How to solve any problem?

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For more texts on memory, learning, sleep, creativity, and problem solving, see SuperMemo Guru

Problem solving

You probably read this text because you have already doubted your problem solving powers at some point. This is the text to help you dispel the doubt. However, if you are a prolific problem solver, I still hope to inspire, e.g. with the mental computation model, or details of the natural creativity cycle (incl. the computational role of sleep). Naturally, there are problems that are not solvable or seem insolvable, but even if you plan to work on teleportation or immortality, you can hopefully benefit from this text.

This article is NOT about fast problem solving. Fast problem solving is based on expertise and speed. It can often be reduced to quick choices among available options. This article is about tough problems that are based on vast and complex knowledge that may be full of falsehoods and contradictions. Slow problem solving may span decades. It can be used, for example, to find major answers in science.

Why we struggle with problem solving?

It is popularly believed that problem solving is primarily about the brain power or IQ. It is not. In particular, in healthy people, it is rarely about the inborn brain power. Our usual endowment is good enough for major inventions and scientific breakthroughs. Failure to solve problems usually comes from three primary factors: distraction, energy and skillset.

Distraction: problem solving proceeds best with a round-a-clock investment in the process. This is not just about time invested. It is about the nature of mental computation. This is what a vast majority of population can't do or can't afford to do. In modern world, most people are reduced to solving minor problems they encounter on their way, or problems that seem major but are otherwise simple to solve in terms of complexity.

Energy: bad sleep, high levels of stress, and often imperfect health are highly efficient nullifiers of major problem solving efforts. For those who can eliminate distraction, and can close themselves in a well-isolated peaceful place, low mental energy may be the main bottleneck. Paradoxically, a degree of stress keeps most of people on the run despite sleep deprivation. However, they are on the run while employing a brain that is not best primed for problem solving. When they are about to slow down to think, they are more likely to think about sleep.

Skillset: After decades of schooling, a great proportion of people miss on the practice of problem solving. Many years of being told what to do has an awful side effect: when you are the boss of your own mind, you suddenly feel helpless and lost. Schooling also results in knowledge that is missing on coherence and abstraction. You may remember the equations, but you never ponder in which context they might be deployed. You do not need extensive knowledge to be a fantastic problem solver. What you need is a knowledge that lives in positive feedback with the learn drive. It is not about books, reading, or decades of investment. Teens can solve major problems facing mankind. It is more about "hungry knowledge", i.e. the type of knowledge that keeps asking for more.

After many years of distraction, low mental energy and limited learning, many people turn into experts in solving minor problems in their lives. This is a form of brain neglect that may become irreversible at some point. This explains why Nobel-level achievement is rare. This also explains why Nobel-level achievement is not reserved to a limited group of people. It is within reach.

It helps to take inspiration from people like Eric Kandel. Kandel spent most of his waking time thinking about a little snail Aplysia. He did not mind travelling to France or working in the field. Each time he found an obstacle, he would look for people who could help him jump over. After decades of effort, neuron after neuron, molecule after molecule, he build a perfect model of a little nervous system of the slug. With so much problem solving in the field, Kandel can take any problem in neuroscience and make a good guess after just a few minutes of reading. He turned into a human generalizing expert system. In 2000, he received a deserved Nobel Prize for his work.

Great achievement is less about the brain power. It is more about giving the brain a chance

The tools

All sane individuals know how to solve problems. The greatest problem solvers have an excellent toolset without ever formalizing it. However, a simple formal model of problem solving translated into a formula expressed in a simple language can make a world of difference for everyone. In my formula below, I will introduce a couple of concepts that many problem solvers sense intuitively, but do not employ extensively or systematically for these concepts are not defined clearly enough in their mind. Here are the most important problem solving tools I would like you to carry away from this text:

  • mental computation, i.e. simple language explanation how the brain goes about solving problems (no anatomy or neural network knowledge needed)
  • protected mental computation, i.e. how to give the brain perfect peace time for thinking
  • disruptive learning, i.e. how to disrupt brain's peace for the sake of a creative breakthrough
  • brainstorming and incremental brainstorming, i.e. how to employ other people's brains in the job
  • creative forgetting, i.e. how to forget old wrong solution pathways to find the right one
  • natural creativity cycle, i.e. how to solve problems in sleep without lifting a finger

How does mental computation work?

In problem solving, it is helpful to understand the concept of mental computation. There are many models of computation, e.g. a classic and highly inspirational model is that of a Turing Machine. Here I propose a simpler model that is more illustrative. It requires no formal language of mathematics. It will help me explain how the brain works in problem solving and why the discussed techniques work.

Concept: when you think of someone, a concept will light up in your head. By concept I will mean a set of brain cells that get activated upon recognition of a specific object or idea. When you think about a forest, a forest concept in your brain will get activated.

Concept link: two concepts can be connected by a link. The link is an association between concepts. In the brain, it is a simple connection between concept networks or cells. The concept of a forest is linked with a concept of the tree because forests are made of trees. The concept of a tree is linked with a concept of a branch. When you think of a tree, you increase the probability of thinking about the forest because the two form a concept link.

Concept map: a set of concepts connected by links forms a concept map. When you think about having nice time in a forest, your brain will light up a concept map that will include concepts like: forest, self (i.e. you), pleasure, and perhaps other associated concept like: blue sky, warmth, butterflies, bird song. A concept map represents the current status of your thoughts. Good time in forest may feel complex and complete, full of details and experiences. However, in reality, it is just an amazingly small number of concept maps lighting up in the brain. Our minds convert map simplicity into a perceived complexity of experience.

