Naked Science - Intelligence

Copyright Michael Curtis 2020

All rights reserved


Part 5

Problem Solving



Some people are much better than others at using logic to solve problems. The first barrier to problem solving is the process of making mistakes - some people generally seek less proof for what they choose to believe than others. So, they can end up trying to solve a problem by using mistaken facts and mistaken rules. Beyond making mistakes, problem solving is difficult if you do not know about the properties common to good problem solving. That is what the rest of this chapter covers

(some problem solving ideas to play with, anyway).



Problems are best solved by covering every possibility. For instance, what is the chance of throwing a coin three times and getting heads each time? A way to solve this problem is to write out every possible combination of throws. e.g. ‘Heads, Heads, Tails’ , ‘Heads, Tails, Heads’, etc.

There are eight possible outcomes - only one of which is ‘Heads, Heads, Heads’. For this reason, the answer to the question is that the chance of getting three heads in a row is one in eight. By being exhaustive (exploring every outcome), a clear insight into the answer is arrived at.

Compare that with the question of how often you would get 2 heads and one tail. You might imagine a static  image of 2 heads and one tail and know that there are 8 throw outcomes. “The answer is 1 in 8”, you might say but that is not certain unless you seek 2 heads and 1 tail in the remaining 7 outcomes. Eg. Heads Tails Heads.

All of that was covered in maths lessons at school. I would hesitate to say that studying math improves one’s thinking skills: your mind has to take the leap and absorb the concept, in this case, of ‘exhaustiveness’. Otherwise, you would be good at answering ‘heads and tails’ questions but not apply the principle of exhaustiveness to other fields. And this course of heuristics is about stepping back from an experience and taking away a broader more generalised lesson from it. Play with an idea; try to imagine it not as heads and tails but as cards in a pack or red cars and blue cars. Invent maths questions to yourself based on the variations. Memory requires revision. So revisiting the concept of exhaustiveness must surely help to plant it in your behaviours.

Heuristics’ is a great word isn’t it? You can find books on the subject of problem solving.


Playing / Trial and Error

Sometimes, it is good to play around with a problem and then a solution might make itself apparent. A feature of a problem might simply look interesting. By taking the time to play with it and to explore it, new ideas might occur to you. For example, in a game of chess, where there are many possible combinations of moves which can be made, you might examine a king because it is an important piece and notice something about its position that can lead to you winning the game.

Another example is with graphs with vertical (y) and horizontal (x) axis and you have to plot the path of an equation on that graph. Sometimes, by imagining where y is when x is nearly infinite, you get an idea of the general shape of the graph.

Playing around with a problem and solving it by trial and error works largely because it encourages you to notice the characteristic details of the problem. Once you begin to see the nature of the problem, ideas about how to solve it can spring to mind.

Another reason for trial and error working is that sometimes there is a small number of possibilities of how the elements of a problem can be arranged. By fiddling about with those elements, there is a fair chance of getting the right combination. e.g. A bicycle chain that just needs a nudge to engage properly again.

Note: Limited playing with a concept may lead to limited understanding of the concept.

e.g. Watching a flock of birds for only part of the year might not show the flock’s tendency to migrate both ways.


Qualities of a Solution

Early on in solving a puzzle, someone can make statements about what the solution will be like – without knowing the specifics.

For example, someone solving a Rubik’s cube might think that the cube has to get a little scrambled, some significant change needs to happen, and then it would be good to get the cube largely back to the order it was in – with the small change applied.

Someone might assume that they can make one corner piece rotate its colours while leaving the rest of the cube unchanged; and then explore ways to make that happen – but maybe it is not possible to make a corner rotate without some knock-on effect on another piece. So the statements one states about what the solution will look like and what manipulations are possible will have a huge effect on the solvability of the puzzle.


Eliminator Questions

These days, I work with spreadsheets a lot. Sometimes, I apply a filter to select just the rows of the sheet of rows which are the ones that matter to me.

Sometimes, a problem with a large number of combinations or, let’s say, rows to explore, can have a huge number of rows eliminated by a simple filter: an eliminator.

