mathematics

The Fascinating Mathematics Behind How to Cut It Perfectly in One Slice

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Imagine having a sandwich in front of you with two slices of bread and in the middle a slice of ham, which you want to cut with a single straight cut with a knife, so that the top bread, the bottom bread and the ham are divided exactly in half. This is something that makes us wonder: would it always be possible no matter how the ingredients are arranged? And this, which for our intuition has a relatively simple answer, for mathematicians has been the basis for creating the ‘ham sandwich theorem‘. His story. Although the name sounds like a university hallway joke, we are dealing with a very serious classical mathematical theorem. For trace its originwe must travel to 1938, where there is the first known evidence of this problem and which appeared in a note by the Polish mathematician Hugo Steinhaus in the magazine Mathesis Polska. The difference is that at that time slices of bread were not used, but rather he simultaneously bisected meat, bone and fat from a ham with a single flat cut. Although it must be detailed that Steinhaus proposed his conjecture, but it was the mathematician Stefan Banach who managed to prove it, making it exist for a long time. many doubts about who had the attribution of this theorem. How it was demonstrated. To see that the perfect cut really exists, the mathematician had to resort to topology and reduce the problem to the so-called Borsuk-Ulam theorem. Without going into the formulas for this, Banach’s proof uses the unit sphere. To give us an idea, if we consider all the possible directions in which we can point our knife on the sandwich, we can define a continuous mathematical function that evaluates what volume fraction of each ingredient remains on the “positive side” of the knife. Here the Borsuk-Ulam theorem guarantees us that, in any three-dimensional sphere, there are always two opposite points that will have the same result. In summary. In practical terms, this means that mathematics assures us that there will always be an exact angle and position for the knife that will balance the volumes of bread, ham and cheese in an exact 50% ratio. What is it for? Beyond the anecdotal aspect of seeing how a sandwich stars in a mathematical theorem, it is important to highlight that it has applications in geometry and computer science to create complete algorithms to be able to process a large amount of data. Today, this problem is a classic in mathematics teaching. Dissemination platforms in Spanish, such as the well-known blog Gaussians or the channel Smyth Academyit they use usually to explain advanced topological concepts in an intuitive way, and all ultimately due to this concept that we have all done at some point in the kitchen. Images | Suea Sivilaisith In Xataka | This ranking shows which scientists have saved the most lives with their discoveries

