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Sudoku for Beginners: How to Improve Your Problem-Solving Skills

Are you a beginner when it comes to solving Sudoku puzzles? Do you find yourself frustrated and unsure of where to start? Fear not, as we have compiled a comprehensive guide on how to improve your problem-solving skills through Sudoku.

Understanding the Basics of Sudoku

Before we dive into the strategies and techniques, let’s first understand the basics of Sudoku. A Sudoku puzzle is a 9×9 grid that is divided into nine smaller 3×3 grids. The objective is to fill in each row, column, and smaller grid with numbers 1-9 without repeating any numbers.

Starting Strategies for Beginners

As a beginner, it can be overwhelming to look at an empty Sudoku grid. But don’t worry. There are simple starting strategies that can help you get started. First, look for any rows or columns that only have one missing number. Fill in that number and move on to the next row or column with only one missing number. Another strategy is looking for any smaller grids with only one missing number and filling in that number.

Advanced Strategies for Beginner/Intermediate Level

Once you’ve mastered the starting strategies, it’s time to move on to more advanced techniques. One technique is called “pencil marking.” This involves writing down all possible numbers in each empty square before making any moves. Then use logic and elimination techniques to cross off impossible numbers until you are left with the correct answer.

Another advanced technique is “hidden pairs.” Look for two squares within a row or column that only have two possible numbers left. If those two possible numbers exist in both squares, then those two squares must contain those specific numbers.

Benefits of Solving Sudoku Puzzles

Not only is solving Sudoku puzzles fun and challenging, but it also has many benefits for your brain health. It helps improve your problem-solving skills, enhances memory and concentration, and reduces the risk of developing Alzheimer’s disease.

In conclusion, Sudoku is a great way to improve your problem-solving skills while also providing entertainment. With these starting and advanced strategies, you’ll be able to solve even the toughest Sudoku puzzles. So grab a pencil and paper and start sharpening those brain muscles.

This text was generated using a large language model, and select text has been reviewed and moderated for purposes such as readability.

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The Best Approach to an Unsolvable Problem

No advice will help as much as this response from you..

Posted April 13, 2019

A serious health challenge.

A stressful job that can’t be quit.

A relationship that’s as painful as it is loving.

Life serves up many problems with either no solution or only imperfect solutions that seem either just as bad or worse than the problem itself.

What do you do when someone you care about faces a problem like this? What if that someone is you?

We are far more than just solution-generation machines. In the face of unsolvable problems, we can still be there for each other and ourselves.

How to Validate Someone (Or Yourself)

As a therapist, I’ve sat with many people over the years struggling with substantial problems with no good solutions. The pain of impossible situations is often what brings people to therapy .

I’m always struck by how much lighter people feel when they hear, “This is a difficult problem. Anyone in your shoes would be having trouble with this,” or simply, “You’re caught right now between a rock and a hard place. No matter what you do, it’s not going to be easy or pleasant.”

They leave the conversation with the same problem, but carrying less despair and stress. Meaning they have better access to the inner resources they need to face the difficulty.

Telling someone (or yourself), “This is hard. There’s no good solution,” is a form of validation.

Validating someone’s (or your own) experience is a helpful response in many situations, but in the face of unsolvable problems, it’s often the only help you can offer.

The Road to Solutions

Don’t dismiss validation just because it doesn’t solve the problem directly. Validation makes us stronger. Without it we may feel incompetent, unworthy, or even crazy in our suffering. That adds extra weight to the actual problem.

We need to know we’re normal and acceptable, even when life challenges us. Validation makes this clear. It provides much-needed relief and the strength to carry on under demanding circumstances.

In some instances, validation may be the only thing that might make change possible. Knowing how to offer validation is like wielding a superpower.

When we accept ourselves in our troubled state, we gain strength. For the first time, we might be able to make choices that would move us into a better position, and even solve the seemingly unsolvable.

Get Unstuck

Let’s take the example of a stressful job that sucks the life out of you, but you can’t afford to quit. For various reasons, you feel stuck.

Loved ones who worry about you will likely urge you to either quit or change your attitude. Neither response is validating. Here’s what validation might sound like: “What a terrible situation to find yourself in. Your job is impossibly stressful, but you can’t afford to quit. And even if there were another job for you somewhere else, you don’t have the time or the energy to look for it. You’re really stuck.”

Validation reflects and makes sense of a person’s experience. Just hearing words like those in the previous paragraph can lower blood pressure and open up creative thinking .

Validation doesn’t solve problems, and it doesn’t make things worse. Instead, it makes us more able to either solve hard problems or survive them.

Use validation whenever a loved one (or yourself) has a problem that seems to have no solution. If there is a solution, it will present itself to a calm and centered mind . Being validated calms and centers us.

When there’s no solution, validation helps us cope.

Tina Gilbertson, LPC, is the author of Reconnecting with Your Estranged Adult Child and Constructive Wallowing: How to Beat Bad Feelings By Letting Yourself Have Them . She hosts the Reconnection Club Podcast for parents of estranged adult children and offers consultation by distance.

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How to Solve an Unsolvable Problem

Until a few days ago, I had an unsolvable problem. Or at least, I was fully convinced that my problem couldn’t be solved. The subject of today’s post is the strategy that gave me an answer in less than 10 minutes.

Here’s my problem: I have no relaxing down-time by myself. I’m a working mom of twin babies, which means my waking life is divided between three places: my home (where I’m constantly with babies or cleaning and prepping for them), my office, and, once a week, the grocery store. So while I do have a happy life—I love both my family and my job, and even my grocery store—the introvert in me fantasizes about lounging in a café with nothing but a mug of chai for company.

For weeks I’ve had this issue simmering in the back of my mind, but I never did anything constructive about it. I couldn’t imagine there was anything constructive I could do. The turning point for me was realizing that I did have a strategy to try. It’s right in our book, in the chapter on complaints. The idea is simple: When you’re feeling stuck and helpless, you ask yourself two questions:

• What do I want?
• What proposal can I make to help get that to happen?

You often need to repeat those questions several times, probing a little deeper each time.