The brain is a parallel machine, i.e. it can do many things at the same time. It can activate many concept maps at the same time. However, in the thought process, parallel activations are focused into a single window of attention. While thinking, you follow a specific reasoning pathway that rarely branches out to start two or more parallel active computations. In general, it is not possible to seed your brain with a couple of problems to solve and work on them consciously in parallel. It is possible to switch, to multitask, but this is not true parallel computing. The reason for that approach may in part be energetic. Thinking is pretty costly in terms of energy use. However, a great deal of parallel processing happens during sleep, and energy is not the only factor. The second reason is that the brain evolved as a central command organ. Its purpose is not to solve dozens of independent problems based on mental computation. Its main purpose is to decide the next step for the animal to make. For that sake, it collects available data from the senses, integrates it, and executes a plan. For the brain, the main goal is to find the next step in the currently executed plan. Parallel processes will be autonomous and/or subservient. Parallel processing may convert visual signals to a concept map representing an apple, its location, and physical availability. This translates to a concept "food within reach", which helps execute the next step for a hungry animal: reaching for food. The same animal might have had some reproductive plan on its mind, but the brain will always go for a specific executive action. The reproductive plan and the corresponding concept map may remain in the state of high activation, but it won't be part of conscious processing, unless on some multitasking principle. In all likelihood, well activated map will make the animal return to the reproductive plan once the apple has been reached.

Active map is any concept map in the state of higher activation. It is sending signals to other portions of the brain. There may be dozens or hundreds of maps active at the same time. It is hard to say. However, very few of those maps will affect conscious experience, and only one map will play a truly central role: the focus map.

Focus map is an active concept map hat has a direct access to the chief executive. That map represents what you are currently thinking about.

Brain state is a set of active concept maps with one map in focus.

Brain state transition is a change of the focus map, or the activation of new maps based on spreading of activation with gradual fading of all activations left behind.

Mental computation (i.e. thinking) is the sequence of brain state transitions. Hopefully, that sequence leads to a solution.

Breakthrough association is a new link established between concept maps. This link is an equivalent of a piece of knowledge that determines the solution.

Computation in the brain is the transition between active maps with one map in the focus at any time. Can we choose and decide how to run the computation? I do not think so! Our conscious planning of the computation is just an illusion. In reality, the computation is the expression of maps we wake up with, our general goals that are represented by maps that easily light up, and sub-goals that might be even easier to activate. On waking, great morning idea may pop up first in the brain, then, unpredictably, most excitable active maps will light up. This first morning map may have a form of "my usual morning routine", "my usual basic goal(s)", etc. Then there are things that come up because of news on TV, or messages on the phone, or aches and pains in the body, or worries, which have a form of easily activated maps that can invade your thoughts at any minute.

The thinking process is nothing else than a computation. We move between different brain states.

Let's call the concept map associated with the problem to solve, a problem map. First thing we can do for problem solving is to keep the problem map active. If this is the first map that wakes up with you, you are already on a good path to a solution! If you need a reminder, you will not do that well. All your daily activities, all knowledge, all inspirations, must be in close connection with your map. The problem map needs to be like a predator ready to snatch any inspiration and mold itself upon the influence. If you see Trump in the news, you see him as a component of your problem solving. If this is a health problem, you might wonder if Trump is suffering too. If this is a technical problem, you might wander what Trump arsenal or people around him could do if they faced a similar issue.

The more processing time you provide, and the more inspiration contexts you feed to your problem map, the greater the chance a given day will be notched as an incremental step towards the solution. When you go to sleep, your brain is actually going to work. Your brain will consume what your produce during the day. It is helpful to go to sleep in peace, and without influence of medication, alcohol, and the like.

Mental computation is a simple chain of connected concepts streaming through your mind. Even the most complex problems can be reduced to simple links between simple ideas

Protected mental computation

Protected computation is the thinking process in which you activate concept maps of your choice and nothing in the world has the right to disrupt your focus. If you live and sleep with a specific problem, its central problem map will come to your mind in the morning and will keep coming up during the day. It is impossible to keep a perfect purity of mind in which you only focus on solving the problem. You need to eat. You need to use the bathroom. It is always recommended to exercise. Those activities will throw in some impurities in the protected computation process. However, if you limit the impurities to things that are easy to ignore or be solved on autopilot (e.g. coffee, table, mouse, spoon, etc.), you will be able to keep your mind focused like a razor. If you happen to work in a busy lab, or a busy corporation, or a cubicle, launch a protest. If you are to solve problems, you cannot be subject to disruption by noise, phone or e-mail. Those form the antithesis of protected computation. When Alan Turing got to solving the Enigma problem, he insisted he wanted to work solo. Luckily, he was indulged. Nicholas Carr is right: if you let the net invade your day, "internet will make you stupid".

Newton had his best time in plague days. Darwin had his best time on the Beagle. Einstein did great in his peaceful patent office. All great breakthroughs came from a moment of deep contemplation. For 20 years now I have worked in a locked room with a bed away from the office. When in protected mental computation mode, I am unavailable to the world. I reserve the right of absolute physical and mental isolation. This does not mean cutting yourself from the world. Only toughest problems are best solved on creative vacation (i.e. away from everyone). Protected mode requires just a few hours. It is your own processor that will overheat and terminate that period. Social interaction may then come as major relief.

