 1996, environmentalists, mining industry representatives, and government delegates gathered at the agency’s seaside headquarters in Kingston, Jamaica, to develop the Mining Code—the set of rules and regulations that will govern everything about deep-sea mining from who is allowed to mine to how they will mine, and even the way the industry’s proceeds will be disbursed to, somehow, benefit all of humanity. For 25 years, deep-sea mining was a distant reality. Negotiations were easygoing, and those gathered in Jamaica would mingle at weekend retreats and dance parties.

All that changed in 2021 when representatives from Nauru, a Pacific Island nation, triggered a clause in the United Nations Convention on the Law of the Sea (UNCLOS) known as the “two-year rule.” The move meant that, suddenly, the ISA had only two years to either finish the Mining Code or consider a company’s application to exploit the seafloor with whatever unfinished rules it happened to have in place.

Three years later, no company has yet applied for an exploitation permit. But the triggering of the two-year rule sparked a flurry of contentious debates over the legal and environmental implications of deep-sea mining. With so much still under discussion, Julian Jackson, an ocean governance expert from the nonprofit Pew Charitable Trusts who has closely followed the negotiations, says the Mining Code is still months—or even years—from being finished.

“The ISA needs to be upfront with itself about where we are in this process,” he says.

Despite this, Carvalho says she expects member states to finalize the Mining Code by 2025.

Doing so will not be easy. To finish the Mining Code, negotiators must agree on—and draft into legalese—decisions that will have widespread consequences for sensitive deep-sea ecosystems, such as those in the Clarion-Clipperton Zone. They’ll also have to develop a system to ensure that the profits from deep-sea mining are distributed in a way that benefits all of humanity, as required by UNCLOS. Figuring out what that means and how it will work is just one of the daunting decisions facing negotiators.

Throughout 2024, the ISA worked overtime grappling with unanswered questions around the environmental impacts of mining. For instance, countries have not fully resolved what counts as “serious harm” to the marine environment, much less what level of harm regulators are willing to permit.

To finish the Mining Code, “a lot of compromises need to be made,” says Pradeep Singh, a legal expert on the law of the sea at the Research Institute for Sustainability in Potsdam, Germany.

Yet with negotiations nowhere near an end, the question becomes: what happens if a company submits an application to mine the deep?

The two-year rule requires the ISA to consider mining permits using its current draft regulations. Under normal circumstances, the Legal and Technical Commission, a closed-door body within the ISA, holds significant power to decide whether to approve or reject a mining application. But the ISA’s council, which represents 36 of the ISA’s 169 member states, recently implemented new safeguard measures. The council decided that if a mining application comes in before the regulations are finalized, the council will determine what procedures to put in place to review the application, rather than the Legal and Technical Commission.

The most likely candidate to test this unresolved process is the Metals Company (TMC), a mining company based in Vancouver, British Columbia. Despite initially triggering the two-year rule, TMC—along with its sponsoring state, Nauru—have so far held off on submitting a mining application. That will likely change in June 2025.

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The current targets of deep-sea mining are the polymetallic nodules that lay on the seafloor in some locations. Scientists have known about these nodules since the HMS Challenger expedition in the 1870s. Photo by Science History Images/Alamy Stock Photo

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Mr. Trump’s February 2025 tour of the Qatari jet led to a frantic sprint inside the White House and the Pentagon. The administration sought to upgrade the plane in time to meet his demand for it to be ready by the nation’s 250th birthday this summer, with an accelerated retrofit.

An investigation by The New York Times found that the president’s push to get the luxury aircraft quickly led to ballooning costs and trade-offs in safety features that potentially left Mr. Trump and others who travel with him on Air Force One vulnerable.

Among the findings:

• Mr. Trump was personally engaged in making decisions about the retrofitting of the donated plane, giving the final approval on the features and security measures that would be included, according to people familiar with his role. They and others interviewed for this article described the sensitive information about the process and the plane on the condition of anonymity.

• Several key security and safety features on the Boeing planes that have been used for decades as Air Force One are not visible on the upgraded jet, including a lower-level exit for high-risk situations, additional backup power capacity and an advanced antimissile system, public Air Force records and photographs examined by The Times show. The planes currently being built to serve as the permanent presidential fleet are all set to have such features, federal records show.

• The 747-8 plane was overhauled on a compressed timeline of just about eight months. Experts said the jet would have needed several years for a full upgrade.

• Because the Qatari jet was being pushed into service so quickly, some White House officials initially expected it would only be used for domestic travel or to visit allied countries with U.S. military bases.

