Thinking outside the box: One half of thirteen is eight

outside-the-box-xl

With our pattern thinking, we easily get caught up in a habitual way of approaching a problem. In school, we have also been trained to think convergently, i.e. to look for the one and only correct answer, instead of taking a step back, observing the world with non-judgmental eyes and reflecting on unexpected solutions to the problem. I have previously highlighted this phenomenon in the blog post Thinking outside the box: The Barometer question.

With this strategy, we often come up with appropriate solutions to a problem, but these are often neither very odd nor innovative. Since our brain wants to use as little energy as possible, we usually stick to the first answer that fits our question, and then stop thinking.

The opposite way to address a problem, is instead to try to find as many divergent solutions as possible to a problem that at first glance appears to be convergent. To succeed, we would need to consider the problem from completely new perspectives, i.e. what we sometimes call lateral thinking or thinking outside the box.

An illustrative example of this is the question: “What is one half of thirteen”? Our immediate intuitive answer is “6.5”, which is probably the only reply that would render us a correct answer in the school’s math classes. But if we were to take the challenge from another perspective, the answer is not as obvious.

By halving 13 expressed as Arabic or Roman numerals or letters, half of 12 may also be:

13    ⇒     1 and 3

XIII = XI and II   ⇒     11 and 2

Thirteen = Thir and teen   ⇒     4 and 4

XIII =    XIII        ⇒     VIII = 8    (upper part of the half)

Albert Einstein once said: “Most people stop looking when they find the proverbial needle in the haystack. I would continue looking to see if there were other needles.

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Tänka utanför lådan: Hälften av elva är sex

outside-the-box-xl

Genom vårt inlärda mönstertänkande så fastnar vi lätt i invanda sätt att angripa ett problem. I skolan har vi dessutom tränats i konsten att tänka konvergent, d.v.s. leta efter det enda rätta svaret i stället för att ta ett steg tillbaka, se världen med nya ögon och reflektera över oväntade lösningar på problemet. Jag har tidigare belyst det här fenomenet i blogginlägget Tänka utanför lådan: Barometerfrågan.

Med denna strategi kommer vi ofta fram till ändamålsenliga problemlösningar, men dessa är ofta inte särskilt udda eller innovativa. Eftersom vår hjärna vill använda så lite energi som möjligt så nöjer vi oss oftast med det första svaret som passar till vår fråga och slutar sedan att tänka.

Det motsatta sättet att angripa ett problem är i stället att försöka hitta divergenta (så många som möjliga) svar på problem som vid första anblicken verkar vara konvergenta. För att lyckas med detta så behöver vi ofta betrakta problemet från helt nya synvinklar. Det vi kallar för lateralt tänkande eller att tänka utanför lådan.

Ett bra exempel på detta är frågan: ”Vad är hälften av elva”? Vårt omedelbara intuitiva svar är “5,5”, vilket sannolikt varit det enda svaret som gett oss rätt i skolans matematikundervisning. Men om vi skulle anta utmaningen från ett annat perspektiv är svaret inte lika självklart i sin ensidighet.

Genom att halvera 11 uttryckt som arabiska eller romerska siffror eller bokstäver kan hälften av 11 också vara:

11    ⇒    1 och 1

XI    ⇒    X och I   ⇒    10 och 1

XI =  XI     ⇒   VI = 6 ( övre delen av XI )

Elva   ⇒    El och va   ⇒    2 och 2

Albert Einstein har sagt: “De flest människor slutar leta när de funnit den berömda nålen i höstacken. Jag skulle fortsätta att leta för att se om det fanns fler nålar.”

united-kingdom-flag-1- This blog post in English

Trees falling without a sound, quantum mechanics, and creative problem-solving

Falling tree

If a tree falls in a forest and no one is around to hear it, does it make a sound?” This is a well-known philosophical question, which many persons have been pondering on.

Physically, a sound is generated when a wave movement is propagated through a medium (in this case the air) and then reaches a hearing organ of a biological being and then perceived as sound. Everything that is experienced happens therefore within the individual who experiences it, as I wrote about in a previous blog post.

One way to consider the falling tree in the forest is that if no one is there to observe the event, there is also no safe way to say whether the event occurred or not. Therefore, the event then becomes identical to an event that never occurred.

Similarly, according to quantum mechanics, as described by Niels Bohr and Werner Heissenberg in the 1920s, a subatomic elemental particle can be described both as a particle and a quantum wave and paradoxically be in both these states simultaneously.