Major portion of your time spent on problem solving should be spent in protected mental computation mode where all distractions should be considered mental pests

Creativity

Creativity model

I presented "rat maze" model of creativity here and it seems to have helped some readers understand the process. Here I will use our mental computation model. Creativity is largely about the brain state. It is not a computational brain state as discussed in the model of mental computation above, but a neurohormonal state. In a highly creative state, individual concept maps show a reduced activation threshold. This means that a random map may pop up in your mind at any time. That random map may associate with the focus map forming a link that may turn out to be your creative breakthrough. That random map may also become a new focus map and form association with all recently active maps. Randomness is actually an illusion. All forms of mental impurities will cause maps to pop up in the creative mode. This is why isolation is so important. Even a minor change to an object in your environment may attract attention and reshuffle map activations in the brain. This may be welcome later at the disruptive stage. Early in the process, you need your creativity to revolve around the problem map.

High creativity is based on an easy and unexpected activation of concept maps in mental computation

In natural creativity cycle, the brain gradually moves from the state of high creativity to the state of low creativity. This has nothing to do with physiological value of such creative decline for survival. It is a bug, not a feature. This decline is probably a sheer expression of the fact that creativity and mental computation are costly in terms of energy and in terms of neural network capacity. To prevent catastrophic interference, the brain uses load and store approach: first it loads memories in short-term memory, then it unloads them to the long-term storage. In the process, it slowly approaches its maximum short-term capacity. This triggers fatigue and sleep. The cycle begins on the next day or after a siesta.

The timeline of the process indicates that we have evolved for high creativity mode, not for high focus. The latter is simply a side effect of fatigue. If we push kids to school benches early in the morning, we deny them what is natural: high creativity. That morning creativity, should be combined with motor activity, socializing, self-learning, etc. School could actually come at the time when kids are just about to slow down. School could bore them to early bedtime, and perhaps might capture some attention for actual learning.

High creativity means higher excitability of individual concept maps. High excitability may be disruptive. This is why creative kids are labelled ADHD. If new concept maps keep popping up for seemingly unrelated reasons, the kids seem unable to focus. The attention gets hijacked by ideas that might be born spontaneously. However, in problem solving, we should welcome the chaos of creativity. We should let the brain do the computation at will.

Creativity and attention stand in contradiction. In the waking state, we gradually lose creative power, but gain on the ability to focus

Creativity, madness, and drugs

Creativity may be cranked up to the point of madness. This is why creative genius and mental disease are not far apart. Creativity may be boosted artificially with drugs, of which coffee is my favorite. Different types of drugs can have different impact on individual brain centers depending on the types of receptors used and even depending on the individual. We are all unique in terms of actual receptor type use and their distributions. LSD has been famously touted as a creativity drug. However, for problems solving, we need creativity that can be harnessed in the problem solving process. Randomization of thought cannot go as far as to make you lose your waking coherence. In healthy individuals, a cup of coffee is probably just enough to go high enough and still stay within limits of sanity.

Even alcohol can, in tiny doses, boost some forms of creativity by disinhibiting portions of the brain while suppressing other portions of the brain. If you look at young people socializing in a bar, you can see that after a beer, their socializing becomes more creative. They perk up, chat more, get excited, and social disinhibition makes them forget that red pimple at the top of their nose. That's good for creativity. Naturally, if there is more beer involved, creativity gets quickly suppressed, and creative socializing turns into a slur. Alcohol is actually a depressant and can only form a temporary boost by its differentiating impact on different portions of the brain.

Absent-minded professor

Unusual creativity and focus abilities lead to a term "absent minded professor". An absent minded professor is not as absent minded as the name implies. He would not be a professor. His brain is simply able to intensely focus attention on the chain of ideas that come from creative activity in a number of twinkling concept maps. Professor seems absent because he lives in the world of his ideas. A shot of a gun is bound to instantly set a specific focus map in the brain. This would quickly demonstrate that professor's mind is not really absent.

An absent-minded professors is able to employ protected computation in extreme conditions, e.g. on a busy street. He can do it through sheer will and curiosity! However, for a hard problem that does not budge, don't try to be a professor. Free yourself from distractions! That's the easy way to get the same result.

The ability to hyperfocus is trainable, but why would you ram through an open door? In an ideal case, all you need to train is your ability to avoid distraction by petty things. In perfect isolation on an uninhabited island, your brain would probably quickly run out from petty things to think about. However, if you live in the modern society, the battle for your attention is relentless. If you are not in the driver seat, others will drive your life and you may as well forget about efficient problem solving.

Extending human memory

For complex problems, you need to extend your memory with notes. These can be paper notes or computer notes. Great problems solvers of human history are well-known for their own elaborate systems of note taking, organizing notes, and writing. Writing can be considered a hyperfocusing tool. To write a text, you have to follow a specific line of thought. You will always diverge creatively, however, you can always come back to your last sentence written. This leaves a permanent record of your focus map. While writing, you never get lost in a forest of new ideas. The low speed of writing has its advantages. It helps you organize your thoughts. Naturally, nothing beats the speed of thought. Solutions to problems probably rarely come first while writing. It is the random association that matters, while writing helps you consume and consolidate hazy ideas that you generate in the creative process. Note taking is also essential for long-term perpetuation of thought. If the problem solving extends over weeks, it can easily turn into a self-sustaining process in equilibrium: you forget as much as you create. In the long-run, the process can be convergent only with a permanent record that expands the limits of human memory. Obviously, there is also another remedy: cut down on the level of complexity. This article is not about fast problem solving though. Cutting down on complexity would take away the fun and joy of coming up with a solution to a problem nobody else have solved thus far. More importantly, the value of the solution is often related to its complexity (see: Problem valuation).

