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Propositional logic is a formal language that treats propositions as atomic units. A typical propositional logic word problem is as follows:

A, B, C, D are quarreling quadruplets. If A goes to the party, then B will not go. If C goes to the party, then B will not go. What is the largest possible number that will go to the party?

Logic is the study of valid reasoning. It is applied not only in studies but also in our day-to-day lives. Using simple logical reasoning and deduction, we are able to obtain information from a certain premise. Likewise, we can also verify or disprove statements.

We will be discussing ways of identifying common mistakes and how to avoid them. We will also talk about different proof techniques, such as using Venn diagrams and analogies so that you have a toolkit for solving logic word problems.

## What's wrong with this proof?

Proof by venn diagram, proof by analogy.

In this section, we will use familiar notations used in propositional logic. You might want to familiarize yourself with Propositional Logic first.

Like solving any other questions, we should always ask ourselves what we can and can't do when writing out our reasoning. The first step to learn how to solve propositional logic problems is to list out what can't be done or what is not a possibility so we can narrow down what the possible scenarios are. Remember that it is very easy to fall into an erroneous conclusion based on faulty reasoning. Take the statements below as an example, if the first statement is true, is the second statement also true?

$\begin{array} &\text{"If it's raining, then I can't play soccer".} &\text{"If I can't play soccer, then it's raining."} \end{array}$

It is pretty clear that the issue here is: there can be other reasons why I can't play soccer, which doesn't necessarily depend on the weather. If we make such simple errors in reasoning when the context is very clear, just imagine what will happen when you are less certain about statements that are more vague. In the next paragraph, we will be introduced to these errors.

## Converse and Inverse Errors

As a beginner, the most common mistake you can make is to assume that the converse and/or inverse of the original statement is also true. Take a look at the two sections below:

Introduction to Converse Error with erroneous reasoning:

Premise : If it's raining, then I can't play soccer. Conclusion : If I can't play soccer, then it's raining. Explanation : From the first statement, we are given a condition and a result: "raining" as a condition and "I can't play soccer" as a result. The entire premise is phrased in such a way that if the condition is fulfilled, then the result will occur. However, the conclusion shows that if the result is fulfilled, then the condition will occur. This does not make sense because it is not necessary for the condition to take place if the result occurs first. This is known as a converse error. In a general form, the argument for a converse error is as follows: If P occurs, then Q occurs. Q occurs. Therefore, P also occurs.

Introduction to Inverse Error with erroneous reasoning:

Premise : If it's raining, then I can't play soccer. Conclusion : If it's not raining, then I can play soccer. Explanation : From the first statement, we are given a condition and a result: "raining" as a condition and "I can't play soccer" as a result. The entire premise is phrased in such a way that if the condition is fulfilled, then the result will occur. However, the conclusion shows that if the condition does not occur, then the result does not occur either. This does not make sense because there can be other reasons/factors such that the result does occur. This is known as an inverse error. In a general form, the argument for an inverse error is as follows: If P occurs, then Q occurs. P does not occur. Therefore, Q also does not occur.

It may now be abundantly clear that it is easy to identify we've made an erroneous reasoning. However, what if the statements given appear more vague? This is the reason why we introduce the two errors above (converse error and inverse error) to show that not all wrong statements are easily identifiable. Simply put, the relationship between two events do not necessarily imply that one causes the other. In short, we are pointing out the common fact that "correlation does not imply causation".

Now that we have seen these mistakes first hand, let's do another example to remind ourselves that they are mistakes and we can hopefully avoid them in the future. Keep in mind that some of the converse/inverse statements can appear ridiculous but some do not.