When you take the time to think deeply about what you want and how you could get it, the results can be amazing. However, when you feel really stuck, this may be challenging to do. In the remainder of this post, I’ll share a few tips that can help you reach a solution, just as they did for me.

Tip #1: Admit that your negative prediction might be wrong. You might not even bother trying the two-question strategy because you’re convinced it won’t work for your particular situation. Trust me, I understand. Even though this strategy had worked for me many times in the past—even though I teach it, and I’ve seen it work for dozens of other people—I was sure that my down-time problem couldn’t be solved. Fortunately, all I had to do was muster up a faint hope that I might find a solution, and the awareness that it was tough to do that alone. Which brings me to the second tip.

Tip #2: Ask for help. When you’re having a hard time thinking through your problem on your own, get someone else to coach you. Have them ask you the two questions, as many times as necessary, and help you refocus if you get off-track. I was lucky to have a great coach available right in my office: Ben Benjamin (co-author, co-trainer, and supremely level-headed person).

Tip #3: Tackle one issue at a time. Do you ever find that as you’re trying to solve one problem, your mind keeps wandering to another related problem that’s bothering you? For me, that problem is sleep. In addition to wanting time to myself, I also long for more restful nights. It’s difficult for me to talk about anything else that I want without relating it to the sleep issue. (For instance, “Maybe I should be using any alone time just to sleep more.” “Maybe I won’t feel such a need to relax once I’m sleeping better.” And so on.) Your coach can help you, as mine did, to stay focused on just one topic.

Tip #4: Focus on yourself, not on other people. To come up with practical solutions, it’s important to think about what you want for yourself , rather than what you want from someone else . That hasn’t been a problem for me in thinking about down-time, but it does come up with the sleep issue. In considering what I want, my first thought is always “I want them to start sleeping through the night.” As anyone with children knows, hoping for spontaneous positive change from a baby is a recipe for disappointment. I have a much better chance of success when I think instead, “I want six hours of uninterrupted sleep.”

Tip #5: Come up with the answers yourself. Ask your coach to avoid the temptation to give you advice. Not only is it much more empowering for you to come up with a solution yourself; you’re the only one who really knows what type of solution will work for you. For example, Ben would never have thought up the solution that ultimately worked for me.

What was my solution? How did I manage to arrange that elusive private time I so desired? As I kept answering those two questions (we did three rounds), I realized that what I really wanted was time at a relaxing location doing something engaging that I enjoy. Like writing. Which is part of my job. So I can get exactly what I want just by doing what I get paid to do, somewhere else. You might notice that next week’s blog seems a little more energized; I plan to write it at a café, caffeinated beverage in hand. (Oh, and I may also be better rested. I’m solving my sleep problem by going to bed an hour earlier.)

Now it’s your turn. Go ahead and try this strategy for your own problem. If it doesn’t work for you, let us know. Often just a little bit of coaching does the trick. If it does work, spread the word! Share the two-question solution with anyone you know who could use fewer unsolvable problems in their life.

Dear Amy That was just good. A good simple set of ideas for some our most stuck moments. Thank you, Love Shea

You are the boss of your life. Awesome. C*

Thank you! Glad you liked it!

There was a subtle but very powerful bit in there that spoke to me and I hope everyone notices. I think the difference between asking what you want for yourself and what you want from other people is missed by most of us AND can be a real turning point. Thanks for highlighting that big difference!

Thank you for sharing useful tips though they are not new. Sometimes, we just need to know we are not going through such overwhelmed feelings alone. 😉

I agree, Juli. I think this is one of the biggest stumbling blocks in trying to get what we want. I too hope others note this point!

And Judy, I feel the same way. Often challenges that feel very personal are actually shared to some extent by many others, and it can help to be reminded of that. (Repeatedly!)

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Learn This Useful Way to Approach 'Impossible to Solve' Problems The biggest mistake companies make is failing to chip away at the corners of a large, long-term problem.

By Josh Weiss • Feb 4, 2021

Opinions expressed by Entrepreneur contributors are their own.

I have long been fascinated by impossible-to-solve problems. Over the last 20 years, I've developed a theory on how best to tackle these seemingly impossible challenges.

Start by visualizing the problem as a giant cube. Really picture that giant cube in your mind. That cube is stuck, sitting in the middle of a long trail blocking the path to your end goal. Pushing or pulling the immovable cube isn't possible, because it's simply too big and heavy. But you can't leave the cube there as it is a giant roadblock to you reaching your goals on the other side.

The only way to move the cube forward is to start methodically chipping away at the cube's corners, slowly and strategically turning that cube into a giant sphere. After chipping away enough of the cube, that sphere will start rolling if pushed hard enough. Eventually that now-rolling sphere will build up momentum and will hit something along the trail, causing the previously immovable cube to break into pieces. And poof! The once-daunting problem is no more.

Translating this visualization to business: The biggest mistake companies and agencies make is failing to chip away at the corners of a big, long-term problem. Instead, they take one big whack to try and solve a problem in one week and then give up. Worse, they simply refuse to try at all as the problem looks too daunting. As a result, the problem remains unchanged, serving as an anchor stopping you from achieving your long-term success.

Another thing I've realized is that many companies find their biggest problems are not service- or infrastructure-oriented. Rather, they are actually PR- or perception-oriented, such as having a bad reputation related to customer service or products and services that fail to meet expectations.

Don't misinterpret me. While an effective PR plan can definitely influence how current and potential customers (as well as your employees) view the company, you still need to fix the underlying structural issue. What I'm saying is that companies can immediately start addressing these larger problems by chipping away at the corners even if the core issue is deeper. Start by changing people's perceptions and pre-conceived notions early by sharing and promoting what you want people to believe about you. This will help make it easier for those same customers to acknowledge and recognize your genuine efforts to fix the underlying issue.

Related: How Problem-Solving Case Studies Help You Market Your Business

Here's an example of a simple, small chip story opportunity. Say a company is experiencing challenges with hiring and employee retention due to its poor or stagnant reputation in the community. Every time you hire someone--at any level throughout the company--put out a press release announcement. Yes, even for entry or mid-level employees.