In a detective movie, they might use a police database to find people with a particular colour and type of vehicle within a 50 km radius of an incident; then the tv detective goes door to door to interview the small list of people who match those criteria. So he has massively reduced the number of people to investigate albeit on a questionable assumption that 50km is significant…

Then one further filter observation might eliminate most of that list. A typical movie device is that a particular sound can be heard when a phone call is made. That sound can only occur in a few places in the city.

And sometimes, the line of inquiry is not to catch a suspect but to talk to a witness; and then the detective discovers that the witness is the villain. So you can stumble across the right answer by having a play around with the information.


Thinking about the Goal First of All - and Related Information

y plus x equals 4. y is 3. What is x? The mind can go blank when looking at a question. A way to start yourself off in answering the question is to ask, "What is the goal?". Here, the goal is to find x. It can be expressed as being 4 - y (in the same way that 1 is 4 - 3).

Next, you look for any information that refers to x (in this case, y plus x equals 4). The y is a problem: we want x to equal a number and not something with a y in it. So, the new goal is to find what y equals.

Next, you look for any information that refers to y. We are told that y equals 3. Now that y can be expressed as being 3, we can go back to the goal of finding out what x is. Since x is 4 - y, and since y is 3, x is 4 - 3. So, x equals 1. That’s a lot of wordy sentences but algebra makes it clearer to express.

That is an example of solving a problem by looking at the goal and any information which refers to the goal. Many people have a problem solving mathematical problems because they do not have this habit of arranging the information given in a way that links up with the goal: start at the goal and work backwards from there.

Similarly, real world problems are solved by linking the goal to known information. If your goal is to find a friend’s telephone number then you consider what is needed to achieve that goal; so you think of any context in which the person’s phone number occurs (a public telephone book, your own address book, letter stationery, internet, etc.) and then the next goal is to locate one of those contexts.

The ‘telephone number’ example is easy because it is a problem which occurs in everyday life and so the solution is as much recalled as it is ‘figured out’. For people who learn mathematics, the maths problem is easy too since they see the same sort of problem every day. It is when we encounter new territory that the idea of identifying the goal and building bridges to it is the only way that the answer can be figured out.

If you can not find your friend’s phone number then you might contact a mutual friend who does have that person’s phone number: a bridge is being built across what seemed like a dead end. Nobody has thrust the answer towards me. I have just had to play around with things I know about phones and see if any of it forms a bridge to the answer.

Once a particular fast solution method has been decided upon, there may be efficient ways and inefficient ways of following the method. For instance, a person might have many phone numbers in a list (home, work, holiday numbers). By pausing to think, "Is the person likely to be at home or at work or on holiday?", you might decide upon a sensible order in which to try each phone number.

Looking back at the x and the y algebra, a lot of people feel uneasy about algebra but it is an uncluttered way of playing with the facts of a problem. If you agree that clutter is a drain on concentration and can lead to mistakes being made then how clean is it to have a language of algebra to bring clarity (I claim!) to a question?

Having said that, a maths teacher might use real world situations to make students instinctively get a feel for the maths. The heads and tails game is an easy to imagine situation and is a nice way into the world of maths probability; but the empowering aspect is the abstraction of a principle (neatly expressed in algebra) so that you can do more than make predictions about games of heads and tails.

Maybe some brains cope with algebra better – in the same way that some people think in words, others in pictures. We do not all process information in one style.

That was a bit of a detour! I was trying to talk about starting with the goal and to frame a problem in terms of the goal.

Bolt on to that, “What are the eliminators?”: what can reduce the size of this problem?

And, when you reach a solution, see if a small test can challenge the accuracy of your solution. For instance, if the detective is seeking a man and the only relevant vehicle driver in the vicinity is a woman then he would look silly arresting her.

It is also good to ask a fresh pair of eyes to check your work. Often people see things we have somehow missed.

I know I am digressing but good thinking is a combination of behaviours.


Sweeps and Sleeps

Sometimes, the mental activity needed, to explain to someone what your problem is, forces you, unexpectedly, to state the solution out loud. You are revisiting old material but it is somehow connecting up in your mind in a new way that makes sense.

People say that they will ‘sleep on a problem’: they believe that coming back fresh to a problem will somehow bring them closer to solving it. When you consider the low Miller number of items we can juggle in our minds to problem solve, it may be that having a rest causes an absent item to be added into the mix of what you are using to solve a problem. There may be many reasons why it can work.