We are given the following statement: If today is Sunday, then the weather is sunny.  $$\qquad \text{ (i)}$$ Write the inverse and converse of this statement. $$\qquad \text{(ii)}$$ Identify which of these statements you have made is not logical and explain why. $$\text{(i)}$$ Inverse and Converse Inverse: If today is not Sunday, then the weather is not sunny. Converse: If the weather is sunny, then today is Sunday. $$\text{(ii)}$$ Logical or Illogical Though they are the inverse and converse of the original statement, we must keep in mind that they might not necessarily be an error. However, there is no harm in checking whether they are correct or not. The inverse statement implies that the day has a direct relation on the weather being sunny or not, which is ludicrous because there can also be non-sunny days which do not fall on a Sunday. The converse statement implies that only if the weather is sunny then the day is Sunday, which is also ludicrous because they can also have a sunny weather on days not falling on a Sunday. $$_\square$$

## Pinpoint the exact error

Now that we can identify how the errors occur, let's take a step further and apply these techniques so that we can pinpoint exactly where the error occurs. Note that the easiest way to identify where the error arose is to convert logical statements into symbolic forms (like P implies Q). Let's try the following example.

Taking the long view on your education, you go to the Prestige Corporation and ask what you should do in college to be hired when you graduate. The personnel director replies that you will be hired only if you major in mathematics or computer science, get a $$\text{B}^\text{+}$$ average or better, and take accounting. You do, in fact, become a math major, get a $$\text{B}$$ average and take accounting. You return to Prestige Corporation, make a formal application, and are turned down. Did the personnel lie to you? Let's list down the requirements to be hired: $$\begin{array}{r r l} & \text{(i)} & \text{Major in mathematics or computer science}\\ & \text{(ii)} & \text{Get a } \text{B}^\text{+} \text{ average or better}\\ & \text{(iii)} & \text{Take accounting}\\ \end{array}$$ Since you became a math major, criteria $$\text{(i)}$$ is satisfied. Since you got a $$\text{B}$$ average instead of a $$\text{B}^\text{+}$$ average, criteria $$\text{(ii)}$$ is not satisfied. Since you took accounting, criteria $$\text{(iii)}$$ is satisfied. Since you did not satisfy all the criteria and were turned down, the personnel didn't lie to you. $$_\square$$

Now that you're familiar with writing out these statements and identifying possible errors, let's try another example that uses such a property!

A store has been raided by looter/s, who drove away in a car. Three well-known criminals Satvik, Krishna and Sharky are brought to the police station for questioning. Inspector Aditya of the police extracts the following facts:

$$(1)$$ None other than Satvik, Krishna and Sharky was involved in the robbery. $$(2)$$ Sharky never does a job without using Satvik (and possibly others) as accomplices. $$(3)$$ Krishna doesn't know how to drive.

Find the person who, in any case, is guilty.

## This problem is the part of my set "Is This What You Call Logic?"

Formal terminologies.

In the previous sections, we have learned the two most common errors that students will make when solving a logical reasoning problem. However, we did not formally touch on the terminologies for those terms: converse error and inverse error. Let's begin!

Contrapositive : A statement is logically equivalent to its contrapositive. The contrapositive negates both terms in an implication and switches their positions. For example, the contrapositive of "P implies Q" is the negation of Q implies the negation of P.

Converse : The converse switches the positions of the terms. The converse of "P implies Q" is "Q implies P".

"If and only if", sometimes written as iff and known as equivalence, is implication that works in both directions. "P if and only if Q" means that both "P implies Q" and "Q implies P".

Let's try a few examples that cover this area!

$$\text{ (i)}$$ Write down the contrapositive statement for $\text{"If you are human, then you have DNA."}$ $$\text{(ii)}$$ Write down the two if-then statements for $\text{"A polygon is a quadrilateral if and only if the polygon has 4 sides."}$ $$\text{ (i)}$$ contrapositive $$\qquad$$ If you do not have DNA, then you are not human. $$\text{(ii)}$$ if-then statements $$\qquad$$ If a polygon is a quadrilateral, then it has 4 sides. $$\qquad$$ If a polygon has 4 sides, then it is a quadrilateral. $$_\square$$

Simple, isn't it? Let's try some problems that apply the techniques we have learned above.

If Jeff spends 5 hours playing video games, then he cannot finish his math homework.

If Jeff finishes his math homework, then he will do well on his next math test.

Based on this information, which of the following is logically correct?