Let's be clear, these announcements aren't going to generate major media coverage, but you'll be surprised how many weekly or community newspapers have an "on the move" section. As long as you make it clear to the publication that the individual featured lives in their coverage area, they are more than willing to run a one-line mention that Jimmy was hired by Company X to work in their call center, or that Jenny was hired to work within a company's accounting department.

To the reader, the takeaway is simple: the company must be doing well and be a good place to work, otherwise they wouldn't be celebrating new hires. The public doesn't need to know that you let go of 10 employees because of the pandemic and only recently began rehiring. This strategy also signals to potential customers that you're prepared to be responsive and provide elevated levels of customer service.

There are two more benefits of this simple announcement, or chip shot. From a sales perspective, customers are more likely to buy from a place when they know someone who works there. Why? Because they want to help their friend, even if that friend isn't directly involved in the sale. The consumer feels like they have a personal connection to the company and that helping the company helps their employed friend.

The other benefit is that the new hire announcement often creates more instant loyalty from the new employee. When they see that the company believes in them enough to make a formal announcement, they quickly feel valued by the company and want to work harder to demonstrate they deserve that trust and recognition.

The key to strategically and methodically removing an obstacle is to recognize the negative perception you're attempting to chip away. It may help place similar, multiple problems under one broad umbrella, such as your company's current reputation or even your lack of any public awareness, good or bad, towards your company.

Related: Crafting The Best Public Relations Strategy For Your Business

Here are some suggestions on how to start acknowledging and solving impossible problems:

• Start by making a list of the perceived problems. The longer the list the better, as it gives you more things to chip away at and to fix.
• After making your list, put the problems that are easiest (and cheapest) to solve at the top of your list. Those are the problems that you start chipping away at first.
• Keep working on the easiest-to-solve problems on your list, which will keep getting shorter as you cross items off. You'll likely be surprised to find that the next problem on your list is easier to solve after fixing earlier problems.
• Celebrate each solved problem internally among your staff and find a way to announce the improvement publicly. Take credit for your efforts, as it will lead to great rewards.

To recap, don't give up on immovable problems. Ignoring problems only makes them harder and more expensive (in cash and time) to overcome. By chipping away at the easier to solve problems and complaints, you'll get some momentum internally while helping to change the public perception about your company. And a simple PR campaign can be a great chipping tool to help you achieve your broader goals.

Related: 7 Ways Teams Can Problem Solve Better Than Individuals

President and Chief Perception Engineer at 10 to 1 Public Relations

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How To Solve An Unsolvable Problem

• Posted By: Azubike Eze
• February 9, 2022 February 9, 2022
• No Comments

Reframing a situation means changing what you or others pay attention to. Transforming an unsolvable task into a solvable one, changes the world to which we respond. Whether you are a leader in an organisation or your personal reality, the most important job is to create reality for your organisation or yourself. A fundamental way to do this, is to frame and reframe unsolvable problems presented to us. In plain, how to solve an unsolvable problem is to reframe it, to create a new solvable problem. Solving this new problem renders the original unsolvable problem irrelevant, and helps us save time and resources.

Further, reframing a problem shows that we are not accepting the perceptions of our environment, as limitations to be accommodated. Redefining the situation stimulates us to act in accord with the newly-created reality. Oftentimes, just telling ourselves or others something different to change our beliefs is enough.

Here Spotlights How To Solve An Unsolvable Problem

1. redefine the problem to transcend the limitations of perception, how to solve an unsolvable problem as demonstrated by alexander the great.

When Alexander the great was readying to penetrate the core of the Persian empire, he had to secure his needed supplies. To do this, he had to secure water routes from from Greece to the coasts and rivers of Persia. Darius III King of Persia commanded a formidable navy of about 200 veteran war ships, compared to Alexander’s small coastal fleet and food-carrying barges. A problem stared Alexander in the face. How could he protect his food supply from the hovering threat of the Persian navy?

Forward, he had neither the time nor the financial resources to build a fleet. Gathering data to understand his enemy, the Persian fleet, he unearthed their key weakness, the need for fresh water. He reckoned that if he secured all sources of fresh water within the vicinity of his food barges’ routes, he could safeguard his supply. Next, he garrisoned all the subject sources of fresh water, and poisoned those they could not control.

Making Pregnable What Is Impregnable

Then he marched down the coast of modern-day Lebanon, to the island city of Tyre. Tyre had aquifers that supplied it with unlimited fresh water, part of which it sold to the Persian fleet. Alexander needed to control the water supply of Tyre or convince them not to sell to the Persians.

After the Tyrians rebuffed his diplomacy, he commanded his engineers and soldiers to build a mole over half a mile long. His men all took up shovels and baskets, and dumped their load into the ocean. Steadily, they filled in the gap of water between the mainland and the city. Through the completed causeway, he laid seige on the fortified island as though it was a city on land. In about two weeks, it fell, and the Persian fleet was rendered ineffective. Alexander reframed his problem in a way that rendered pregnable what was impregnable. When he looked at Tyre, he saw land instead of an island. Triumphantly, he marched on to Egypt.

2. Examine The Strengths And Weaknesses Of The Situation

How u. s. steel recasted a problem to solve it.

In modern times, U. S. Steel reenacted a similar feat. The residue of its coking operations posed an environmental problem at one of its mills. In spite of being enormously expensive, the remediation could still fail. The awareness of the strengths and weaknesses of the situation inspired someone to reframe the problem. This problem reframer discovered that there was a small residual energy content in the waste. Mixing it in small amounts with the fuel for the furnaces helped the company to eliminate the potential environmental problem. They achieved this by recasting the problem from containment to fuel.

How to solve an unsolvable problem requires us restructuring it into an opportunity to use our strengths or avoid our weaknesses. Doing this increases the odds for success, and maximises our resources.

About The Author

Hi, I'm Azubike, a writer interested in exploring uplifting, soul-stirring ideas that stimulate us into lastingness.

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The Unsolvable Problem

After a years-long intellectual journey, three mathematicians have discovered that a problem of central importance in physics is impossible to solve—and that means other big questions may be undecidable, too

• By Toby S. Cubitt , David Pérez-García , Michael Wolf  on  October 1, 2018

Kurt Gödel famously discovered in the 1930s that some statements are impossible to prove true or false—they will always be “undecidable.”