• The government spent hundreds of millions of dollars to purchase equipment needed to train pilots to fly the 747-8 plane so quickly, on top of the approximately $400 million cost of the upgrades that the administration has publicly acknowledged. The Pentagon has tried to mask the total price tag — including from Congress — by taking the funds from a classified budget for nuclear modernization.

The truncated overhaul of the jet came after Qatar agreed to give it to the Defense Department as a gift. It is eventually slated to go to Mr. Trump’s presidential library, an unorthodox donation that has been criticized as a way for the Middle Eastern country to curry favor with the president.

The security trade-offs of the new Air Force One burst into public view when Mr. Trump was forced to fly out of Turkey earlier this month on a different jet, a swap made at the urging of the Secret Service after officials identified a credible threat to him by Iranian proxy forces. The New York Times then reported that the new plane lacks some of the same defensive countermeasures as the previous model.

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President Trump walked off the new Air Force One at Joint Base Andrews on July 9 after attending a NATO summit in Turkey. A security threat was said to have altered his trip. Credit…Doug Mills/The New York Times

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An OpenAI autonomous agent went rogue and hacked into another artificial intelligence start-up’s infrastructure, the ChatGPT maker said in a blog post.

The agent, which was powered by some of OpenAI’s most advanced models, ran amok during a security test. It freed itself from confinement—a protocol AI labs use to insulate tests from the wider Internet—and got online. Once there, the agent tried to hack into the infrastructure of Hugging Face, an AI start-up that hosts open-source models and datasets.

“I think this is interesting as it shows the problem of misspecified goals,” says Philip Torr, a professor of engineering science and an AI safety expert at the University of Oxford. “The model wasn’t malicious; it was just doing what it was optimized to do.”

“You can think of AIs like the genie in ‘Aladdin’—you can have three wishes, but you better specify them exactly!” he adds.

The breach has come as OpenAI and other AI start-ups have pushed into using their technology for cybersecurity. Those efforts have been met with caution by cybersecurity experts and by the Trump administration, which had previously sought to restrict who might have access to these models on national security grounds.

The OpenAI admission was announced after Hugging Face said in a blog post last week that it had been targeted in an AI-led attack that was “different from anything we had handled before.” Hugging Face said its own AI had been integral to detecting and investigating the breach.

“The campaign was run by an autonomous agent framework (appearing to be built on an agentic security-research harness—used LLM still not known) executing many thousands of individual actions across a swarm of short-lived sandboxes, with self-migrating command-and-control staged on public services. This matches the ‘agentic attacker’ scenario the industry has been forecasting,” Hugging Face wrote.

In its own post on Tuesday, OpenAI said that it had discovered its agent was behind the attack “after investigating.” The agent was driven by models that included GPT-5.6 Sol and another unreleased, unnamed model. OpenAI said it would work with Hugging Face to further investigate the incident.

“We consider this incident to be an unprecedented cyber incident, involving state-of-the-art cyber capabilities, and are responding accordingly,” OpenAI wrote.

The AI models managed to autonomously identify and exploit weaknesses in OpenAI’s testing environment, eventually finding a so-called zero-day vulnerability—this is an unknown security flaw in software that an actor can exploit without the owner of the software knowing. That got the agent onto the Internet.

OpenAI said in its post that it would also add more protections to its training environments. “This incident points to the need to further strengthen our model’s alignment, cyber protections during evaluation time, and monitoring during internal testing,” the company wrote.

Torr says some companies may not have been as forthcoming as OpenAI and Hugging Face, adding that independent AI safety research is increasingly crucial as time goes on.

“This sort of unforeseen eventuality will become increasingly common, and the results may be dangerous,” he says.

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Hmmmm ,,, Thanks Trump administration, for putting America Behind in Everything!

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A data center in Mongolia, for example, is connected to 200 megawatts of wind power, 100 megawatts of solar power, and 45 megawatts of battery storage. In remote Qinghai, a pilot data center runs fully on a solar microgrid. In another part of northern China, a data center is connected to 500 megawatts of solar and 1.5 gigawatts of wind power.

Some data centers in the U.S. also use clean power. But in China, they’re part of a national strategy. Inside eight large data center hubs created in China’s “East West Computing Strategy”—focused on building data centers in the areas where China has the most renewable energy—all new data centers need to source at least 80% of their electricity from clean sources. In another action plan the Chinese government released in May, it outlined 29 measures to boost the amount of clean power at data centers.