In computer coding teams, there is ‘code walk through’ where an author of code talks through what he/she has by talking to a colleague. This overlaps with the idea of “Don’t be a silo” and with this idea of seeing a subject in a new light as a result of revisiting it. Sometimes, one of the two people in the ‘code walk through’ spot a mistake in the code.

Another way in which bugs are found in code is through the good practice of documenting how a computer program works. In doing so, the author might challenge an idea which he had certainty about at the time the code was written but which does not seem valid at the present time.

The drinks machine or water cooler that causes staff to congregate and chat; or someone revisiting their work to write documentation, these are examples of how revisiting a system and/or its context can help you to think through the system better.


Causation and Correlation


If I press a button and a drink appears out of a vending machine then I might naturally assume that the pressing of the button causes the vending machine to provide me with a drink.

What if the button actually has nothing to do with the machine’s drink-serving function?

What if the button is relevant but only if a coin has been put into the machine?

If the button CAUSES the drink to appear then that is different to a random trend that each time I press the button, by some unrelated process, a drink has also happened to appear. Maybe it is programmed to release a drink once every 60 seconds; and, by chance, I am pressing the button once every 60 seconds. That is CORRELATION.

It is easy to be ‘caught out by correlation and to mistake it for causation.


Unexpected Results because of an UNKNOWN


An UNKNOWN is something that you did not consider in your evaluation of a problem. It might be something you know about but you did not include it in your thought process.

For example, you know that sometimes we get bad rain. You might make an investment based on a statistic that, over a 10 year period, a farming investment has a good rate of return; but you have no knowledge that this year will be very wet and spoil the crop; and you only intended to invest for one year.

So, the UNKNOWN is the bad weather; and the UNKNOWN makes a mess of your plans.

Add to that, you might have invested in a ten year farming plan; and then we get a ‘once a century’ freakish weather event – a year after the last ‘once a century’ freakish weather event happened. This shows that maths about the risk of something happening can be correct and wrong at the same time!

Since we have a strong desire to arrive at concrete solutions, I think that we have a tendency to ignore the possibility of UNKNOWNs. We can not visualise the things we do not know; so maybe the brain’s design has a frailty when it comes to weighing up UNKNOWNs.

An exercise to improve at this way of thinking might be to list all future possibilities of something everyday; e.g. Which cashier might serve you at the grocery store today? What is in your left pocket right now? How warm will the water coming out of your cold tap (faucet) be today? Which actors you already know of will be in the movie you are going to watch tonight?

With these questions, there is the chance of the reality being outside the set of possibilities you list; and that could raise your awareness of UNKNOWNs. Going back to the example of the jogger, when he turns into his driveway, that is a kind of UNKNOWN; or if his friend hails him and he stops to chat to his friend then that is an UNKNOWN too. The end of life of a device that you read or watch this tutorial from is an UNKNOWN; remember the Miller idea of a small number of ideas that we can work with well at any one time. If we added to that mental load the job of imagining UNKNOWNs possibilities then how would work get completed accurately? So an awareness of UNKNOWNs in general might typically be the insight which we bring to our thinking.

However, an ability to think of plausible UNKNOWNs is probably an empowering skill too.




If what is ‘unknown’ are the facts of the situation and one believes that one knows the facts of a system and its context then REDEFINITION is needed: redefinition of the system or its context.

So, if the jogger turns out to be someone running away from an assailant then there is redefinition if ‘jogger’ had the connotation of ‘person doing exercise’.

With problem solving, the barrier to a solution can be this lack of understanding of the nature of the problem.

Earlier, with UNKNOWNs, I was saying that it is good to know that one's list of choices (to evaluate) may be missing some choices.

In the 'Redefinition' situation, it is a different type of unknown information because it is not your store of choices that has possible unknowns but, rather, the system you are considering the choices of.


Applying Experience

Once a solution to a mathematics problem is found, that method of solving the problem can be memorised so that any similar problems can be solved in a similar way. For instance, there are some questions in which you have to find x, you are given four pieces of information – some of which are not needed to answer the question. If the examiners set a question of this style regularly then it makes sense to memorise general qualities of the solution.

So although problems can be solved by applying logic, it is good to use your experience to recall the method of solving a problem - when that way of solving the problem is appropriate.