Selena, Jennifer and Miley wear a blue dress, yellow dress, and green dress in an unknown order. It is known that:

1) If Selena wears blue, then Jennifer wears green. 2) If Selena wears yellow, then Miley wears green. 3) If Jennifer does not wear yellow, then Miley wears blue.

What is the color of the dress Selena is wearing?

Now that we have mastered these techniques, let's move on to the following section for other cool proof techniques!

In this section, we will be applying some basic rules of set notations. You might want to familiarize yourself with sets and Venn diagram first.

In the previous section, we have learned the most common ways in identifying and pinpointing errors. In this section, we will apply the use of Venn Diagram as an alternative proof in solving logical reasoning problems. But what's the benefit of this? Well, it's simple: We do not need to verbalize these statements and we can use the visual aids to guide us to solve these problems.

## Recap of set notations and Venn diagram 

Let's do a brief recap for the application of Venn Diagrams by taking the following as an explicit example:

Consider $$W,X,Y,Z$$ as sets, each with their own elements in them. Then by interpreting the Venn Diagram, we can obtain information like:

• All elements in set $$W$$ is in set $$Y$$.
• All elements in set $$X$$ is in set $$Z$$.
• Not all elements in set $$Y$$ is in set $$W$$.

## How to use Venn diagrams to solve a logical word problem 

Let us consider the following statements and deduce whether the conclusion is true or false by Venn diagram:

True or false?  It is given that all birds have wings. All chickens are birds. Therefore, all chickens have wings. Explanation : By Venn diagram, the statement "A chicken is a bird." implies that the set "all chickens" is a subset of "all birds." Thus we can say that all chickens have the same characteristics as a bird. Because it is given that all birds have wings (a characteristic), all chickens have wings too. Thus the conclusion is correct. Note : We should keep in mind that this only works if the premise is true. For example, if we replace the word "wings" by "forearms" in the first statement, then the conclusion of "All chickens have forearms." will inevitably be true despite its ridiculous claim.

Food for thought : If all phones have batteries and I have a phone, does it mean that my phone has a battery?

## Careful! There are other ways of drawing out Venn diagrams!

Though it may appear very simple to set up a Venn diagram, the setup may not necessarily be unique. Let us consider a revised version of the statements above and deduce whether the conclusion is true or false by Venn diagram.

True or false?  It is given that all birds have wings. All chickens are birds. Therefore, all birds are chickens. Explanation : By Venn diagram, the statement "A chicken is a bird" implies that the set "all chickens" is a subset of "all birds." Thus we can say that all chickens have the same characteristics as a bird. However, it is not necessarily true that all birds share the same characteristics of a chicken. (Sounds familiar? It's converse error.) So the claim "All birds are chicken" must be false. Note : To fix the conclusion, you should say " Some birds are chickens" instead of " All birds are chickens."

Now that we know the fundamental applications of proof by Venn diagram, let's apply these knowledge we learned on the following examples:

True or False?

$$\quad$$ All pangs are pings. $$\quad$$ Some pings are pongs. $$\quad$$ Therefore, some pangs are pongs.

Johannes has several written publications on his bookshelf. Albert notices that all of the comics are paperback books, and that some of the paperback books are manga. Are all comics manga?

## Image Credit : Wikimedia Johannes Jansson

Isn't the proof by Venn diagram fun? You don't need to use actual words to formalize these statements. Looks very unusual, right? But it works. Speaking of unusual, is it possible to solve these logical statements if we were to spice things up by dramatizing out the statements? Yes, we can! Proof by analogy is another proof technique to solve logical problems. See the following section:

## How do we solve texts that are seemingly hard to decipher?

All pangs are pings. Some pings are pongs. Therefore some pangs are pongs.

Consider the logical statements given above. Since we can't relate to or identify what pangs, pings, or pongs are, it will appear that these terms are vague or all too similar. How are we supposed to solve problems like this if we have little to no clue to what is going on? Well, proof by analogy will be useful here: this is when we dramatize or caricaturize the terms used.