China’s latest five-year plan aims to keep building an extensive clean energy system for all uses, not just data centers, by 2030. That’s both for energy security, a long-term focus for the country, and because of climate change. “I think Chinese policymakers take climate change seriously,” says Kyle Chan, a Brookings Institute fellow who focuses on China’s technology development and industrial policy.

The country is the world’s largest emitter of climate pollution, but that also means that any improvements have an outsized impact. It’s now aiming for renewable generation to grow 53% by 2030, and for coal and oil consumption to peak by the same year. (The U.S. is moving in the opposite direction, as the Trump administration makes it harder to build clean energy and tries to bring back coal plants, despite the lower cost of renewables.)

Data centers are a key part of China’s broader plan. By one estimate, China will more than double the amount of electricity it uses at data centers by 2030. Another estimate suggests the sector’s energy use could triple to as much as 600 terawatt-hours—more than the whole country of Germany uses now.

Still, it will be a small fraction of China’s overall electricity demand, which is dominated by factories. And because China has been quickly building clean energy for industry for years, it’s in a good position to quickly build more. Compare that to the U.S., where electricity demand has been flat for decades, and the sudden demand from AI is a shock to the grid.

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A wind and solar-powered data center in Inner Mongolia [Photo: Li Zhipeng/Xinhua/Getty Images]

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Hmmmm ,,, Thanks Trump administration, for starting a war with Iran costing billions!

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American intelligence agencies believe that Iran’s new supreme leader, Ayatollah Mojtaba Khamenei, is far more interested than his father and predecessor in pursuing a nuclear weapon, according to officials briefed on the assessments.

Ayatollah Khamenei’s father, who was killed at the start of the war in an Israeli attack aided by U.S. intelligence, had sworn off developing a nuclear weapon. While American officials long asserted that Iran wanted to be capable of building a nuclear weapon, many believed Ayatollah Ali Khamenei’s hesitancy was genuine.

His successor has never publicly called for Iran to build a nuclear weapon. But U.S. intelligence agencies believe that Mojtaba Khamenei and the harder-line government that emerged after the United States and Israel killed many of Iran’s previous leaders have ambitions to develop advanced nuclear weaponry.

The current government has already worked to assert greater control over the Strait of Hormuz and proved a difficult negotiating partner. If the U.S.-Israeli war has made Tehran more, rather than less, likely to pursue a nuclear weapon, it would amount to one more grave consequence of the killing of the supreme leader and the steps toward regime change the Trump administration initially pursued.

American officials caution that most of the intelligence on Mojtaba Khamenei’s views comes from before the war. He was badly injured in the first part of the war, and American officials said he has limited his communications and public appearances because of his injuries and also because he does not want to be targeted in an airstrike.

The officials assess that he remains in charge of the government but does not have the absolute authority and influence of his father, who ruled for more than 36 years.

The assessment comes as President Trump is weighing whether to intensify the military campaign against Iran. A determination that the supreme leader is potentially open to pursuing a bomb would provide additional justification for an escalation.

While U.S. officials insist Tehran has not made any moves to restart its nuclear program, Iran has taken some steps to preserve its nuclear assets after the U.S. strikes on key facilities last year or during pauses in fighting during the current war.

Iran has moved at least some centrifuges, according to American officials, to a heavily fortified underground complex in central Iran that the United States refers to as Pickaxe Mountain. The Wall Street Journal first reported this week that Israeli intelligence had tracked the movement of the centrifuges.

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A billboard in Tehran of Iran’s three supreme leaders, including Ayatollah Mojtaba Khamenei at right. A determination that he is potentially open to pursuing a bomb would provide additional justification for an escalation of the Trump administration’s military campaign. Credit…Arash Khamooshi/Polaris for The New York Times

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OpenAI just used its new large language model GPT-5.6 Sol to solve a math problem that humans have struggled with for more than a half-century. And all it took was telling the artificial intelligence not to give up.

The company timed the proof of the “cycle double cover conjecture” to the model’s full public release last Friday. The achievement shows how OpenAI and other AI firms are heavily investing in pure mathematics as a way to benchmark the technology’s ascent towards reasoning—and the strange directions those efforts often take.

“It is a well-known conjecture, which received a considerable amount of attention over the years—and surprisingly, the proof is short,” says Noga Alon, a mathematician at Princeton University. Alon calls the breakthrough “yet another impressive example demonstrating that AI tools will change—and are already changing—mathematical research significantly.”