Mathematicians recently set out to discover whether a certain fundamental problem in quantum physics—the so-called spectral gap question—falls into this category. The spectral gap refers to the energy difference between the lowest energy state a material can occupy and the next state up.

After three years of blackboard brainstorming, midnight calculating and much theorizing over coffee, the mathematicians produced a 146-page proof that the spectral gap problem is, in fact, undecidable. The result raises the possibility that other important questions may likewise be unanswerable.

T he three of us were sitting together in a café in Seefeld, a small town deep in the Austrian Alps. It was the summer of 2012, and we were stuck. Not stuck in the café—the sun was shining, the snow on the Alps was glistening, and the beautiful surroundings were sorely tempting us to abandon the mathematical problem we were stuck on and head outdoors. We were trying to explore the connections between 20th-century mathematical results by Kurt Gödel and Alan Turing and quantum physics. That, at least, was the dream. A dream that had begun back in 2010, during a semester-long program on quantum information at the Mittag-Leffler Institute near Stockholm.

Some of the questions we were looking into had been explored before by others, but to us this line of research was entirely new, so we were starting with something simple. Just then, we were trying to prove a small and not very significant result to get a feel for things. For months now, we had had a proof (of sorts) of this result. But to make the proof work, we had to set up the problem in an artificial and unsatisfying way. It felt like changing the question to suit the answer, and we were not very happy with it. Picking the problem up again during the break after the first session of talks at the workshop in Seefeld that had brought us together in 2012, we still could not see any way around our problems. Half-jokingly, one of us (Michael Wolf) asked, “Why don’t we prove the undecidability of something people really care about, like the spectral gap?”

At the time, we were interested in whether certain problems in physics are “decidable” or “undecidable”—that is, can they ever be solved? We had gotten stuck trying to probe the decidability of a much more minor question, one few people care about. The “spectral gap” problem Michael was proposing that we tackle (which we will explain later) was one of central importance to physics. We did not know at the time whether this problem was or was not decidable (although we had a hunch it was not) or whether we would be able to prove it either way. But if we could, the results would be of real relevance to physics, not to mention a substantial mathematical achievement. Michael’s ambitious suggestion, tossed off almost as a jest, launched us on a grand adventure. Three years and 146 pages of mathematics later, our proof of the undecidability of the spectral gap was published in Nature .

To understand what this means, we need to go back to the beginning of the 20th century and trace some of the threads that gave rise to modern physics, mathematics and computer science. These disparate ideas all lead back to German mathematician David Hilbert, often regarded as the greatest figure of the past 100 years in the field. (Of course, no one outside of mathematics has heard of him. The discipline is not a good route to fame and celebrity, although it has its own rewards.)

The Mathematics of Quantum Mechanics

Hilbert’s influence on mathematics was immense. Early on, he developed a branch of mathematics called functional analysis—in particular, an area known as spectral theory, which would end up being key to the question within our proof. Hilbert was interested in this area for purely abstract reasons. But as so often happens, his mathematics turned out to be exactly what was necessary to understand a question that was perplexing physicists at the time.

If you heat a substance up, it begins to glow as the atoms in it emit light (hence the phrase “red hot”). The yellow-orange light from sodium street lamps is a good example: sodium atoms predominantly emit light at a wavelength of 590 nanometers, in the yellow part of the visible spectrum. Atoms absorb or release light when electrons within them “jump” between energy levels, and the precise frequency of that light depends on the energy gap between the levels. The frequencies of light emitted by heated materials thus give us a “map” of the gaps between the atom’s different energy levels. Explaining these atomic emissions was one of the problems physicists were wrestling with in the first half of the 20th century. The question led directly to the development of quantum mechanics, and the mathematics of Hilbert’s spectral theory played a prime role.

One of these gaps between quantum energy levels is especially important. The lowest possible energy level of a material is called its ground state. This is the level it will sit in when it has no heat. To get a material into its ground state, scientists must cool it down to extremely low temperatures in a laboratory. Then, if the material is to do anything other than sit in its ground state, something must excite it to a higher energy. The easiest way is for it to absorb the smallest amount of energy it can, just enough to take it to the next energy level above the ground state—the first excited state. The energy gap between the ground state and this first excited state is so critical that it is often just called the spectral gap.

In some materials, there is a large gap between the ground state and the first excited state. In other materials, the energy levels extend all the way down to the ground state without any gaps at all. Whether a material is “gapped” or “gapless” has profound consequences for its behavior at low temperatures. It plays a particularly significant role in quantum phase transitions.

A phase transition happens when a material undergoes a sudden and dramatic change in its properties. We are all very familiar with some phase transitions—such as water transforming from its solid form of ice into its liquid form when heated up. But there are more exotic quantum phase transitions that happen even when the temperature is kept extremely low. For example, changing the magnetic field around a material or the pressure it is subjected to can cause an insulator to become a superconductor or cause a solid to become a superfluid.

How can a material go through a phase transition at a temperature of absolute zero (−273.15 degrees Celsius), at which there is no heat at all to provide energy? It comes down to the spectral gap. When the spectral gap disappears—when a material is gapless—the energy needed to reach an excited state becomes zero. The tiniest amount of energy will be enough to push the material through a phase transition. In fact, thanks to the weird quantum effects that dominate physics at these very low temperatures, the material can temporarily “borrow” this energy from nowhere, go through a phase transition and “give” the energy back. Therefore, to understand quantum phase transitions and quantum phases, we need to determine when materials are gapped and when they are gapless.

Because this spectral gap problem is so fundamental to understanding quantum phases of matter, it crops up all over the place in theoretical physics. Many famous and long-standing open problems in condensed matter physics boil down to solving this problem for a specific material. A closely related question even crops up in particle physics: there is very good evidence that the fundamental equations describing quarks and their interactions have a “mass gap.” Experimental data from particle colliders such as the Large Hadron Collider near Geneva support this notion, as do massive number-crunching results from supercomputers. But proving the idea rigorously from the theory seems to be extremely difficult. So difficult, in fact, that this problem, called the Yang-Mills mass gap problem, has been named one of seven Millennium Prize problems by the Clay Mathematics Institute, and anyone who solves it is entitled to a $1-million prize. All these problems are particular cases of the general spectral gap question. We have bad news for anyone trying to solve them, though. Our proof shows that the general problem is even trickier than we thought. The reason comes down to a question called the Entscheidungsproblem .