the mathematics of the crisis no longer add up

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We have been in the Third Gulf War for a month and what everyone feared has happened: the WTI barrel has broken the $100 barrier. It is normal for panic to spread. Experts get tired of repeating that, if we add inflation, in previous crises things looked worse. Okay, they’re right. But the drama right now isn’t what’s ticking on the quote screen. The problem is that there is no raw physicality. The ships do not arrive, the inventories are emptied and, no matter how much we stretch the numbers, the mathematics of oil has been blown up. A chasm of 8 million. The reality of the current oil market is summarized in a subtraction that continues to produce negative balances. The closure of the Strait of Hormuz has led to the immediate loss of 20 million barrels per day (b/d) of crude oil and refined products. So far, the market has managed to absorb this blow, but the accounts are clear and terrifying: after exhausting the first defense shields, the world faces a net deficit of approximately 8 million barrels per day. To put it in perspective, that figure exceeds the combined consumption of Germany, France, the United Kingdom, Italy and Spain. The mirage of temporary patches. If the world has lost a fifth of its oil and gas supply, why haven’t we seen a total collapse from day one? The answer lies in a series of emergency “patches” that, although effective, have an expiration date. According to Bloombergthe market has been protected by several layers of cushioning. The first has been the diversion of routes: Saudi Arabia and the United Arab Emirates have been quickly redirected part of its exports through pipelines that bypass Hormuz to exit through the Red Sea and the Gulf of Oman. The second has been the record release of 400 million of barrels of the strategic reserves of the member countries of the International Energy Agency. Furthermore, the United States temporarily lifted sanctions on Russian and Iranian oil that was stored in ships on the high seas, injecting more crude oil into the market. The efforts are finite. As energy analyst Javier Blas explainsthese joint measures have managed to absorb 60% of the loss of supply (about 12 million barrels per day). But the reserves are empty. As Paola Rodríguez-Masiu, chief analyst at Rystad Energy, warned: “The system has gone from being cushioned to being fragile.” The black hole is not going to stop, because the escalation does not let up. An Iranian drone recently attacked to the Kuwaiti supertanker Al-Salmi, fully loaded, in the same anchorage area of ​​the port of Dubai, showing that no vessel is safe. Besides, as reported exclusively The Wall Street JournalPresident Donald Trump has told his advisers that he is willing to end the US military campaign without reopening the Strait of Hormuz. The administration assesses that forcing the sea lane to open would extend the military mission beyond its planned time frame of four to six weeks. We enter “demand destruction.” For Washington, the strait is a bigger problem for Europe and Asia than for the United States. With no new supplies in sight and reserves running out, the market has only one way out, and it is the most painful of all: demand destruction. If there is no oil for everyone, someone has to stop consuming it. The impact will be brutally uneven. How to analyze Financial Timesdeveloping economies are the first to fall. Unlike advanced economies, which rely more on the service sector, the developing world depends on manufacturing that is highly energy intensive. When prices rise, rich countries simply pay more and hoard available cargoes on the spot market, leaving poorer nations in the dark. Rationing has already begun. On the Asian continent, the lack of liquefied natural gas (LNG) from the Middle East is forcing machinery to stop. In fact, the blockade has already exploded the myth of LNG as a “bridge fuel”, forcing Asia to burn coal in desperation and resurrect nuclear energy to avoid a large-scale blackout. Pakistan has closed schools to save energy, the Philippines has experimented with shorter work weeks, and across the region, fertilizer, steel and ceramics factories are shutting down their kilns. But the energy shock will not stay in that part of the world, but it is traveling towards the West. Europe faces an imminent shortage of diesel – the fuel that powers the global economy – in the coming weeks. If the Strait of Hormuz remains closed until the second quarter of the year, oil could skyrocket to $200 per barreltriggering a stagflation shock (high inflation with economic stagnation) not seen since the 1970s. The final verdict. A month into the crisis, the consensus in the energy industry is terrifying: the world has not yet understood the gravity of what is coming. Government tools to cushion the blow have been exhausted. As Mike Sommers confessedCEO of the American Petroleum Institute: “The playbook is pretty empty right now.” The crude oil does not flow and the mathematics is exact. There is a shortage of 8 million barrels a day in the world. As Jeff Currie crudely summarizedformer director of Goldman Sachs and current strategist of the Carlyle Group: “The main message is that the energy transition is going to be imposed on us in a very painful way and that it is going to happen very quickly.” That is to say, the leap towards a world without oil will not come in a planned and orderly manner, but rather through blackouts, inflation and the forced closure of industries. There are no longer valid patches; The global market is about to crash into a wall of physical scarcity. Image | Unsplash and Unsplash Xataka | That there are dozens of A-10s heading to Iran suggests something: the US has rescued its most “brute” plane for an impossible mission

We thought that the heart of the Milky Way was an immense black hole. Mathematics has changed this idea for us

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Science advances, and this also means rewriting what we believed to be ‘absolute truth’ within different fields of knowledge. For example, for decades the scientific consensus has been unwavering in pointing out that in the heart of the Milky Way, about 27,000 light years from Earth, there is a huge supermassive black hole. But now this is not so clear thanks to a new study who has “seen” something even more interesting in this location. Breaking rules. It has been a study published this year the one who has proposed that the “monster” that governs our galaxy is not a black hole, but an ultradense core of dark matter. A compact object of almost four million solar masses that a priori would be composed entirely of fermionic dark matter. How do they know it? To support this bold claim, researchers have used the RAR model. This is very important, since, unlike the classical theory, which separates the central black hole from the halo of dark matter that surrounds the galaxy, this new approach unifies both concepts into one. In this way, it is proposed that dark matter particles are highly concentrated in the galactic center, forming a compact and massive nucleus, while on the outskirts they are diluted, forming the well-known and extensive dark halo. The big question. If it’s not a black hole, why does it “look” like one? And it is something normal that passes through our minds, especially after the year 2022 when the Event Horizon Telescope (EHT) gave us the first “photograph” of Sgr A* where a bright ring could be seen surrounding a deep central darkness. And although this could be definitive proof that there is a black hole at the center of our galaxy, this is not the case. This is where previous key work published in 2024 comes into play, which pointed out that a dense core of fermions illuminated by an accretion disk generates a “shadow” visually indistinguishable from that cast by a classical black hole. That is, this dark matter is disguised to be able to deceive our telescopes when taking different measurements. Mathematical tests. In addition to this interesting theory, the scientific team has subjected it to a rigorous statistical examination using complex simulations and Bayesian analyzes to verify its robustness. Here they have shown that this dark matter core perfectly explains, for example, the orbits of the S stars that orbit the galactic center. But this unified model also fits precisely with the most recent data on the galaxy’s outer rotation curve provided by the Gaia DR3 mission. You have to look better. Although the mathematics add up and the model passes the statistical tests with flying colors, dethroning a supermassive black hole from the scientific imagination is not an easy task. And it is somewhat relevant, since the dark matter core lacks an event horizon, which is the absolute gravitational boundary of no return from which any element would be absorbed by the black hole. To know once and for all whether we are dealing with a black hole or a giant ball of dark matter, astronomers are aiming for the next generation of observations. We need to track what happens a little closer to the absolute center and future data of the GRAVITY interferometer (installed on the Very Large Telescope) will be key to detecting the subtle orbital deviations in the closest stars that would end the debate. Images | Dns Dgn BoliviaIntelligent In Xataka | We have a serious problem in our plans to colonize Mars: the astronauts’ blood is mutating