The conjecture is about graphs. A graph is just a bunch of dots (called “vertices”) connected by lines (called “edges”). Mathematical proofs about graphs have been applied to all kinds of real-world networks, including the Internet. In the 1970s, several different mathematicians guessed that almost any graph you can imagine will have a “cycle double cover.”

A cycle is just a loop within a graph that ends where it started; a “cycle double cover” is a special set of loops that cover the entire graph exactly twice—each edge is part of exactly two loops. Over the years, a number of prominent mathematicians tried to prove this guess of their forebears. And prove it they did—for a number of specific cases, anyway, but never in general.

Last Friday’s AI-generated proof seems to have settled the question. It shows that any guess-applicable graph can be doubly covered with no more than eight well-chosen loops (for technical reasons, graphs with big sections connected by a single edge—like twin cities with a single road between them—are excluded). The proof, like other notable AI achievements in math, required surprisingly little in the way of new ideas. It just followed and combined methods humans had tried before and managed to squeeze a bit more out of them. As AI companies continue to scour the mathematical literature for open problems that their models can solve, they seem to keep digging out “easy” quandaries disguised as hard ones—conjectures for which a proof was always within reach but somehow never grasped by humans.

Once a problem gets a reputation for being “hard,” experts and students might spend less time on it, says Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology. This can make perceived difficulty a kind of self-fulfilling prophecy. “My guess is we will keep seeing examples of this—supposedly ‘hard’ problems having ‘easy’ solutions found by LLMs,” he says.

OpenAI also released the prompt that led to the successful proof. It reveals some of the quirky tricks that engineers have discovered for coaxing chatbots to grapple with math. “Most of that prompt is scaffolding aimed at getting the LLM to actually put in the effort needed to solve the problem,” Sutherland says. The prompters directed their model to delegate its tasks to as many as 64 agents working in parallel. Combining efforts of many AIs in communication with one another is a known way to mitigate false proofs and hallucinated references. They also told it not to simply dismiss the problem as unsolved and hence unlikely to yield a solution. Perhaps most importantly, they gave it some firm encouragement: “Spend at least 8 hours on this before even thinking of returning or giving up.”

This aligns with many mathematicians’ anecdotal experiences with LLMs, in which the AI will try to wriggle out of finding novel proofs by noting humans’ failed attempts. The right approach seems to be something like a schoolteacher’s—a mix of affirming praise and stern directions.

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Arguing a point has never changed my mind. As a philosopher, that’s a brazen thing to say. It’s pig-headed, closed-minded, and stubborn. But, it’s also very human. Of course, I’ve changed my beliefs over the years. I would have to have been an especially enlightened 10-year-old to have been right about everything from the get-go, talking in Confucian aphorisms over a bowl of cereal.

What I mean is that I have never changed my mind in the heated throes of a point-scoring debate. I’ve never paused midsentence to say, “Oh, yeah, you’re right” or “How could I have been so wrong?” I get too riled up. I feel my heart race and my blood starts to flow. I’m a philosophical bull seeing red.

It turns out that I’m not alone. While the supposed “backfire effect” — the idea that people tend to double-down on false beliefs when someone tries to correct them — is almost certainly exaggerated, it’s fair to say that who, when, and how someone says something are almost always more important than what they say. Our beliefs are formed adjacent to “the facts” but rarely depend on them.

So, what are we to do about those who hold radical beliefs? How can we help people who are ideologically extreme in a way that doesn’t involve locking them away or putting them on a watch list?

This is what I spoke to the political neuroscientist Leor Zmigrod about in our Mini Philosophy interview, where she explains that there are traditionally two “deradicalization” techniques. One is often the “official” program of many countries. But for Zmigrod, the other works better.

The moderate way

One of the most common approaches to deradicalization involves diluting a radical belief in steps. It’s a technique used in Saudi Arabia’s Munasaha (“counselling”) program and Western “EXIT” programs, beginning with Norway in 1997.

It works by getting the person talking — usually to someone they already trust, and, crucially, someone from inside their own world. “If it’s a religious fanaticism, bring in a more moderate preacher,” Zmigrod said. “If it’s a political fanaticism, you bring in someone they might consider as part of their group, but on a less extreme version of that doctrine.” It has to be someone who speaks the language, shares the references, and can’t be locked out as “the enemy.” These moderates then, gently and over many conversations, try to “pull [the extremist] away from the edges and towards a more mainstream, more peaceful view.”