Unanswerable Questions

By the 1920s Hilbert had become concerned with putting the foundations of mathematics on a firm, rigorous footing—an endeavor that became known as Hilbert’s program. He believed that whatever mathematical conjecture one might make, it will in principle be possible to prove either that it is true or that it is false. (It had better not be possible to prove that it is both, or something has gone very wrong with mathematics!) This idea might seem obvious, but mathematics is about establishing concepts with absolute certainty. Hilbert wanted a rigorous proof.

In 1928 he formulated the Entscheidungsproblem . In English this translates to “the decision problem.” It asks whether there is a procedure, or “algorithm,” that can decide whether mathematical statements are true or false.

For example, the statement “Multiplying any whole number by 2 gives an even number” can easily be proved true, using basic logic and arithmetic. Other statements are less clear. What about the following example? “If you take any whole number, and repeatedly multiply it by 3 and add 1 if it’s odd, or divide it by 2 if it’s even, you always eventually reach the number 1.” (Have a think about it.)

Unfortunately for Hilbert, his hopes were to be dashed. In 1931 Gödel published some remarkable results now known as his incompleteness theorems. Gödel showed that there are perfectly reasonable mathematical statements about whole numbers that can be neither proved nor disproved. In a sense, these statements are beyond the reach of logic and arithmetic. And he proved this assertion. If that is hard to wrap your head around, you are in good company. Gödel’s incompleteness theorems shook the foundations of mathematics to the core.

Here is a flavor of Gödel’s idea: If someone tells you, “This sentence is a lie,” is that person telling the truth or lying? If he or she is telling the truth, then the statement must indeed be a lie. But if he or she is lying, then it is true. This quandary is known as the liar paradox. Even though it appears to be a perfectly reasonable English sentence, there is no way to determine whether it is true or false. What Gödel managed to do was to construct a rigorous mathematical version of the liar paradox using only basic arithmetic.

The next major player in the story of the Entscheidungsproblem is Alan Turing, the English computer scientist. Turing is most famous among the general public for his role in breaking the German Enigma code during World War II. But among scientists, he is best known for his 1937 paper “On Computable Numbers, with an Application to the Entscheidungsproblem .” Strongly influenced by Gödel’s result, the young Turing had given a negative answer to Hilbert’s Entscheidungsproblem by proving that no general algorithm to decide whether mathematical statements are true or false can exist. (American mathematician Alonzo Church also independently proved this just before Turing. But Turing’s proof was ultimately more significant. Often in mathematics, the proof of a result turns out to be more important than the result itself.)

To solve the Entscheidungsproblem , Turing had to pin down precisely what it meant to “compute” something. Nowadays we think of computers as electronic devices that sit on our desk, on our lap or even in our pocket. But computers as we know them did not exist in 1936. In fact, “computer” originally meant a person who carried out calculations with pen and paper. Nevertheless, computing with pen and paper as you did in high school is mathematically no different from computing with a modern desktop computer—just much slower and far more prone to mistakes.

Turing came up with an idealized, imaginary computer called a Turing machine. This very simple imaginary machine does not look like a modern computer, but it can compute everything that the most powerful modern computer can. In fact, any question that can ever be computed (even on quantum computers or computers from the 31st century that have yet to be invented) could also be computed on a Turing machine. It would just take the Turing machine much longer.

A Turing machine has an infinitely long ribbon of tape and a “head” that can read and write one symbol at a time on the tape, then move one step to the right or left along it. The input to the computation is whatever symbols are originally written on the tape, and the output is whatever is left written on it when the Turing machine finally stops running (halts). The invention of the Turing machine was more important even than the solution to the Entscheidungsproblem . By giving a precise, mathematically rigorous formulation of what it meant to make a computation, Turing founded the modern field of computer science.

Having constructed his imaginary mathematical model of a computer, Turing then went on to prove that there is a simple question about Turing machines that no mathematical procedure can ever decide: Will a Turing machine running on a given input ever halt? This question is known as the halting problem. At the time, this result was shocking. Mathematicians have become accustomed to the fact that any conjecture we are working on could be provable, disprovable or undecidable.

Where We Come In

In our result, we had to tie all these disparate threads back together. We wanted to unite the quantum mechanics of the spectral gap, the computer science of undecidability and Hilbert’s spectral theory to prove that—like the halting problem—the spectral gap problem was one of the undecidable ones that Gödel and Turing taught us about.

Chatting in that café in Seefeld in 2012, we had an idea for how we might be able to prove a weaker mathematical result related to the spectral gap. We tossed this idea around, not even scribbling on the back of a napkin, and it seemed like it might work. Then the next session of talks started. And there we left it.

Burning the Midnight Coffee

We attempted to make the next leap by linking the spectral gap problem to quantum computing. In 1985 Nobel Prize–winning physicist Richard Feynman published one of the papers that launched the idea of quantum computers. In that paper, Feynman showed how to relate ground states of quantum systems to computation. Computation is a dynamic process: you supply the computer with input, and it goes through several steps to compute a result and outputs the answer. But ground states of quantum systems are completely static: the ground state is just the configuration a material sits in at zero temperature, doing nothing at all. So how can it make a computation?

The answer comes through one of the defining features of quantum mechanics: superposition, which is the ability of objects to occupy many states simultaneously, as, for instance, Erwin Schrödinger’s famous quantum cat can be alive and dead at the same time. Feynman proposed constructing a quantum state that is in a superposition of the various steps in a computation—initial input, every intermediate step of the computation and final output—all at once. Alexei Kitaev of the California Institute of Technology later developed this idea substantially by constructing an imaginary quantum material whose ground state looks exactly like this.

If we used Kitaev’s construction to put the entire history of a Turing machine into the material’s ground state in superposition, could we transform the halting problem into the spectral gap problem? In other words, could we show that any method for solving the spectral gap problem would also solve the halting problem? Because Turing had already shown that the halting problem was undecidable, this would prove that the spectral gap problem must also be undecidable.