But another type of memory store is a store of solution options. Yes, you have a method that will certainly get you the right answer with 5 minutes work but is there another solution in your tool kit which will solve the problem in even less time? It is an impressive mind that can access a store like that!


Improving Strategy

It is possible to play chess or any game for years without improving much. What is needed for there to be improvement is the habit of analysing a previous game and thinking about one’s weaknesses.

(And seeking out slightly superior competitors to accelerate that process.)

In chess, one might discover that a big weakness is the ease with which the player loses his queen. A conscious decision can be taken from then on to never make even the most simple looking move without thoroughly investigating whether the queen could be lost as a direct result of that move.

This idea of analysing your performance goes beyond games. The theme of this book is that your intelligence can be improved by analysing where you are going wrong, and focusing your mind on good habits. More generally, you can make yourself more efficient in everyday activities by analysing your methods and trying to make them better. In this way, you can find better solutions to problems.

One of my favourite comic heroes is Batman. Very typical in a Batman story is that he is bested in the first half of a story, he has a little think and then he comes back for the rematch but armed with the right tools. There is great satisfaction in outsmarting a foe or a system (the satisfaction of solving a tough puzzle).


More Than One Answer

Sometimes problems have more than one answer. So, there may be two phone numbers for ringing a friend who is at home. Maybe one phone is off the hook [most of this work is from a time when phones had hooks]. You might see that phone’s number in your address book, ring it, find that it is not getting through, and give up. What is better, is the habit of thinking, "Is there more than one answer?", and in that way looking at the address book to see if there is a second phone number.

In a mathematical situation, 2 times 2 is 4, and -2 times -2 is 4; when asked, "What when multiplied by itself gives the answer 4?", it is incomplete to give 2 as being the answer. Ie. because -2 is also an answer. If a person is in the habit of expecting only one solution then he is limiting his ability to be thorough.

If a person does not see his discipline as an integrated syllabus of approaches then he can only grasp at solutions in the current chapter he is studying. The great problem solvers see the big picture and thus have a much more powerful tool kit for solving problems.

In mathematics, people from different mathematical backgrounds might meet to attack one riddle. They acknowledge that, despite their own excellence, this pooling and integrating of abilities leads to solutions which one person could not realistically arrive at alone.

So, in talking about seeking more than one solution, I am overlapping the idea of using the ‘big picture’ as an enabler.


When Stuck, Check Assumptions

Some problems are hard to solve because you are not looking at the problem correctly. For instance, if you misread a mathematics question in which you are meant to convert one expression into another, it may be impossible if you copied the question down wrongly.

On a computer, you may be typing that the date of arrival of an order is 31st September and then the computer rejects the entry. You might think that the solution is to type September 31st or 31/9 or 9/31 but all the time the problem is that September only has thirty days. Here, the assumption that presentation is the obstacle to success leads to a blindness to the real problem.

Maybe it is worth checking how many days September has before checking the presentation of the date since, if the number of days is wrong then you might say that there is little point experimenting with the layout: there seems to be a directional flow to the things that might be causing the error.

But a counter-argument to that is “How much time does it cost me to explore avenue A rather than avenue B?”: a wrong format or a wrong date are both possible sources of failure; maybe prefer to check validity of the data primarily because it requires less thinking effort and because it should take less time than playing with several number formats.

Then again, the directional flow might lead to clarity of thought in diagnosing a problem: you don’t want to overcomplicate things.

I suppose you just weigh up your options and hope for the best!

Another example is where one’s key does not fit in your car at a car park - and it is because a car just like your own is parked near your own car and it is that car which you are trying to unlock. One might think that one’s key has been bent out of shape or that a vandal has damaged the lock by trying to force it open. The assumption that the car is one’s own causes a blindness to the actual problem.

What is required is the ability to seek proof for what seems at first glance to be obvious. Instead of thinking, "Why is my car not unlocking with my key?", there needs to be an abstraction to, "Why is this car not unlocking with this key - that is assuming that I am correct in my assumption that the lock has not yet been unlocked."

By abstracting and seeking proof for what is seemingly obvious, there is a better chance of reaching a solution and reaching it soon.

As an aside, it is perhaps good to limit use of the phrases “Of course” and “Obviously” from your vocabulary. They blinker your visibility of what might be the cause of an error.