It is only when the system is observed that the probability is reduced to one of these two possibilities immediately after the measurement. Therefore, reality depends on whether it is observed or not.

This paradox has been illustrated by physicist Erwin Schrödinger in the classical thought experiment Schrödingers cat, where a cat trapped in a box can be both dead and alive at the same time.

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In the experiment, the death would be triggered by a single atomic decay in a radioactive material in the box (which, according to the quantum mechanics, could simultaneously take place and not). Whenever such a decay is recorded by a Geiger meter, a poisonous gas would be discharged, immediately killing the cat. However, it is only when someone opens the box and observe the cat that the two outcomes are reduced to a single one – a dead cat or a living cat.

schroedingers-cat

Both the quantum mechanical paradox of Schrödinger’s cat and the story of the falling tree have a direct parallel in our minds when we solve problems using our associative ability.

Every second there are millions of subconscious associations where different thoughts, concepts, feelings, symbols and memory fragments constantly bounce against each other in combinations that are tested against the problem we are trying to solve without us being aware of it.

As the brain focuses on unresolved issues, only one or a few of these combinations of different thoughts and memories will reach our consciousness as an insight or an aha experience. This would also require that we have a mental readiness to capture the insight and that we are not in an environment where we are distracted.

All the other millions of different thought associations taking place in our brains will never reach our consciousness, and in the same way as the falling tree in the forest, they may either have happened or not and we have no way to decide which.

As long as our consciousness has not perceived a thought from our subconscious brain part, this thought is thus identical to a thought that never occurred, and therefore lacks all meaning and relevance to us.

We can never know what thoughts exist within us, if they don’t reach our consciousness. However, we can train our ability to open up for the wealth of creativity we carry within us.

This can be done both by practicing our association skills and by staying in stressful environments, meditation and mindfulness, and tuning down the filters and opening our window to our subconscious minds and recording in a non-judging manner everything that comes up and down.

We can also denounce this creative gift by blindfolded stressing through life and further rejecting all the odd and unexpected ideas that do not directly fit our world view.


Note: Einstein, who with his work laid the theoretical foundation for quantum mechanics, could never accept the underlying paradox described above. He expressed his views as : “God does not play dice with the universe.”

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Ljudlöst fallande träd, kvantmekanik, och kreativ problemlösning

Falling tree

“Om ett träd faller i en skog och ingen finns där som lyssnar hörs det då något ljud?” Detta är en känd filosofisk fråga som många funderat på

Rent fysiskt så uppstår ett ljud när en vågrörelse fortplantas genom ett medium (i detta fall luften) för att sedan nå ett hörselorgan hos en biologisk varelse och hos denna uppfattas som ljud. Allting som upplevs sker därför inom den individ som upplever det, vilket jag skrivit om i ett tidigare blogginlägg.

Ett sätt att då betrakta det fallande trädet i skogen är att om ingen (eller inget) finns där som kan observera händelsen, så finns det heller inget säkert sätt att säga om händelsen ägt rum eller inte. Händelsen blir då identiskt med en händelse som aldrig ägt rum.

Ett liknande tankedilemma finns inom kvantmekaniken, såsom den formulerats av Niels Bohr och Werner Heissenberg redan på 1920-talet. De fann att en subatomär elementarpartikel både kan beskrivas som en partikel och en vågrörelse, och paradoxalt nog befinna sig i båda dessa tillstånd samtidigt.

Det är bara om man observerar systemet som sannolikheten reduceras till en av dessa två möjligheter omedelbart efter mätningen. Verkligheten blir alltså beroende av om den observeras eller inte. Den här paradoxen har illustrerats av fysikern Erwin Schrödinger i det klassiska tankeexperimentet Schrödingers katt. I tankeexperimentet kan en katt instängd i en låda vara både död och levande samtidigt.

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I det här fallet skulle döden ske genom att ett atomiskt sönderfall i ett radioaktivt material i lådan (som enligt kvantmekaniken samtidigt både kan ha skett och inte skett) registreras av en Geigermätare och ett utslag resulterar i att en giftgas släpps ut i lådan och omedelbart dödar katten. Det är då bara om man öppnar lådan och tittar efter som de båda utfallen reduceras till ett enda – en död katt eller en levande katt.

schroedingers-cat

 

Både den kvantmekaniska paradoxen med Schrödingers katt och historien med det fallande trädet har en direkt parallell i våra hjärnor när vi löser problem med hjälp av vår associativa förmåga.