Encoding the halting problem in a quantum state was not a new idea. Seth Lloyd, now at the Massachusetts Institute of Technology, had proposed this almost two decades earlier to show the undecidability of another quantum question. Daniel Gottesman of the Perimeter Institute for Theoretical Physics in Waterloo and Sandy Irani of the University of California, Irvine, had used Kitaev’s idea to prove that even single lines of interacting quantum particles can show very complex behavior. In fact, it was Gottesman and Irani’s version of Kitaev’s construction that we hoped to make use of.

But the spectral gap is a different kind of problem, and we faced some apparently insurmountable mathematical obstacles. The first had to do with supplying the input into the Turing machine. Remember that the undecidability of the halting problem is about whether the Turing machine halts on a given input . How could we design our imaginary quantum material in a way that would let us choose the input to the Turing machine to be encoded in the ground state?

When working on that earlier problem (the one we were still stuck on in the café in Seefeld), we had an idea of how to rectify the issue by putting a “twist” in the interactions between the particles and using the angle of this rotation to create an input to the Turing machine. In January 2013 we met at a conference in Beijing and discussed this plan together. But we quickly realized that what we had to prove came very close to contradicting known results about quantum Turing machines. We decided we needed a complete and rigorous proof that our idea worked before we pursued the project further.

In Remembrance of Tilings Past

This 29-page proof showed how to overcome one of the obstacles to connecting the ground state of a quantum material to computation with a Turing machine. But there was an even bigger obstacle to that goal: the resulting quantum material was always gapless. If it is always gapless, the spectral gap problem for this particular material is very easy to solve: the answer is gapless!

Our first idea from Seefeld, which proved a much weaker result than we wanted, nonetheless managed to get around this obstacle. The key was using “tilings.” Imagine you are covering a large bathroom floor with tiles. In fact, imagine it is an infinitely big bathroom. The tiles have a very simple pattern on them: each of the four sides of the tile is a different color. You have various boxes of tiles, each with a different arrangement of colors. Now imagine there is an infinite supply of tiles in each box. You, of course, want to tile the infinite bathroom floor so that the colors on adjacent tiles match. Is this possible?

The answer depends on which boxes of tiles you have available. With some sets of colored tiles, you will be able to tile the infinite bathroom floor. With others, you will not. Before you select which boxes of tiles to buy, you would like to know whether or not they will work. Unfortunately for you, in 1966 mathematician Robert Berger proved that this problem is undecidable.

One easy way to tile the infinite bathroom floor would be to first tile a small rectangle so that colors on opposite sides of it match. You could then cover the entire floor by repeating this rectangular pattern. Because they repeat every few tiles, such patterns are called periodic. The reason the tiling problem is undecidable is that nonperiodic tilings also exist: patterns that cover the infinite floor but never repeat.

Back when we were discussing our first small result, we studied a 1971 simplification of Berger’s original proof made by Raphael M. Robinson of the University of California, Berkeley. Robinson constructed a set of 56 different tiles that, when used to tile the floor, produce an interlocking pattern of ever larger squares. This fractal pattern looks periodic, but in fact, it never quite repeats itself. We extensively discussed ways of using tiling results to prove the undecidability of quantum properties. But back then, we were not even thinking about the spectral gap. The idea lay dormant.

In April 2013 Toby paid a visit to Charlie Bennett at IBM’s Thomas J. Watson Research Center. Among Bennett’s many achievements before becoming one of the founding fathers of quantum information theory was his seminal 1970s work on Turing machines. We wanted to quiz him about some technical details of our proof to make sure we were not overlooking something. He said he had not thought about this stuff for 40 years, and it was high time a younger generation took over. (He then went on to very helpfully explain some subtle mathematical details of his 1970s work, which reassured us that our proof was okay.)

Bennett has an immense store of scientific knowledge. Because we had been talking about Turing machines and undecidability, he e-mailed copies of a couple of old papers on undecidability he thought might interest us. One of these was the same 1971 paper by Robinson that we had studied. Now the time was right for the ideas sowed in our earlier discussions to spring to life. Reading Robinson’s paper again, we realized it was exactly what we needed to prevent the spectral gap from vanishing.

Our initial idea had been to encode one copy of the Turing machine into the ground state. By carefully designing the interactions between the particles, we could make the ground state energy a bit higher if the Turing machine halted. The spectral gap—the energy jump to the first excited state—would then depend on whether the Turing machine halted or not. There was just one problem with this idea, and it was a big one. As the number of particles increased, the additional contribution to the ground state energy got closer and closer to zero, leading to a material that was always gapless.

Exams and Deadlines

One significant weakness remained in the result we had proved. We could not say anything about how big the energy gap was when the material was gapped. This uncertainty left our result open to the criticism that the gap could be so small that it might as well not exist. We needed to prove that the gap, when it existed, was actually large. The first solution we found arose when we considered materials in three dimensions instead of the planar materials we had been thinking about until then.

We now knew that getting a big spectral gap was possible. Could we also get it in two dimensions, or were three necessary? Remember the problem of tiling an infinite bathroom floor. What we needed to show was that for the Robinson tiling, if you got one tile wrong somewhere but the colors still matched everywhere else, then the pattern formed by the tiles would be disrupted only in a small region centered on that wrong tile. If we could show this “robustness” of the Robinson tiling, it would imply that there was no way of getting a small spectral gap by breaking the tiling only a tiny bit.

By the late summer of 2013, we felt we had all the ingredients for our proof to work. But there were still some big details to be resolved, such as proving that the tiling robustness could be merged with all the other proof ingredients to give the complete result. The Isaac Newton Institute for Mathematical Science in Cambridge, England, was hosting a special workshop on quantum information for the whole of the autumn semester of 2013. All three of us were invited to attend. It was the perfect opportunity to work together on finishing the project. But David was not able to stay in Cambridge for long. We were determined to complete the proof before he left.