The case of mathematics shows that the hype threatens to explode in their faces

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A group of OpenAI researchers claimed to have “found solutions to 10 previously unsolved Erdös problems, and progress has been made on 11 others.” The statement seemed to indicate that GPT-5 had made an important qualitative leap in the field of mathematics, but the reality was very different. In fact, it all turned out to be an exaggeration that may harm OpenAI’s reputation going forward. what has happened. The OpenAI engineers’ claim was promising, but exaggerated. The original message from Mark Selke, one of them, was added to those of other researchers such as Boris Power—who he apologized after realizing that they had screwed up—or Sebastian Bubeck—who also ended up modifying the tweet and acknowledged the error—. The original tweet seemed to make it clear that GPT-5 had managed to solve several of the famous Erdös mathematical problems. I hadn’t really solved them. GPT-5 served to find solutions. The mathematician Thomas Bloom, who is precisely in charge of managing the website where all these open problems are managed, quickly clarified the situation. As explained on X/TwitterOpenAI’s claims were “a dramatically misinterpretation.” When he talks about “open” problems on the website, what he means is that he doesn’t know the solution, not that the problem has not been resolved. The only thing GPT-5 did was find recent research and studies that Bloom had not found. Here we must say that AI has managed to make striking mathematical advances recently: Meta AI, for example, managed to generalize the Lyapunov function. Demis Hassabis and Yann LeCun criticize OpenAI. Demis Hassabis, CEO of DeepMind, indicated in X that this event had been “shameful”, while Yann LeCun, one of the top AI managers at Meta, highlighted how OpenAI had believed its own hype sales message with the message “Hoisted by their own GPTards”, which plays on GPT and “tards” (a suffix derived from “retards”), in reference to the gullible expectations that OpenAI usually sells. Expectations are everything. Although OpenAI researchers and engineers admitted their mistake, what we see here is a dangerous pattern: one in which even the company’s own employees—or the enthusiasts who follow it—can end up falling victim to those expectations. It is very likely that internally the pressure to achieve great advances with their models is enormous, but that can lead to oversights and exaggerations like this that can cost the company’s reputation dearly. GPT-5 didn’t do badly at all. Although the role of GPT-5 in this process was exaggerated, what must be recognized is that this model demonstrated its ability to become a very valuable assistant for researchers. Thus, this AI model can search the Internet and scientific study libraries in a very powerful way, and can “find solutions” already published where academics had not yet seen them when trying to solve related problems. Research assistant. For mathematician Terence Tao, this is precisely a very striking element of these AI models: they may not solve the most complex mathematical problems, but can speed up tedious tasks such as those of the search for academic literature that helps solve them. For this expert, AI can help “industrialize” mathematics and act as a catalyst or “lubricant” for mathematicians’ hypotheses and theories. But this is important. OpenAI is a machine for creating expectations, and its CEO, Sam Altman, does not hesitate to make vague and impossible to verify promises to attract more interest in his generative artificial intelligence models. A year ago promised that the AGI would arrive “in a few thousand days”something that sounds like one of those “Musk’s promises”. risky bet. In recent weeks we have seen how OpenAI has reached unique circular financing agreements with NVIDIA, amd either Broadcom to create data centers, but the reality is that all these projects focus on one promise: that AI will be a fundamental part of our lives sooner rather than later. That can happen, of course, but if it doesn’t, the domino effect can be an absolute catastrophe given the tens of billions of dollars invested in such projects. Image | Vitaly Gariev In Xataka | If the question is whether there is an AI bubble, Sam Altman has just given the answer. One with which he wins