Det sker varje sekund miljontals undermedvetna associationer, där olika tankar, koncept, känslor, symboler och minnesfragment hela tiden studsar mot varandra i kombinationer som omedvetet testas mot det problem vi försöker lösa.

Då hjärnan fokuserar på olösta problem så är det bara en eller ett fåtal av alla dessa kombinationer av olika tankar och minnen som når upp till vårt medvetande som en insikt eller en aha-upplevelse. Detta kräver dessutom att vi har en mental beredskap att fånga upp insikten och att vi befinner oss i en miljö där vi inte är distraherade.

De övriga av dessa miljontals olika tankeassociationer kommer aldrig att nå vårt medvetande och på samma sätt som det fallande trädet i skogen kan vilken som helst av dem antingen ha skett eller inte skett och vi har inget sätt att avgöra vilket.

Så länge en tanke från vårt undermedvetna inte har nått upp till vårt medvetande så är den alltså identisk med en tanke som aldrig ägt rum, och den saknar därmed all mening och relevans för oss.

Vi kan aldrig veta vilka tankar som finns inom oss men som inte släpps upp till vårt medvetande. Vi kan däremot träna upp vår förmåga att öppna upp oss för den rikedom vi bär inom oss.

Detta kan ske både genom att öva upp vår associationsförmåga och att genom vistelse i avstressande miljöer, meditation och mindfulness och tona ner filtren och öppna upp vårt fönster till våra undermedvetna tankar och på ett icke bedömande sätt registrera allt som då kommer upp och ut.

Vi kan också frånsäga oss denna skatt genom att stressa genom livet med skygglapparna på och kritiskt förkasta till intighet alla udda och oväntade idéer som inte direkt passar in i vår världsbild.


Not: Einstein, som med sina arbeten lade den teoretiska grunden till kvantmekaniken, kunde aldrig acceptera den underliggande paradoxen som beskrivits ovan. Han uttryckte det som: “Gud kastar inte tärning med universum”.

united-kingdom-flag-1- This blog post in English

The monk and the mountain climb

Munk

When we face a tricky problem, we often try to attack it with our logical thinking. This will often lead us right, but sometimes it may be worth taking a step back to consider the problem from another point of view.

Problem 1: Precisely at sunrise, a monk begins to climb a high mountain on a small path to reach a temple located on the top. The walk is laborious and he has to stop and rest several times. He arrives to the mountain top late in the afternoon and spends the night in the temple.

Exactly at sunrise the following morning, he begins to walk down the mountain again along the same path. He is now rested and is walking downhill, so the walk goes easy.

The question is whether there is a single point along the path which he would pass at exactly the same time both days.

Consider the problem for a while and then proceed to Problem 2. Answers can be found at the bottom of the page.

Problem 2: You are responsible for a football elimination tournament. Twenty-seven teams have signed up and for your planning of fields and referees, you need to figure out how many matches will be played before the winning team can be selected.

soccer

Solution to problem 1: Since we do not know the distance nor the walking speed of the monk, there is no way to mathematically figure out where he will be at any single moment. If we then try to intuitively reason, it would seem highly unlikely that this could happen.

But if we instead visualise two different monks doing the same two walks during the same day, then we would quickly realize that they must definitely meet somewhere along the path no matter how fast or slow they walk.

Solution to problem 2: Most people facing this problem would adress it by drawing a chart of all the matches, from the first to the last one. Others with better math skills may try to set up a formula to calculate the required number of matches.

The solution is much easier than that. The answer is 26. Because at the end of the tournament there can only be one winning team, so by that time there must have been 26 defeated teams. And as in an elimination tournament you have to leave after your first loss, there will need to be as many matches as teams to be eliminated.

What may make the thought go astray in this example, is that we are more likely to focus on the winners than the losers. By reversing the reasoning and looking at the losses, the problem imediately becomes easier to solve.

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Munken och bergsvandringen

Munk

När vi står inför ett klurigt problem så försöker vi ofta först att angripa det med vårt logiska tänkande. Ofta leder det oss rätt, men ibland kan det vara värt att ta ett steg tillbaka för att betrakta problemet ur en annan synvinkel.

Problem 1: En munk börjar exakt vid soluppgången gå uppför ett högt berg längs en stig för att nå upp till ett tempel beläget högst uppe på berget. Vandringen är mödosam och han får stanna och vila flera gånger. Han är framme sent på eftermiddagen och tillbringar natten i templet.