You can use mindmapping, conceptualizing, or note-taking software. However, I don't think anything can compete with incremental writing. In incremental writing, you combine note taking with creative elaboration of ideas. The essential component of incremental writing is the consolidation of models over time. This requires incremental approach and spaced repetition. In addition, you can employ strictly targeted creativity in a problem solving process that will involve only your own notes or literature dedicated to the problem.

Incremental writing involves: (1) writing, (2) review, (3) learning, and (4) creative elaboration. All those processes are derived from incremental reading, however, the focus shifts freely from learning to creativity to problem solving.

Incremental writing makes it easy to control the portion of the "big picture" that lives in your mind as opposed to the portion that lives in your database. This is important for complex problems that involve a great deal of knowledge. The solution will always happen in the mind, but the size of the knowledge structure is limited by available time. Long-term memory consolidation and building coherence take time. The bigger the structure, the more time is needed. The model of the problem will gradually emerge by shifting knowledge resources from your database to simple, well-structured, well-polished, and durable knowledge in the brain.

Employ incremental writing to organize mental computation and extend it beyond the limits of human memory. Incremental writing helps you creatively elaborate on your ideas and keep track of all your thoughts

Natural creativity cycle

Human brain is subject to the influences of the circadian cycle. This implies that its operations can be optimized to match the creativity cycle. For problem solving, in addition to good sleep, the most critical part is the morning. This is the time when an average individual can gain most in problem solving by employing judicial strategies.

Waking up

The active problem solving process can begin even before the point of waking up! While dozens of concept maps might be still active in your brain following the last dream in your last REM-rich stage of sleep, your attention is still not there. You may have some vestigial capacity to introduce a bit of thinking at that "waking" stage. Your brain is in the process of setting up the focus map that begins your conscious thinking process. Ideally, the waking moment is like a light switch in which you move fast from sleep to a fully conscious mind. In reality though, this may take a while. The brain can actually capture and lose focus, while still remaining in an unstable state between sleep and waking. You can then use those first glimpses of clarity to capture concept maps still active in the wake of nighttime memory optimization. The moment when your focus map lights up clearly with waking might be the most likely time for a breakthrough idea. There is also a period of increased creativity following that moment. When great discoverers speak of inventing things in sleep or upon waking, they rarely make it specific, and possibly they do not even know if the lightbulb turned on with the first captured focus map, or perhaps only in the minutes that follow. In the end, it won't matter as long as you consciously protect that period from external disruption.

Morning creativity is the time of best ideas and this is the time when modern lifestyle does most of damage. Morning is the time of maximum protection. If you begin the day by turning on the news, you already miss on your problem solving capacity. This changes the constellation of active concept maps in your brain and takes your mental computation away from the problem map.

It is less obvious that capturing nighttime inspirations in the very early morning is the key to a major breakthrough. Morning routine and all problem solving that follows, should all be run in protected mental mode away from all distractions.

Morning routine

Morning routine is extremely important. The reason for this is that most major breakthroughs happens through re-assembling a skeletal framework of an idea built through knowledge optimization in sleep, esp. its terminal REM-rich stages.

If you follow biographies of great inventors and discoverers, you will often find the power of the daily routine. You will see how those great people adjust minor details of their daily plan to maximize their output. They may change a tiny detail in the routine, and then see how it impacted their productivity (e.g. as measured in pages of a novel in the process of writing). Those themes show up again and again: long walks in nature, siesta, specific creative hours, good sleep, writing letters, healthy lifestyle, etc.

Morning routine provides the best time for coming up with the outline of a solution. Its details should be polished for maximum effect. You should develop your own routine by months and years of self-experimentation. If walking a dog happens to be helpful, you will probably notice. After a month on a highly repetitive routine, you will be well armed to observe the effect of minor adjustments. After decades, it may even become easier.

I will tell you my own routine for the record, but this is not a universal panacea. Your best routine will likely differ. On waking, I avoid all forms of distraction: conversation, TV, etc. I don't have a phone and I work from home, so it is really easy to keep a focused mind. For breakfast, I just slowly drink coffee with a tiny bit of milk, and keep pacing around while ruminating the problem to solve. On a good day, I jot down ideas, and hopefully, never have time to consume them all. The best ones will add up to a solution. Sooner or later. That's it. My morning is mostly about me and my brain.

Your routine will differ. Research says that not eating breakfast is not a good idea. However, it is not a good idea on average. If you are younger, you may need more energy. If you are on a weight loss diet, your optimization will differ. If you get hungry after just 1 hour, you obviously need to power your brain with a bit of glucose.

That walking part may also be highly individual. Perhaps you prefer to walk or bike to the office? Or walk the dog? Many high-octane executives start their day with vigorous exercise. This helps them hit the ground running. However, this might be more about energizing the brain, e.g. due to early waking, than about using its own natural creative energy for problem solving. Circadian cycle seems to point to pre-siesta time as better suited for serious sports.

Whatever you choose, protect your morning mental computation with good sleep and minimum distraction. If you need 1-2 hours for commuting or delivering kids to daycare, your best time of the day may be gone. If you do not use up too much mental or physical energy in that time, there is still some hope. The window of solid alertness and creativity may cover the first 2-4 hours of the day.

Morning routine protects the best 2-4 hours for creativity, and problem solving

Fatigue

Mental fatigue is a bit like computer overheating. All networks have their workload capacity and maximum sustainable activity period. Problem solving is costly in terms of energy. The harder the problem the faster the fatigue will set in.