How to create mathematics exercises for children using artificial intelligence

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Let’s explain How to create mathematics exercises for children wearing artificial intelligence. Thus, if you want to reinforce your knowledge or put some extra tasks or exercises, you will be able to make them easily even if you do not have many knowledge or mathematics or didactics. Let’s start by giving you A series of previous tips With things you should take into account before creating exercises with AI. Then, we will tell you how to do it with a generative AI. Then we will continue with a list of other tools based on AI that you can use, and we will end up explaining how to use two of them. Before starting, some notes Before starting, you should have some things into account. First, you should know that IA may have errors When creating exercises. These algorithms can have what we call hallucinations, and put meaningless things as if they were real. That’s why, It should be reviewed all exercises To check that they are well before giving them to children. When you are going to start these exercises, the first thing you should know is What area or competition you want to work. You have to think if you want varied exercises, or if you want to focus on sums, subtraction, logic, geometry, fractions, or any other specific area. This you will have to specify it when asking to create them. You must be clear about the age strip with which you are going to work. Here, the simplest thing will be to specify the age for which they are aimed at AI, but it is also convenient that you know that you should avoid very technical words or that you must get the exercises to be brief and concrete. Try to give variety to exercises so that they are not always the same. As you create them, try that the structures or operations to be performed are not repeated, and that if you use any type of examples, they are not always the same. In addition to this, If you are going to use the AI to correct The exercises, you should ask that simple explanations to know why something is wrong, or that it is not limited to giving the answers. In addition, you can ask you to give you clues or progressive help. Children’s exercises using generative The easiest and most generic way to create mathematical exercises for children is to use general and generative artificial intelligence chats, such as Chatgpt, COPILOT, Gemini, Deepseek or similar ones. Here, The important part will be the command either Prompt that you give the AI. For example, you can use a prompt like this: “I need you to create 10 sum exercises for 6 -year -old children with fruit drawings as a context. Shows the step -by -step solution.” Using this prompt, the AI will generate a series of exercises using fruit emojis, or the subject you ask. The important part is Specify age of the children for whom the exercise is directed, and explain both the quantity and the type of exercises. If you do not want them to be alone, you can tell you to be varied, but specifying that they are mathematical and the age for which they are directed. Be careful with the subject of solutions. With the prompt that we have used, these will appear under each exercise. However, you can Ask the exercises to be on one sheet and solutions in anotherso that they are separated and the little ones do not be tempted to skip steps. Finally, remember that You can ask the AI to convert the answer into a PDF So you can print everything. You can also ask you to make it a PNG image, specifying the size format, such as in size of an A4 sheet. Use a specialized online tool There are several online tools that They allow you to create exercises with artificial intelligence. Here you have a list with the main ones that you will be able to use, as well as its characteristics: Teachermatic: A virtual classroom by the in which you can create all kinds of school materials. You can choose the level, and what it creates is totally printable and multilingual. In the free account you can generate 5 documents per day. Link: teachermatic.com. Conker AI: A free platform for teachers, with which you can generate questionnaires for AI, including mathematical content. Here, you can specify age, topic or type of questions. The bad news is that it is in English, although you can generate Spanish content. Link: Conker.ai. Curipod: It serves to create educational presentations and activities, being able to use it to generate exercises, games, surveys or mathematical problems with explanations. Link: Curipod.com. Magicschool.Ai: Another portal where you can create all kinds of exercises and evaluations with AI, from spreadsheets to mathematical problems. You can use it in Spanish, but the web is English. Link: App.Magicschool.Ai. Khanemigo: A Khan Academy website, where its GPT -based educational assistant allows you to generate exercises or help a student solve them step by step. It is not available in all countries. Link: khanemigo.ai. Create exercises with Teachermatic This has seemed a great tool to create school materials with AI. It has a giant index with all kinds of filters, but to create mathematical exercises you can Choose the option Worksheetwhich is to create all kinds of exercises. Now, you will go to a previous screen where you can Set the type of exercises you want create. You can specify the subject Ly the activities, as well as specify the age for which they are directed and also give many details about your students and their level to give context. When you specify everything, the AI will generate the worksheet. Will do it in a format that You will be able to print To use it in your class, and at all times you can regenerate it and ask for changes. Create Exercises … Read more

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