Exakt vid soluppgången morgonen därpå börjar han gå nerför berget igen längs samma stig. Han är nu pigg och utvilad och har nedförsbacke så vandringen går lätt.

Frågan är om det finns en plats på stigen som han passerar vid exakt samma klockslag de båda dagarna.

Fundera ett slag och gå sedan vidare till Problem 2. Svar finns längst ner på sidan.

Problem 2: Du är ansvarig för en utslagningsturnering i fotboll. Tjugosju lag har anmält sig och för din planering av planer och domare behöver du nu räkna ut hur många matcher som måste spelas innan det segrande laget kan utses.

soccer

Lösning på problem 1: Då vi varken vet hur snabbt munken går (och hastigheten dessutom inte är konstant) eller sträckan han tillryggalägger så kan vi inte räkna ut var han befinner sig i något givet ögonblick. Om vi i stället försöker tänka intuitivt  så förefaller det högst osannolikt att detta skulle kunna inträffa.

Men om vi i stället visualiserar två olika munkar som gör samma vandring under samma dag, så inser vi snabbt att de absolut måste mötas någonstans på vägen oavsett hur fort eller sakta de går.

Lösning på problem 2: De flesta människor som ställs inför det här problemet angriper problemet genom att rita ett diagram över alla matcherna, från de första till den sista. Andra som har mer fallenhet för matematik kanske försöker sätta upp en formel för att räkna ut det nödvändiga antalet match.

Lösningen är mycket enklare än så. Svaret är 26. Eftersom det i slutet bara kan vara ett vinnande lag, så måste det när turneringen är slut ha funnits 26 förlorande lag. Och eftersom man i en utslagsturnering bara kan förlora en gång innan man åker ut så kommer det i turneringen att behövs lika många matcher som lag som ska slås ut.

Det som lätt för tanken fel i det här exemplet är att vi gärna lägger fokus på vinnarna och inte på förlorarna. Genom att vända på resonemanget och betrakta förlorna så blir problemet genast lättare.

united-kingdom-flag-1- This blog post in English

The importance of a good night’s sleep

sleep

Sleep is the vital balm that restores us after a long day of work and play, and prepares us for the next. How we feel at day is largely determined by how well we sleep at night.

Sleep has a vital role for our health and creative energy. Sleep is involved in how we heal and repair our blood vessels and sleep deprivation increases the risk of cardiovascular disease, kidney problems, high blood pressure, and diabetes.

Sleep is also involved in the balance between the two hormones that make us hungry (ghrelin) and saturated (leptin) and sleep deprivation increases the risk of obesity in both young and adults. Sleep also facilitates normal growth in adolescents and by affecting our immune system and our hormone balance, it helps keep us healthy throughout the years.

Sleep is not a constant and passive state but is characterised by four different stages of a sleep cycle; falling asleep, stable sleep, deep sleep and dream sleep or REM sleep (REM stands for rapid eye movements).

Each such sleep cycle is approximately 90 minutes long, and repeated during the night with longer periods of REM sleep in each new cycle; from 5-10 minutes at the beginning of the night to half an hour or more in the last cycle before waking up.

The longer you have been awake, the faster you fall asleep and deeper sleep. The body recovers most during deep sleep, as the brain’s activity is at low speed and the secretion of stress hormones is the lowest. Good sleep also leads to improved memory features, a good mood and increased creativity.

During sleep, we consolidate our memories from the previous day, but we also integrate these memories in a broader context and create new links to previous similar memories. If we have been to a concert, the brain, when we are asleep, connects these memories to our experiences from previous concert visits or the music to the memories of other music we have heard. If we’ve been there with a certain person, there are also connections to memories of other contacts we had with this person.

Consolidation of our memories is a complicated process over the course of several nights. During sleep, the different memory fragments are built into meaningful concepts, but memories can also be repaired and enhanced so that when we wake up, we’ve actually improved our ability, whether it’s factual skills, mathematical problem solving, or music or sport skills. The sleep also allows us to distinguish new previously hidden rules or abstractions, such as grammatical patterns during language learning. These positive memory effects of sleep are mainly related to activities that have touched and engaged.

This memory consolidation occurs mainly during the first 90-minute cycle and the last period of REM sleep in the morning. Alcohol or sleepiness that affects sleep or premature awakening can seriously interfere with memory consolidation, and what you lose during one night can not be compensated for by more sleep the following nights. On the contrary, one night without sleep also causes memory consolidation for the next 2-3 nights when the brain tries to restore normal daily rhythm.