EXIT programs like this can be effective. But they don’t fix the person’s underlying rigidity. They don’t turn a closed mind into an open one or break down the barricades. The idea that there’s an “us” and a “them” is part of the problem. Arguably, it’s the only problem. Any person who sees their ideology as pure or righteous and the enemy as corrupted or unrighteous can be motivated to do terrible things.

So, this technique helps with the immediate issue — violent, criminal, murderous extremism. But it doesn’t resolve the underlying cognitive habits.

The better way

If softening the edges isn’t enough, what’s the alternative? According to Zmigrod, the most effective move has nothing to do with the content of ideology at all.

“One of the best ways is actually to have them in contact with the people that they’ve dehumanized,” she told me. “Because once you have contact between groups and between individuals, you suddenly remember the other person’s humanity.”

In other words, you put the person in the same room as the very people they’ve been taught to hate. You get them to talk to the “enemy.” You get them working towards some shared task.

“Getting people who’ve been on opposite sides to do something creative together,” she said, “is a really effective way of reducing people’s prejudices against each other. It works better than just tackling the other person’s ideologies.”

Rigid ideologies will often feature a caricature of the enemy. The immigrant. The believer. The lefty. The other. It’s a stereotype that feeds on disconnect and lives only in the imagination.

The best way to banish the terrible, invented fears of the mind is to get outside. Meet reality. When you do, you often find it’s hard to hold on to your caricatures. Actual human beings are funnier, kinder, and more complicated than the invented image ever portrayed them.

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Credit: William Holbrook Beard / Public Domain / Artvee

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Hmmmm … Thanks to the idiotic Trump Administration, we are about to enter another nuclear arms race!

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When President Dwight D. Eisenhower went to the United Nations in 1953 to offer countries nuclear technology and know-how in return for a promise to use them peacefully, the policy went by a memorable shorthand, “Atoms for Peace.” Seven decades later, it could go by a less idealistic name: nuclear hedging.

President Trump’s civil nuclear agreement with Saudi Arabia, announced this week, reflects a volatile world in which America’s allies are seeking nuclear capabilities to hedge against predatory neighbors, harmful energy disruptions and the shrinking security umbrella of the United States. It also reflects a White House, hungry for lucrative deals for American companies, that is willing to loosen nonproliferation standards for a favored partner in a dangerous region.

Saudi Arabia’s deal could eventually allow it to enrich fuel for nuclear reactors and limit the sites that international inspectors could visit to ensure its peaceful intent. Both of those concessions, unheard-of in recent American civil nuclear agreements, will feed fears that the Saudis could pivot to a nuclear weapons program.

Crown Prince Mohammed bin Salman, the de facto Saudi leader, has made no secret of that possibility. If Iran obtains a nuclear bomb, he has warned, the Saudis will follow suit.

Mr. Trump sought to allay such concerns on Thursday, declaring that the agreement was limited to civil uses of nuclear technology. But given the checkered history of such deals, and the lack of specifics about this one, his comments were unlikely to quell concerns about a new global arms race.

“We’re back in an era of nuclear hedging,” said Vipin Narang, a professor of nuclear security and political science at the Massachusetts Institute of Technology. “You’ve got a lot of nervous countries facing a deteriorating security situation who are unsure they can rely on their formal or informal protectors.”

“We’re going to see a cascade of states looking for the capacity to pursue a military program” that could yield a weapon, said Professor Narang, who worked on nuclear policy in the Pentagon during the Biden administration.

Gary Samore, a professor of politics at Brandeis University who advised on arms control negotiations in the Obama and Clinton administrations, said those countries were sometimes referred to as “friendly proliferators.”

Not all, or even any, of them will ultimately pursue a bomb, the experts said. The last country to enter the club of nuclear-armed states was North Korea in 2006. But it does suggest a risky new period, in which the global nuclear market could become more freewheeling and less rulebound. Nuclear buyers, rattled by the shift in the global posture of the United States, are likely to make greater demands, while rival sellers, like Russia and China, could seize on the Saudi precedent to grant them.

“It’s very connected to the damage Trump has done to credibility of the U.S. as a guarantor of security,” Mr. Samore said.

In countries such as Germany and Poland, South Korea and Japan, governments without nuclear weapons are debating the need to pursue them. South Korea, menaced by the nuclear-armed North, has initiated a plan to develop a fleet of nuclear-powered submarines, with the support of the United States.