If you are someone who thinks in words then you might be limited in the extent to which you naturally think deeper about what a word means. By seeing the visual system in your mind, there is more chance of a voice in your head saying, “Hey! There’s an inconsistency here!”. For instance, you are carrying a rote instruction in your head: “Give the man some oil and add wood to the fire.”; but you lose concentration and recall it as “Give the man some wood and add oil on the fire.”; without a challenger voice in one’s head that thinks beyond the instructions to the implications of adding oil to a fire, there is less chance of realising a mistake is being made here.


Use stores

The choice stores which you have read about already are intended to be drawn upon to help you to solve problems. There are useful lists in them, useful methods for solving problems, and useful habits for not making bad starting assumptions.


Other aspects of being clever




This work has been about the thinking process. Intelligence can cover other topics which are really valid but are going off piste a bit from the thinking process emphasis I am giving here.




I wrote a separate piece of work about doing well in exams.

It is strange that, despite someone being great at general thinking skills, they can still mess up an exam because of the semantics of what an essay question is asking the person to do.

Eg. ‘Compare’ is not the same as ‘Compare and contrast’. You might think the examiner will love you for not just doing what he asked but adding a little extra on top but the examiner gives you a lower score for ‘failing to understand the instruction’.

Exams are another world!


Where there is a clear overlap with intelligent thinking is the idea, in essays, of defining terms first before discussing the terms.

Moreover, to see that there is a preferred order to the presentation of ideas: to start off with building block definitions, construct arguments and only to delve into comparison of arguments once they have been properly defined.

And to finish off with some kind of conclusion that justifies why you took all that time saying all that stuff: to frame it in the context of the essay question you were given; or, with the topic of intelligent thinking, to have a framework underpinning your thoughts when you try to consider a proposition.

The caches I wrote about earlier are similar to an academic subject’s syllabus: a framework which you should know inside and out because it is going to be so helpful to you in organising your answer to an essay question.

The facts of the academic subject are also much like the cache I wrote about earlier: you take the time to learn something well because it offers so much benefit to you.




Someone can get 8 hours of sleep but it can be low quality sleep.

Good sleep matters and can have a massive effect on the work you are about to do and, also, how well you remember the work you did in the recent past.



Teaching Ability


I hope I have communicated my thoughts effectively.

Sometimes, you encounter teachers who are very clever but they have difficulty in communicating ideas simply.

I have a theory that a new idea should be introduced in a minimalist way: the more material you give people to elaborate on your lesson, the more distraction there is from the core point.

But when I say ‘minimalist way’, there needs to be enough material there for someone to be able to play with an idea and ‘make it their own’; and different students’ brains will prefer to receive a lesson in alternative styles.

The elaboration or sub branches and nuances can be explored on a future date in a fresh lesson.

In the same way that different people’s brains visualise the world differently, maybe the teacher should think about different ways of expressing an idea so that it imprints on different people differently. Maybe a dry principle that suits the mathematically minded could be followed by a dramatic story which illustrates the idea.

Also, a teacher needs an awareness that some people are not ‘keeping up’ well with the jargon being used. Generally, you can only communicate at speed with vocabulary if people have absorbed the meaning of that jargon well in the past.

An awareness of the timescale over which memory weakens is also good for a teacher. He or she can remind people of basic information and jargon so that it is easier for the student to mentally work with whatever fresh material builds on that underlying information.

The quality of being a good listener is desirable in a teacher: to listen enough to criticism to question if the person who is surely wrong is actually correct; and one’s own view of a system needs redefining.


More I could have written


I half wanted to add more material to this book about problem solving but I think that the habit of keeping a mistake diary, etc. will tend to unlock that material anyway.



Other work I have written


I wrote most of my material on intelligence and thinking before I had ever used the internet. I was very interested in the psychology of memory; and people really wanted a technique to memorise a phone number because electronic devices for doing so were pretty much unheard of. Nowadays, you just grab your smartphone! So a lot of my work is from another age!

Despite that, people need good memories to do well in exams and thus to better their positions in life. And, although reading about how to think can be a bit of a stale subject, the benefits of clean logic are invaluable.

The other piece I wrote was about exams and particular thoughts I had about them – having read various books on how to do well in exams; and having done a lot of education in my life.

The current memory work is at my web site.