If the fatigue sets in in the first 2-3 hours, it could be a local overload. Changing the type of activity may be helpful, e.g. from reading to passive listening, or walking and thinking, or a conversation, or just lying with closed eyes. If the fatigue sets in in the 5-6th hour, this could be a brain's signal it is time for a siesta. Those who do not take siesta, can hope for a circadian upshot in brain powers in the evening, however, they will never recapture that morning problem solving power. The higher the creative disruption of the morning, the faster the network fatigue. This is why I am a proponent of siesta for all people involved in problem solving or in creative professions. If you want your problems solved fast, take a nap in the 7-8th hour of the day! 10 minute nap will make a world of difference. 2 hour nap will still pay back handsomely. For longer naps, check your timing and night sleep cycle, long naps may indicate sleep deprivation or circadian phase misalignment. See: Good sleep, good learning, good life

In a healthy lifestyle, there is only one truly effective way to combat mental fatigue: sleep!!!

Evening routine

In the same way as we protect the mental computation in the morning, we should aim at protecting nighttime computation in the evening. In the morning, we avoid disruption to maximize focus on the problem to solve. We also protect the associated rampant creativity. In the evening, the brain is slowing down, and it is not as useful as in the morning. However, it still needs special care to prepare it for sleep. Protecting the last 1-2 hours before sleep is tantamount to protecting the computation that will occur in sleep. Low light is essential to prevent the modern scourge of phase shift and insomnia (see: DSPS). More details on memory optimization in sleep can be found in the chapter on natural creativity cycle. There are also some neural details of the process explained in good sleep. For a problem solver, the model of sleep computation isn't as essential as the waking computation. The computations is largely beyond our control. Sleep proceeds on its own naturally and minimum intervention is actually a good thing.

My favorite morning routine involves walking and so does my evening routine. While walking in the morning helps harness creative activations, evening walking is more about just cooling down and getting a bit of healthy motor fatigue before sleep.

Instead of coffee, which stimulates creativity, in the evening I go for buttermilk. The choice is just a matter of taste. I do not have much physiological backup to explain its particular value. It actually reminds me a bit of a smoker's habit: I like to sip liquids while walking. End of rehydration signals the brain the end of the slot. It is not a conscious choice. It is just a habit.

As morning routine is about a boost to creativity, evening routine is about slowing down. This is why I like to listen to a variety of lectures at that time. It is the exact opposite of what school tries to do. While kids go to lectures to "learn", which actually means "get bored", I choose interesting lectures for the time before sleep to cool down and slow down. While kids get bored in their prime time, and supposedly achieve some learning, I learn a thing or two, while supposedly getting myself bored to sleep.

Despite sub-prime brain, the protected evening routine is never wasted time. Instead of listening to boring curriculum, I chose lectures that best match my current problem to solve. I choose lectures that can provide additional inspiration. The main difference with morning creativity is that, at brain's low point, most of inspiration comes from others, not from within. Creativity is slowly going to sleep, so the brain can focus better. Best ideas I jot down on a piece of paper to consume in the morning.

In the last minutes before sleep, I turn to boring things (e.g. new antics from our beloved politicians). I actually fall asleep with the news on the radio with a specific intent of not thinking about problems to solve. This is my last effort to protect nighttime computation from disruption.

Evening routine should be all about protecting sleep, however, it can also be used for some slow untiring focused work, and passive learning

Creative disruption

Getting stuck in a local minimum

The thinking process will be an iteration between neighboring brain states. Neighboring brain states are those that come from activating neighboring concepts connected to the focus map. New concept maps will get activated and the focus will keep shifting between concepts maps or concepts in a hard to predict waves. Depending on the complexity of the problem, those iterations can easily get repetitive. The brain keeps returning to the same state as if getting stuck in a local minimum. This is the place when disruption is needed.

All forms of optimization can be visualized as walking in the mountains.

Mental computation is like walking the mountains and looking for water. If you always try to walk downhill in search of a stream, you can easily hit a plateau without a stream. That will be your local minimum. Your local lowest valley may carry no water. If there is no stream, the algorithm will send you over the nearest hill. To find the true lowest valley with the stream, sometimes it makes sense to walk uphill in some random direction, over several hills, to check if there is no valley that is even lower. If we discover we keep going in circles, we know we need a major disruption. It is like shooting oneself in a rocket of many hills to an entirely new unpredictable point. That point may be closer or further from the solution. We start again. Usually we use the catapult only in conditions of crisis. If crises repeat, we use a greater catapult distance

In problem solving, the equivalent of the lowest valley is the solution. All those moments of getting stuck in a repetitive state without a solution require walking uphill. This is when creative disruption needs to come in. Creative disruption is any method that will activate a concept map that has not yet been in focus of your active pursuits.

Disruption can have two forms: (1) new knowledge, and/or (2) new activations. Both come best from two sources: (1) learning and/or (2) communication.

Disruption from learning may have a form of reading, watching an interview or lecture, or conversation with a colleague. However, the most powerful disruption that I know can come from incremental reading or neural creativity.

Disruption via communication may come in a conversation, in a phone call, or via e-mail. My favorite forms of disruption via communication are: brainstorming and incremental e-mail communication.

When a new piece of knowledge arrives in incremental reading, it adds to the semantic network of knowledge stores in the brain. It provides new fresh pathways for brain state transitions in computation. New knowledge may be useless, it may be misleading, or it may lead to a breakthrough. Knowledge valuation network whose valuations are highly dependent on the current brain state will instantly tag new pieces of knowledge with their value in the light of the currently active problems to solve. Those highly valued titbits of knowledge will form highly active centers that can powerfully disrupt the trajectory of brain state transitions. The computation changes on arrival of new knowledge. We want new knowledge to be highly relevant or to be highly disruptive. Relevant knowledge brings us closer to a solution. Disruptive knowledge takes us away from the local minimum if there are no solutions in sight.