For example, we may call pangs as humans, pings as apes, and pongs as gorillas. With these new terms, we are able to visualize what they are. Rewriting them into the original 3 statements shows that

$\text{All humans are apes. Some apes are gorillas. Therefore some humans are gorillas.}$

So the given conclusion is wrong because of the ridiculousness of the conclusion "Some humans are gorillas."

However, an important question to ask is why this works. This is too good to be true, right? Or, are we running into some wrong argument? Why does this work?

## Explanation of how this works

The reason why proof by analogy works is because we make an inference that if the objects have multiple similar characteristics, and it is given that you know one of them have an extra characteristics (call it X), then it is not a bad inference to conclude that the other object shares that same characteristic X.

To put it short, the generalized/structured form for proof by analogy is:

P and Q has similar properties $$x_1, x_2, x_3, \ldots, x_n$$. We know that P has a further property $$y$$. Therefore, Q probably has property $$y$$ too.

Now let's try a modified version of the ping-pang-pong question from earlier!

True or false?  $$\qquad$$ All yangs are yengs and yings. $$\qquad$$ Some yengs are yings. $$\qquad$$ Then, all yengs are yangs. This is false. Let "yangs" be defined as "pets", "yengs" as "tigers", and "yings" as "cats". So it is true (or at least still reasonable) that all pets are cats and tigers and that some tigers are cats. But it is not true that all tigers are pets. $$_\square$$ Note: The reason why proof by analogy works best here is because we couldn't label or identify any characteristics for yangs, yengs, and yings. Therefore, a sensible approach is to prove by analogy.

Now that you're ready to solve logical problems by analogy, let's try to solve the following problem again, but this time by analogy!

## Suitability of analogy

Notice from the previous section that we've mentioned that "Q probably has property $$y$$ too." instead of "Q definitely has a property $$y$$ too." This is because the argument may provide what appears to be the right evidence, but the conclusion does not always follow. This subsection explains why this proof (arguemnt) might not always work.

Though it is true that we highlight or amplify parallel characteristics, the differences between things can often overwhelm their similiarities. One might note that it is always possible to extend an analogy to the point of absurdity. For illustration, take the following famous "Information Argument":

DNA is a code. A code requires an intelligence. Therefore, DNA comes from an intelligence.

Yes, this will sound completely logical if we apply the implications approach. That is, it's P implies Q and Q implies R, so P implies R. However, the argument here is not valid because the statement "DNA is a code." is purely an analogy and thus it is not an entirely accurate statement to begin with. Thus, we have started with a wrong premise. So the merits of analogy do not hold. This is further explained in Analyzing arguments from analogy .

We can see that proof by analogy is very useful and can also be used to make incorrect conclusions. Thus, one must be careful in labeling certain characteristics when using this method. Let's see the following examples to see how the proof by analogy backfires:

True or false?  All squares and rectangless are convex, have four sides and form right angles at their vertices. All squares have sides of the same length. Therefore, all rectangles have sides of the same length. This is obviously false because by definition, all rectangles do not have sides of the same length, but only squares have sides of the same length. We make the wrong conclusion that rectangles also have this characteristic because it is known previously that both share a number of characteristics. $$_\square$$

True or false?

It is given that Amy, Bernadette and Penny are good friends of Sheldon and Leonard.

Leslie is a good friend of Leonard.

Therefore, Leslie is a friend of Sheldon as well.

## Image Credit: Wikimedia TBBT. No copyright infringement intended.

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## The Importance of Logic and Critical Thinking

"Critical thinking is a desire to seek, patience to doubt, fondness to meditate, slowness to assert, readiness to consider, carefulness to dispose and set in order; and hatred for every kind of imposture." - Francis Bacon (1605)

As parents, we are tasked with instilling a plethora of different values into our children. While some parents in the world choose to instill a lack of values in their kids, those of us that don't want our children growing up to be criminals and various misfits try a bit harder. Values and morality are one piece of the pie. These are important things to mold into a child's mind, but there are also other items in life to focus on as well. It starts with looking both ways to cross the street and either progresses from there, or stops.