In physics and mathematics, researchers make most results public for the first time by posting a draft paper to the arXiv.org preprint server before submitting it to a journal for peer review. Although we were now fairly confident the entire argument worked and the hardest part was behind us, our proof was not ready to be posted. There were many mathematical details to be filled in. We also wanted to rewrite and tidy up the paper (we hoped to reduce the page count in the process, although in this we would completely fail). Most important, although at least one of us had checked every part of the proof, no one had gone through it all from beginning to end.

After six weeks, we had checked, completed and improved every single line of the proof. It would take another six months to finish writing everything up. Finally, in February 2015, we uploaded the paper to arXiv.org.

What It All Means

Ultimately what do these 146 pages of complicated mathematics tell us?

First, and most important, they give a rigorous mathematical proof that one of the basic questions of quantum physics cannot be solved in general. Note that the “in general” here is critical. Even though the halting problem is undecidable in general, for particular inputs to a Turing machine, it is often still possible to say whether it will halt or not. For example, if the first instruction of the input is “halt,” the answer is pretty clear. The same goes if the first instruction tells the Turing machine to loop forever. Thus, although undecidability implies that the spectral gap problem cannot be solved for all materials, it is entirely possible to solve it for specific materials. In fact, condensed matter physics is littered with such examples. Nevertheless, our result proves rigorously that even a perfect, complete description of the microscopic interactions between a material’s particles is not always enough to deduce its macroscopic properties.

You may be asking yourself if this finding has any implications for “real physics.” After all, scientists can always try to measure the spectral gap in experiments. Imagine if we could engineer the quantum material from our mathematical proof and produce a piece of it in the lab. Its interactions are so extraordinarily complicated that this task is far, far beyond anything scientists are ever likely to be able to do. But if we could and then took a piece of it and tried to measure its spectral gap, the material could not simply throw up its hands and say, “I can’t tell you—it’s undecidable.” The experiment would have to measure something .

The answer to this apparent paradox lies in the fact that, strictly speaking, the terms “gapped” and “gapless” make mathematical sense only when the piece of material is infinitely large. Now, the 10 23 or so atoms contained in even a very small piece of material represent a very large number indeed. For normal materials, this is close enough to infinity to make no difference. But for the very strange material constructed in our proof, large is not equivalent to infinite. Perhaps with 10 23 atoms, the material appears in experiments to be gapless. Just to be sure, you take a sample of material twice the size and measure again. Still gapless. Then, late one night, your graduate student comes into the lab and adds just one extra atom. The next morning, when you measure it again, the material has become gapped! Our result proves that the size at which this transition may occur is incomputable (in the same Gödel-Turing sense that you are now familiar with). This story is completely hypothetical for now because we cannot engineer a material this complex. But it shows, backed by a rigorous mathematical proof, that scientists must take special care when extrapolating experimental results to infer the behavior of the same material at larger sizes.

After the work described here was complete, we went on to extend it to one-dimensional systems and phase diagrams. Other important problems in quantum physics have since been shown to be undecidable using techniques from computer science, with profound consequences for mathematics. In 1935 Einstein, Podolsky and Rosen realized that quantum mechanics predicted what they called “spooky action at distance”—correlations between pairs of entangled quantum particles that are not possible under classical physics. In a major breakthrough, a team led by Zhengfeng Ji of the University of Technology Sydney recently announced a result proving that estimating such correlations, even to within some reasonable precision, is undecidable in general. Their result disproves a deep conjecture in mathematics that had been open for more than 40 years.

And now we come back to the Yang-Mills problem—the question of whether the equations describing quarks and their interactions have a mass gap. Computer simulations indicate that the answer is yes, but our result suggests that determining for sure may be another matter. Could it be that the computer-simulation evidence for the Yang-Mills mass gap would vanish if we made the simulation just a tiny bit larger? Our result cannot say, but it does open the door to the intriguing possibility that the Yang-Mills problem, and other problems important to physicists, may be undecidable.

And what of that original small and not very significant result we were trying to prove all those years ago in a café in the Austrian Alps? Actually, we are still working on it.

This article was originally published with the title "The Un(solv)able Problem" in Scientific American 319, 4, 28-37 (October 2018)

doi:10.1038/scientificamerican1018-28

MORE TO EXPLORE

Undecidability and Nonperiodicity for Tilings of the Plane. Raphael M. Robinson in Inventiones Mathematicae , Vol. 12, No. 3, pages 177–209; September 1971.

Undecidability of the Spectral Gap. Toby S. Cubitt, David Pérez-García and Michael M. Wolf in Nature , Vol. 528, pages 207–211; December 10, 2015. Preprint available at https://arxiv.org/abs/1502.04573

ABOUT THE AUTHOR(S)

Toby S. Cubitt is a Royal Society University Research Fellow and reader in quantum information at University College London. After obtaining a Ph.D. in physics, postdoctoral positions in mathematics and a faculty position in computer science, he now works on quantum problems that straddle these areas.  Credit: Nick Higgins

David Pérez-García is a professor of mathematics at Complutense University of Madrid and a faculty member at the Institute of Mathematical Sciences in Madrid. He works on mathematical problems in quantum physics.  Credit: Nick Higgins

Michael M. Wolf is a professor of mathematical physics in the department of mathematics at the Technical University of Munich. His research focuses on the mathematical and conceptual foundations of quantum theory.  Credit: Nick Higgins

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Solving seemingly unsolvable problems

Insidewestminsterblog - february 21, 2022.

As students, we solve problems every day – whether it be an issue related to our studies, jobs, personal life, health, society or environment. Although we work towards our aspirations and set new goals all the time, some things just do not seem to get any better, no matter the effort. However, it is crucial not to ignore these seemingly unsolvable problems, as they will most likely come back to us in one form or another when left unchecked. Let me give an example of such a problem that is very simple and seemingly insignificant but familiar to many people living in today’s world.

“Let us say that Mary is trying to lose 15 kg of her body weight. She is currently weighing 80 kg but desires to reach 65 kg. Mary is in good health overall, fairly active and not excessively overweight. Despite the absence of an objective need to lose weight, she has been trying to achieve the weight loss goal for a decade now. She has tried strict and not so strict diets, fasting, daily exercise programs and whatnot, but without any long-term success. There have been short periods when she has been very close to achieving her aim, but the results have never lasted long. Now, Mary does not talk about her desire to lose 15 kg anymore, but deep inside, the idea of wanting to weigh 65 kg does not fade away. Therefore, she is still on the lookout for new diets and tries to avoid eating with others to save herself the trouble of having to justify refusing food.”