When you get stuck in problem solving, you can employ disruption in the form of learning. If it does not help, you can leave the protected mental mode and communicate with others or engage in brainstorming

Disruptive learning

If you keep cycling between the same repetitive brain states, and the same set of concept maps gets activated in varied constellations, you need to introduce creative disruption. New learning is the most powerful disruptor. In addiction to communicating with other brains, the best learning tools capable of disruption are incremental reading and neural creativity.

In incremental reading, you build a database of texts with knowledge that you want to master. You process those texts incrementally, generate extracts (i.e. the most important fragments that require special attention) and clozes (i.e. questions about things you must remember for months or years). The main value of incremental reading in the creative process is its high degree of unpredictability. If you get down to reading, you never know which text you will begin with. You only know that high priority texts are most likely to show up first. You can control the priority, or the semantic content of what you read, however, on a daily basis, you are likely to just submit to SuperMemo and trust its choices based on your own priority criteria.

If you are in a creative rut, incremental reading will instantly jerk your mind in different directions. If you work on an engineering problem, you may be disrupted by an article about health. The new article may carry knowledge relevant to your problem, or it just might be helpful by changing your brain states, concept map activations, and interference that can spark a bit of helpful forgetting. If it does not sound as much for you, you really need to give it a try. The power and efficiency of this process is astounding.

Neural creativity is a form of incremental reading where the reading sequence is governed by semantic connections between articles or individual pieces of knowledge. Neural creativity is no more complex than incremental reading, however, for it to work, you need a rich, extensive, and highly processed knowledge database. You may need a few years of incremental reading to make the best of neural creativity. In short, neural creativity works like your accessory brain. Your accessory brain needs to "go to school" first. You teach your "accessory brain" through incremental reading.

For the creative process, neural creativity has the advantage of higher relevance. It is thus more likely to be an inspiration, and it can produce more interference. For those reasons, it is neural creativity that should rather be used first, while incremental reading can be relegated to those moments when you need a bigger disruption.

If incremental reading is not enough, you can randomize the process by ignoring text priorities. This is a bit like random jump in Wikipedia, except every jump will leave permanent traces in your memory and in your "accessory brain" in SuperMemo.

If you get stuck while solving a problem, you first disruptive step is disruptive learning

Brainstorming

Incremental brainstorming with the use of e-mail isn't much different from incremental reading. The main difference is that the articles you read are a result of your direct request for input from somebody else's brain. Those articles are short and often to the point. They can be messy, but they are rich in inspiration. If you have good brainstorming partners, they are a perfect complement to Google. Human brains are in many ways vastly superior to Google. They carry what Google misses: associativity. Your friends can instantly put you on the right track. Google is more like a servant. It will do exactly what you want, but it won't be that influential in taking you to a new track.

Incremental brainstorming isn't much different from face-to-face brainstorming. It just extends over longer time scales and makes it possible to employ some advantages of incremental reading: memory consolidation, forgetting, generalizations, spacing effect, creativity, etc.

A simple conclusion from the above metaphors is that incremental reading is like brainstorming with yourself and with passive contribution from all brains that contributed the materials to the web. The is a new meta-level of human cognition that is likely to bring astronomical acceleration in solving problems in science (esp. on theoretical grounds).

Walk-talking combines walking with brainstorming. It is one of the best ways of brainstorming I know. The value of walking is that it is one of the most natural and autonomous motor programs in human behavioral repertoire. There is a vital contribution of the motor system in organizing a brainstorming sessions. Creativity is like REM-sleep, it generates a great deal of random signaling in the brain. This is why kids do a lot of fidgeting when their brain wanders away from the subject they should focus on in the class. This can be mistaken for ADHD. The same happens to highly creative adults. It happens to me. The more excited I get with a creative idea, the harder it is for me to control fidgeting. If I had a bad luck of hitting some great idea before sleep, I might toss and turn violently. This is a prelude to insomnia. For that reasons, I try to stop solving problems well ahead of my bedtime (see: Evening routine).

Random signaling in creativity and REM-sleep play a vital role. In both cases, it involves forming unpredictable associations between concept maps. In REM-sleep, the brain cuts itself of from the motor system to help you avoid injury. In creativity, it is your prefrontal command and control that does most of that work. This is wasteful. It makes far more sense to enter an automatic motor program like walking, jogging or biking. All creative signaling sent down the motor system will be subject to extinction. It would take a feat of incredible creative catapult to overturn a walking human. I have never heard of such a thing happening to a healthy individual. This is why walking is safe, cheap, and effective way of brainstorming with an added bonus of improved circulation and fresh air that both do wonders to the clarity of thought. Obviously, you can brainstorm with your creative partner, or you can just brainstorm with yourself while walking with your dog friend.

If you ever brainstormed in a stuffy corporate room, you will recall there are always many distractions and little annoyances that can totally ruin the experience. Limited blood supply caused by a tie does not help. When timelines and deadlines enter the room, creativity walks out the door.

Walk-talking is the best form of one-on-one face-to-face brainstorming I know

Disruption via forgetting

When I say, you can solve any problem, I should add that the problem needs to be solvable, you need to have data to process, and you need to have enough time. Given enough time, you should come to a resolution, which does not need to have a form of a solution. Some possible resolutions might have a form of "the problem is not solvable!", or "I do not have enough data to proceed!", or simply "it would take too much time". If the trajectory is too long, instead of solution, you may enjoy "only" a major advanced towards the solution.