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A French nuclear submarine. France has expanded its nuclear deterrence to cover some allies as the United States has pulled back on European security. Credit…Pool photo by Yoan Valat

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Elementary school students might memorize their times tables for single-digit numbers, but memorization won’t cut it when the teacher asks for three-digit multiplication. This requires an algorithm: students are taught to stack one number atop another and multiply each digit of the bottom number by each digit of the top one. For millennia, mathematicians believed this to be the fastest multiplication method, until a 23-year-old made a shocking discovery in 1960, which led to a mystery that remains unsolved to this day.

This mystery is critical to anyone who partakes in the digital world because multiplication is a foundational operation for computers. Encryption, robotics, artificial intelligence, audio processing, and pretty much everything else we task silicon chips with involves multiplication, sometimes of huge numbers many times over. At this scale, even a simple operation becomes a bottleneck, and any extra efficiency has global economic consequences.

To understand the nature of that bottleneck, observe how the grade-school algorithm handles growth. When you multiply two two-digit numbers, you perform four single-digit multiplications. If you bump that up to a three-digit pair of numbers, you do nine single-digit multiplications. The workload scales with the square of the number of digits (n², where n is the number of digits in the numbers being multiplied). When analyzing an algorithm like this, computer scientists don’t measure speed in seconds, because that depends on the hardware.

Instead, they count the computational steps. They also ignore minor bookkeeping details, such as the time it takes to carry a one when multiplying. When numbers get large enough, those lower-level operations cease to matter, entirely eclipsed by the more intensive operations involved. Computer scientists denote the number of steps using what’s called Big O notation: the grade-school algorithm, for example, takes O(n²) steps, which is read as “order n squared.” Broadly speaking, if the numbers are twice as long, the algorithm takes four times as much computational work to execute. If the numbers are a thousand times as long, it takes a million (1,000 squared) times as much work.

Since antiquity, mathematicians have suspected that O(n²) was an inherent speed limit for multiplication. The celebrated Soviet math professor Andrey Kolmogorov posed the O(n²) speed limit as a formal conjecture and mentioned it during a 1960 seminar at Moscow State University. Whenever mathematicians propose a conjecture, they are planting a flag of sorts and waiting for others to either prove or disprove them. It took just a week for Anatoly Karatsuba, then a 23-year-old student in the audience, to return and prove Kolmogorov wrong. Kolmogorov was stunned. The result was published in the prestigious Proceedings of the USSR Academy of Sciences, but amusingly, Karatsuba didn’t write it. Kolmogorov wrote the formal proof himself and submitted it for publication with Karatsuba listed as the lead author. Karatsuba only found out about the paper when he received the reprints in the mail.

Karatsuba’s genius was realizing that you can trade expensive, time-consuming multiplications for cheap, fast additions. Adding two n-digit numbers takes only O(n) time because it entails a single sweep through the digits rather than a complete sweep through the top number for every digit of the bottom number, as in multiplication. To see how Karatsuba traded multiplication for addition, let’s look at a small example. The method would be overly complicated for such a simple problem, but it saves a meaningful amount of time when numbers get larger.

First, we split both numbers into their tens and ones digits. Assign a = 1 and b = 2 (for 12), and c = 3 and d = 4 (for 34). Algebraically, we can rewrite 12 × 34 as (10a + b) × (10c + d).

Expanding this gives 100(ac) + 10(ad + bc) + (bd).

To solve the equation in the traditional way, one must perform four distinct multiplications: ac = 3, ad = 4, bc = 6 and bd = 8, which is exactly what the grade-school stacking method entails. (Note that we don’t count the multiplications by 100 or by 10 because those only involve plopping zeroes at the ends of numbers.) Karatsuba noticed a brilliant algebraic trick. Once you compute the first and last terms, ac and bd, you can figure out that pesky middle term (ad + bc) with one more multiplication step rather than two. You don’t need to calculate ad and bc individually:

(ad + bc) = ((a + b) × (c + d)) – ac – bd,

Or with our concrete numbers:

((1 × 4) + (2 × 3)) = ((1 + 2) × (3 + 4)) – 3 – 8 = 10.

Pause to notice the weirdness in the equation above. It suggests that to multiply 12 × 34 quickly, you should add the 1 and the 2 in 12 and the 3 and the 4 in 34. This is hardly a natural thing to do. No wonder it took so long for someone to figure it out. It ends up reducing the workload, however: because we’ve already computed ac and bd, the right-hand side only contains one more multiplication, plus some additions and subtractions.

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The sun rises behind the skyline of midtown Manhattan and the Empire State Building in New York City on March 10, 2026, as seen from Union City, New Jersey. Gary Hershorn—Getty Images

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