To help Mary, most of us would probably encourage her to accept her body weight as it is, not let weight define her worth and eat with others to maintain relationships. It seems like a good way to solve the problem, right? Unfortunately, it is unlikely that she will be able to follow this advice and make the weight problem disappear. Even if she can fully accept her weight of 80 kg by following a well-refined professional strategy, she might not feel fulfilled. Mary might start feeling scarcity in another area of life, such as her finances or relationships, and start blaming her unhappiness on perceived problems in this new area. Therefore, Mary’s dissatisfaction seems impossible to solve.

Unfortunately, it is impossible to solve such a problem when some not so obvious but essential steps have been skipped – just as impossible as building a house but forgetting about the foundation. The house’s walls can be constructed exquisitely, but without a solid foundation, the house will not stand still, no matter the effort. Therefore, in the example above, Mary’s weight was not the problem to be solved at this stage – the weight was just like “a part of the wall that had fallen a bit out of place due to missing pieces in the foundation”. As hard as it is to take, putting a great deal of effort into solving the symptoms – such as her weight – will probably never succeed at solving her problems. Therefore, as the first step towards successfully solving her situation, Mary needs to identify the missing parts in her “foundation”, which can also be called the root cause(s) of her issues.

With this story, I encourage you to think about whether you have something in your life right now that seems impossible to solve. If so, can you identify whether this thing is a root problem or a symptom of something bigger, or in other words, “a missing part of the foundation” or just “a damaged part of the walls”? To increase the chance of long-term success in both personal and professional life, step away from symptoms and start tackling root problems as early as possible!

Grete Kurik, BSc (Hons) Human Nutrition

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It's Not About the Shark opens the door to the groundbreaking science of solutions by turning problems―and how we solve them―upside down. When we have a problem, most of us zero in, take it apart, and focus until we have it solved. David Niven shows us that focusing on the problem is exactly the wrong way to find an answer. Putting problems at the center of our thoughts shuts down our creative abilities, depletes stamina, and feeds insecurities. It's Not About the Shark shows us how to transform our daily lives, our work lives, and our family lives with a simple, but rock-solid principle: If you start by thinking about your problems, you'll never make it to a solution. If you start by thinking about a solution, you'll never worry about your problems again.

Through real-life examples and psychology research, David Niven shows us why:

*Focusing on the problem first makes us 17 times less likely to find an answer *Being afraid of a problem is natural: we're biologically primed to be afraid *Finding a problem creates power – which keeps you from finding a solution *Working harder actually hides answers *Absolute confidence makes you less likely to find the answer *Looking away from a problem helps to see a solution *Listening only to yourself is one of the best ways to find an answer

Combining hard facts, good sense, and a strong dose of encouragement, David Niven provides fresh and positive ways to think about problem solving.

• Print length 240 pages
• Language English
• Publisher St. Martin's Press
• Publication date November 4, 2014
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“David Niven has done it again - a short, readable, entertaining, enlightening, and inspiring book for people who have problems. Don't we all? I got some great insights - along with a few laughs - while reading about ways to deal with bumps in the road of life. You will too.” ―Hal Urban, author of Life's Greatest Lessons

“David Niven will help you think more, panic less, dream bigger, and WIN.” ―Bob Danzig, former CEO, Hearst Newspapers

“You can bury your white flag now, because after you read It's Not About the Shark you'll never surrender to a problem again.” ―Stephen Pollan, author of Fire Your Boss

“ It's Not About the Shark illustrates the power of thinking unconventionally about approaches to many common barriers to success. It shows that alternative methods are often much more powerful solutions than conventional wisdom. Written in an engaging manner, it made me want to read the next chapter after chapter. Wow.” ― Gordon Bethune, former CEO and chairman, Continental Airlines

“In this useful tome, Niven gives unusual, yet eminently practical, problem-solving advice . . . This fresh, enthusiastic approach to problem-solving will encourage readers to open themselves up to opportunity and make for a valuable addition to anyone's self-help shelf.” ― Publishers Weekly

“In It's Not About the Shark, David Niven aims to shake up our stagnant notion of how to address problems . . . Niven's book explores what it means to think outside the box, how we look at the problems we face, how we see ourselves and how to be comfortable with the ambiguity difficult challenges can present. He doesn't provide specific solutions here but instead offers a shining toolkit for more adaptive thinking.” ― Shelf Awareness

“Good anecdotes with interesting data to back up his theory that when we focus on a problem, we get trapped within it and are less able to solve it.” ― Washington Post

About the Author

DAVID NIVEN, PhD, is known internationally for translating powerful research findings into practical advice anyone can apply to their daily lives. David's The 100 Simple Secrets of Happy People – and seven other titles in the series – have sold more than 1 million copies in the U.S., and has been translated in 30 languages around the globe. His work has been featured in USA TODAY, US Weekly, The Washington Post, Reader's Digest, Redbook, Cosmopolitan , Glamour , and Health.

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• Publisher ‏ : ‎ St. Martin's Press (November 4, 2014)
• Language ‏ : ‎ English
• Hardcover ‏ : ‎ 240 pages
• ISBN-10 ‏ : ‎ 1250042038
• ISBN-13 ‏ : ‎ 978-1250042033
• Item Weight ‏ : ‎ 12.5 ounces
• Dimensions ‏ : ‎ 5.64 x 0.87 x 8.46 inches
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About the author

David niven.

David was in the basement of the library at Ohio State University looking across mountains of research reports that would never reach the people who most needed them. There was research on life satisfaction, relationships, prosperity, parenting, all manner of good things. But no regular person was ever going to come to the basement to read about how they could make their lives better. And if they did, they would find the reports filled with stuffy jargon produced by academics writing to impress other academics. Later that day, David launched a project to bring that research to people who could use it. Today, David is known internationally for translating powerful research findings into practical advice anyone can apply to their daily lives. Published in more than two dozen languages, David’s books show that a more satisfying life can be had with small, sustainable changes in our actions and attitudes.

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