Your quest may end up in a temporary setback. After a longer while of trying, you may conclude you are making no progress. If this is an honest assessment, you may need a larger disruption. If there is no new knowledge that could change your thinking, if conversations keep going in circles, you may need to employ the mega-cannon of forgetting. In our mountain walking metaphor, forgetting is equivalent to being catapulted far enough to have a very limited chance of walking back the same pathways unless there is a substantially increased chance they are indeed proximal to the solution.

If you need to employ forgetting, take a vacation, think about something unrelated, get a new hobby, get another problem to tackle, or two or more. Learn a lot. Employ incremental reading to restructure your thinking via interference. After a longer while, you can return to the problem. Again it is your problem valuation network that can make a good assessment on the required length of the break. It will probably tell you cautiously that your chances are not improving unless forgetting is effective and helps you look at the problem with renewed optimism.

There are several problems I have now been working on for decades. However, none of these problems is stagnant. Each time I return after a longer break, I discover new things, including my own errors in thinking. Problem solving is not easy. Otherwise it would not be called *problem* solving. However, with the right approach, it is always fun!

Severe cases of stagnation in problem solving may require the last resort weapon: forgetting. Forgetting may level out old wrong solution ideas. It may also lead to renewed optimism

Reward in problem solving

Schooling conditions students to believe that hard work and pain are a formula for success in learning. This makes many believe that sweat and pain are supposed to be vital components of problem solving. Elon Musk says "think until you head hurts". I think he means "think a lot", make your computations central and protected. Problem solving cannot be actually unpleasant because this only means the solution is not within reach or not worth your time. The brain has a good way of knowing.

Problem valuation network

As much as pleasure is a good guidance in learning, problem solving is and should be pleasurable for analogous reasons. As much as there is a knowledge valuation network, there is a problem valuation network in the brain. It is actually the exactly same thing, except, outwardly we perceive it differently. Knowledge valuations bring a jolt of pleasure at discovering valuable knowledge. Problem valuations bring the same happiness when we discover a solution to a problem, which in essence is a piece of knowledge that closes the loop. Knowledge that clinches the deal. As much as passions can drive individuals to extreme heights of knowledge, innovation and discovery, they can also help individuals solve problems.

If you hate the problem at the very start, you are in trouble. This will lead to stress and procrastination. If the problem has been imposed on you by others or by circumstances, you may have no way out. Luckily, this does not make the problem unsolvable. It will only make your brain work less efficiently. You should not aim at quantum well if you struggle with fractions. I don't solve puzzles from housewife journals, even if they pose a bigger challenge than my own work.

If you are properly rewarded for the solution, the hate of problem solving may come from problem difficulty or from problem competition (i.e. more interesting problems you would love to solve). The difficulty may lead to procrastination. We may see reaching the solution as unlikely, or requiring a great deal of labor (e.g. plodding through tax laws and regulations, digging into details of some poorly organized technical specification, reams of manual mathematical calculations, etc.).

If you hate the problem, start from looking for the way out. Perhaps you can wriggle out. Perhaps you can replace Problem A with Problem B. Perhaps you can trade? Perhaps you can creatively procrastinate and the problem will self-resolve. Perhaps someone will do it for you? If you hate the problem, I will rather not be able to help you love it. Your problem valuation network is a great predictor, it is strong, it is biological, and it is stubborn. Let's see how it works. Perhaps you will find some room for improvement.

Components of problem solving reward

We are surrounded by problems to solve. We will ignore many for their little value (e.g. how to reduce the noise of wind in the window). We will ignore many for their cost (e.g. sweeping outdoors from autumn leaves). A simple formula is to value tasks by their Value/Time ratio or Value/Cost ratio with Cost and Time interchangeable if you know the cost of time.

SuperMemo insert. What is SuperMemo?
Value/Time formula is also used in SuperMemo tasklists. For example, SuperMemo implementation tasklist is approaching 5000 user requests despite continual progress. This is why tasks valuations and their implementation cost yield Value/Time ratio, which can be used in task sorting. However, it is not always the top task that brings most excitement.

In problem solving, we usually go for high value targets. However, high value of a solution is filtered by the problem valuation network. All problems carry a risk factor. We may fail to arrive at the solution. This will result in frustration that the brain does not like. The brain has been wired for efficiency. It loves productivity as much as it loves learning. These are actually two forms of valuation that evolved the exact same mechanism.

Three components determine problem valuation:

  • Reward based on the value of the solution
  • Probability of success as determined by your problem valuation network
  • Penalty based on the pain of failure

Reward is derived from the value of the solution and increases with problem difficulty. If two problems bring the same benefit, it is the harder one whose solution we will rejoice more. Probability of success decreases with problem difficulty. So does the penalty, i.e. the pain of failure. We can always say I always knew this problem was too hard and sense minimal discomfort. Probabilities are based on: pre-estimations (Have I solved a similar-looking problem?), and actual statistics (mini-failures and mini-rewards on the path to the solution).

The picture is a conceptual simplification of the calculations the brain might be doing when facing a problem:

Problem valuation optimum for problem difficulty

Problem valuation: Harder problems are more rewarding, but are less likely to be solved. This determines optimum difficulty we look for. The expected reward will then be evaluated in the light of execution costs.

The actual valuations in the brain are a bit more complex, and hard to predict. They will change with mood and with arrival of new information. However, there will always be a class of problems that will perfectly fit your optimum valuations.