The need for sleep is for most adults somewhere between six and nine hours, with a greater sleep need in children. We need to sleep, but not necessarily in one stretch. Throughout history, our natural sleep patterns have largely been controlled by the daylight and a sleep pattern with two-night periods of night sleep, combined with a slumber at day, is more natural than the eight hours in a row that has become the norm since the course of industrialisation and the introduction of the electric light.

Sleep deficiency causes slower brainwaves in the browsers, lack of attention, anxiety, memory impairment and an irritated mood. Longer sleep deprivation causes mental, physical and emotional fatigue. People who have been without sleep for 32 hours show a severe and prolonged reduction in ability of diverge thinking compared to a control group who has been sleeping.

But also a few hours lack of sleep is affecting performance the following day in a clearly negative way. We risk dozing off for short micro sleep moments without even being aware of it, which may be fatal in traffic. Many major disasters, such as nuclear accidents in Chernobyl and Three Mile Island, the Exxon Valdezoil tanker spill, have been caused by a lack of sleep among the responsible personnel.

To stay up and study the night before an exam is a bad tactic. But too much sleep can be negative, albeit not as much. So the best habit is to try to have a regular daily rhythm, whether it’s weekdays or weekend, with night sleep that gives enough rest to feel well rested when waking up in the morning.


Tips: There are a number of well-proven tricks you can use if you have trouble sleeping:

  • Go to bed at the same time each evening. We have an internal biological clock that used to be controlled by the seasonal changes before the introduction of electric light. If you have different sleep patterns on weekdays and weekends, you will experience a constant jet lag. Also try to get up at about the same time each morning. The only reason to sleep longer is if you have a “sleep deficit” from earlier for short nights.
  • Take a bath a couple of hours before going to bed. The chill that the body feels when you getting up from the bath signals to the body to slow down. Alternatively take a “light” sauna just before going to bed so that your body is pleasantly warm and relaxed.
  • Avoid mental efforts just before going to bed. Problem solving, detective stories or a horror movie on Tv gets the brain in high speed. Better then to listen to quiet music in front of the fire place.
  • A short walk just before you go to bed will both lower your brain activity and give you some fresh air. On the other hand, avoid a hard physical pass hours before bedtime otherwise you have an excretion of the activation hormone cortisol and adrenaline that aggravate the insomnia.
  • Avoid energicing substances. Coffee, tea and chocolate contain stimulating xanthines. Alcohol may cause drowsiness, but it often leads to a shallow and unrestful sleep. Warm milk containing tryptophan (a substance that the body transforms into the relaxing hormone serotonin) is a better option. Also, avoid eating just before you go to bed – especially food containing fat and sugar.
  • Let the bedroom be a place only for sleep (and sex). If you also use the bedroom and the bed for work and other activities, your brain will also associates it with these habits. Keeping the room “free” from unnecessary activities gives a conditioned reflex to sleep when you lay down. The bedroom may also be a few degrees cooler than the rest of the home.
  • Avoid sleeping pills. Regular medications to fall asleep, almost always lead to addiction. After a while, their initial effect decreases or ceases, as you get accustomed to them, but if you the try to quit them, you may experience worse insomnia than before you started taking medication.

For those who have the ability to control their sleep rhythm, it may be worth exploring the effects of two sleep periods during the night, but also taking the opportunity to have a half an hour nap in the middle of the day. In order for this to work, you will need to reach the stable sleep, which takes about 20 minutes.

The American Space Agency, NASA, has demonstrated that pilots who took a 40-minute nap during long-haul flights increased their ability by 34% and their physical alertness by 100%.

Sources:

  1. National Heart, Lung, and Blood Institute. Why Sleep is important? https://www.nhlbi.nih.gov/health/health-topics/topics/sdd/why
  2. Walker PW & Stickgold R (2010). Overnight Alchemy: Sleep-dependent Memory Evolution. Nature reviews. Neuroscience 11:218.
  3. Home JA (1988). Sleep loss and “divergent” thinking ability. Journal of Sleep Research & Sleep Medicine. 11:528–536.
  4. Levitin Daniel (2015). The organized mind: Thinking straight in the age of information overload. Penguin Books.
  5. Martin Paul R (2003). Counting sheep: The science and pleasures of sleep and dreams.

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