If you stopped explaining the world to your children after they learned to cross the street, then perhaps you should stop reading and go back to surfing for funny pictures of cats. I may use some larger words that you might not understand, making you angry and causing you to leave troll-like comments full of bad grammar and moronic thought processes. However, if you looked at the crossing the street issue as I did – as a logical problem with cause and effect and a probable solution – then carry on. You are my target audience.

Or perhaps the opposite is true, as the former are the people that could benefit from letting some critical thinking into their lives. So what exactly is critical thinking? This bit by Linda Elder in a paper on CriticalThinking.org pretty much sums it up:

Through critical thinking, as I understand it, we acquire a means of assessing and upgrading our ability to judge well. It enables us to go into virtually any situation and to figure out the logic of whatever is happening in that situation. It provides a way for us to learn from new experiences through the process of continual self-assessment. Critical thinking, then, enables us to form sound beliefs and judgments, and in doing so, provides us with a basis for a 'rational and reasonable' emotional life. — Inquiry: Critical Thinking Across the Disciplines, Winter, 1996. Vol. XVI, No. 2.

The rationality of the world is what is at risk. Too many people are taken advantage of because of their lack of critical thinking, logic and deductive reasoning. These same people are raising children without these same skills, creating a whole new generation of clueless people.

Lauren Goode

Angela Watercutter

To wit, a personal tale of deductive reasoning:

Recently I needed a new transmission for the family van. The warranty on the power train covers the transmission up to 100,000 miles. The van has around 68,000 miles on it. Therefore, even the logic-less dimwit could easily figure that the transmission was covered. Well, this was true until the dealership told me that it wasn't, stating that because we didn't get the scheduled transmission service (which is basically a fluid change) at 30,000 and 60,000 miles the warranty was no longer valid. Now, there are many people that would argue this point, but many more that would shrug, panic, and accept the full cost of repairs.

I read the warranty book. I had a receipt that said the fluid was checked at 60,000 but not replaced. A friend on Twitter pointed out the fact that they were using 100,000 mile transmission fluid. So logically, the fluid would not have to be replaced under 100,000 miles if it wasn't needed, right? So why the stipulation that it needed to be replaced at 60,000 and the loose assumption that not doing that would void the warranty? So I asked the warranty guy to show me in the book where the two items are related. Where it explicitly says that if you don't get the service, the transmission isn't covered. There were portions where it said the service was recommended, but never connecting to actual repairs. Finally the warranty guy shrugged, admitted I was right and said the service was covered.

In this case, valid logic equaled truth and a sound argument. I used very simple reasoning and logic to determine that I was being inadvertently screwed. I say "inadvertently" because I truly believe based on their behavior that they were not intentionally trying to screw me. They believed the two items were related, they had had this argument many times before and were not prepared to be questioned. While both the service manager and the warranty guy seemed at least junior college educated, proving my argument to them took longer than it should have between three adults.

However, valid logic does not always guarantee truth or a sound argument. This is where it gets a little funky. Valid logic is when the structure of logic is correct in the way of syntax and semantics rather than truth. Truth comes from deductive reasoning of said logic. For example:

All transmissions are covered parts. All covered parts are free. Therefore, all transmissions are free. This logic is technically valid, and if the premises are true, then of course the conclusion must be true. You can see here however that it's not always true, though in some situations it could be. While the logic is valid, not all transmissions are free, only those covered by the warranty. So based on that, saying all transmissions are free is not sound logic.

To take it one step further:

All Daleks are brown. Some brown things are Cylons . Therefore, some Daleks are Cylons. Sci-fi fan or not, you probably know that this is not true. The basic lesson here is that, while the logic above might seem valid because of the structure of the statement, it takes a further understanding to figure out why it's not necessarily true: That is, based on the first two statements it's possible that some Daleks are Cylons, but it's not logically concludable. That's where deductive reasoning comes on top of the logic. The underlying lesson here is not to immediately assume everything you read or are told is true, something all children need to and should learn.