Your brain is actually a perfectly optimized machine that looks for new knowledge and new problems to solve all the time. It will use knowledge and problem valuation networks to find new optimum actions to take. The valuation optima will provide a nice trajectory through the space of problems and mini-solutions with maximum output in terms of reward, and the best fruit at the end of the journey: a major problem solved.

Having faced dozens of problems, you will nearly always be able to make a statement about your "feelings" about solving a given problem. The feelings will range from explosive enthusiasm to "Hell, no!". I suggest you follow your instincts and go for problems that are fun to solve. This will ensure passion, engagement, new learning, and hunger for more problems to solve. This is the same mechanism as that which determines the learn drive. Good learning makes you hungry for more learning. Good problem solving makes you ask for more.

Hard problems

Some productivity experts tell you to set difficult goals that will make you work harder. I think this is a treacherous advice. You may never climb that wall, and you may lose your self-confidence in the process. You may even lose your health. Arm your problem valuation network with lots of knowledge. Golden knowledge may turn difficult problems into easy problems. Even if you don't know the solution, your network might predict its proximity. If you are eager to get down to action, your brain knows that success is within reach and you will be rewarded. Instead of setting hard goals, follow your natural valuations. Overtime, on an absolute scale, your solved problems will get harder. You will simply develop all the needed habits and knowledge to make the whole process enjoyable.

There are no lazy people! There are only people who hate what they are supposed to do.

Fast problem solving

Fast thinking is based on generalizations, established knowledge, and established routines. Slow thinking is based on exploration, search, and new knowledge. For tough problems based on complex knowledge, you will need to resort to slow thinking. If you can solve problems with your fast thinking arsenal, they do not qualify as tough. Donald Trump is strong at fast thinking. Fast thinking is a habit he inherited from the business world. His formula for problem solving is to rely on other people. He can solve problems fast like students who solve multiple-choice tests. He picks variants. He does not need to ruminate Middle East problems from morning to evening to find the optimum solution. He just picks a variant and moves on. Obama was often accused of being a slow thinker. He would ponder, prepare, and analyze. He was more of a scientists. Unfortunately, presidents or chief executives do not have time to apply slow Darwinian approach to solving problems. If you need to solve problems fast, it helps to have extensive knowledge, it helps to narrow the focus, and build an area of expertise, etc. The fastest way to solve problems is to have a team of good advisors, who can solve problems for you. This is not what this text is about.

The presented approach will be useful for the classes of problems with the following properties:

  • problems that require processing large amounts of knowledge
  • problems that are complex and involve rich branching of the thought process
  • problems where working memory is a bottleneck
  • problems where long-term memory is a bottleneck

The presented approach might be less useful in hurrying a new iPad model, but it would be handy in slowly developing or slowly adopting theories like those of Darwin, Mendel, Wegener, and the like.

Problem solving algorithm

The following algorithm can be used to tackle any problem:

  1. If you hate the problem, start from trying to work around it. Solving problems you hate is very inefficient!
  2. If your problem needs just a few hours, you do not need much help. Pick a good day and invest a few hours. This algorithm is not for you. Here we only discuss hard or complex problems that span over the cycle of many days
  3. Start the problem solving loop that may take days or decades
  4. You need to strive at waking up naturally with the problem on your mind. Any concept map that makes part of the problem should ideally be the first map that battles its way to your conscious focus! Your morning thoughts are hard to control, but with a bit of self-training, you can have some influence
  5. Cut off all external disruption: phone, e-mail, Twitter, radio, etc.
  6. Drink coffee or tea. If you avoid caffeine, rehydrate properly
  7. While drinking do not turn on TV. Do not talk to others. Focus on the problem. This is one of your most precious morning moments that needs best protection
  8. In protected mental computation slot, let your mind go loco with creative inspiration: think, make notes, try neural creativity, etc. It is ok to get very excited, to walk around, to rip hair out of your head. If you want to punch the wall in excitement, you are on a good way. If you are a cool thinker, that's cool too. If creativity does not come, you may start from improving your sleep. See: Good sleep, good learning, good life
  9. Employ incremental writing to organize your mental computation and extend it well beyond the limits of your memory. Incremental writing is the best way to creatively elaborate on your ideas and keep track of all your thoughts
  10. Use creative disruption if you get stuck in a creative groove. Any form of random input is good. Google for a subject. Read an article. I recommend neural creativity first and incremental reading next
  11. Once you start tiring, transition to exercise, socializing, and a meal
  12. Take a siesta to recharge mental batteries. Post siesta creativity is as good as morning creativity and may bring new qualities. If you skip napping, by my guestimate, your creative output for the day will drop by 40-60%. In my own case, 60% is possible because I use a big portion of the noontime for exercise
  13. After a siesta, repeat the morning routine: rehydration, creativity, focus, computation, incremental writing, creative disruption, etc.
  14. Once you start tiring, turn into more passive mode. Deploy your evening routine in low light (e.g. sipping a drink, listening to lectures or interviews in related subjects, etc.)
  15. Get uninterrupted sleep at your favorite time. Don't go to bed until you are tired! This is vital for the quality of sleep! Walking or moderate exercise may help you if your bedtime turns outrageously late
  16. Return back to the beginning of this problem solving loop unless you are so dismally stuck that you need to take a vacation (or work on other projects)
  17. After many loops with no progress, you may feel I failed you with the algorithm? Write to me! It is not the end!

For more texts on memory, learning, sleep, creativity, and problem solving, see SuperMemo Guru