This is the direct lesson that needs to be passed on to our children: that of not accepting the immediately visible logic. While not all problems are complex enough to require the scientific method, some of them need some deduction to determine if they are true. Take the example above — how many kids would immediately be satisfied with the false conclusion? Sure, it's a bit geeky with the examples, but switch out bears for Daleks and puppies for Cylons. That makes it easier, and takes the actual research out of it (to find out what Daleks and Cylons are respectively) but many people would just accept that in fact some bears are puppies, if presented with this problem in the context of a textbook or word problem.

Maybe I'm being paranoid or thinking too doomsday, whatever, but I think this is an epidemic. Children are becoming lazier and not as self sufficient because their parents have a problem with watching a three year old cry after they tell her to remove her own jeans, or ask her to put away her own toys (yes, organizational logic falls under the main topic). These are the same parents who do their kid's science project while the kid is playing video games. These kids grow up lacking the simple problem solving skills that make navigating life much easier. Remember when you were growing up and you had the plastic stacking toys ? Well, instead of toys for early development like that, parents are just plopping their kids down in front of the television. While there is some educational type programming on television, it's just not the same as hands-on experience.

My father is an engineer, and he taught me logic and reasoning by making me solve simple, then complex, problems on my own. Or at least giving me the opportunity to solve them on my own. This helped develop critical thinking and problem solving skills, something a lot of children lack these days. Too often I see children that are not allowed to solve problems on their own; instead their parents simply do it for them without argument or discussion. Hell, I am surrounded by adults every day that are unable to solve simple problems, instead choosing to immediately ask me at which point I have to fill the role that their parents never did and – knowing the solution – tell them to solve it themselves, or at least try first.

One of the things I like to work on with my kids is math. There is nothing that teaches deductive reasoning and logic better than math word problems. They are at the age where basic algebra can come into play, which sharpens their reasoning skills because they start to view real world issues with algebraic solutions. Another thing is logic puzzles , crossword puzzles and first person shooters. Actually, not that last one. That's just the reward.

Since I weeded out the folks that don't teach their kids logic in the first two paragraphs, as representatives of the real world it's up to the rest of us to spread the knowledge. It won't be easy. The best thing we can do is teach these thought processes to our children, so that they may look at other children with looks of bewilderment when other children are unable to solve simple tasks. Hopefully, they will not simply do the task for them, but teach them to think. I'm not saying we need to build a whole new generation of project managers and analysts, but it would be better than a generation of task-oriented mindless office drones with untied shoelaces, shoving on a door at the Midvale School for the Gifted .

h/t to @aubreygirl22 for the logical conversation. Image: Flickr user William Notowidagdo. Used under Creative Commons License.

Scott Gilbertson

Aarian Marshall

Jason Parham

Jennifer M. Wood

## How to master the seven-step problem-solving process

In this episode of the McKinsey Podcast , Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.

Podcast transcript

Simon London: Hello, and welcome to this episode of the McKinsey Podcast , with me, Simon London. What’s the number-one skill you need to succeed professionally? Salesmanship, perhaps? Or a facility with statistics? Or maybe the ability to communicate crisply and clearly? Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

Looked at this way, it’s no surprise that McKinsey takes problem solving very seriously, testing for it during the recruiting process and then honing it, in McKinsey consultants, through immersion in a structured seven-step method. To discuss the art of problem solving, I sat down in California with McKinsey senior partner Hugo Sarrazin and also with Charles Conn. Charles is a former McKinsey partner, entrepreneur, executive, and coauthor of the book Bulletproof Problem Solving: The One Skill That Changes Everything [John Wiley & Sons, 2018].

Charles and Hugo, welcome to the podcast. Thank you for being here.

Hugo Sarrazin: Our pleasure.

Charles Conn: It’s terrific to be here.

Simon London: Problem solving is a really interesting piece of terminology. It could mean so many different things. I have a son who’s a teenage climber. They talk about solving problems. Climbing is problem solving. Charles, when you talk about problem solving, what are you talking about?

Charles Conn: For me, problem solving is the answer to the question “What should I do?” It’s interesting when there’s uncertainty and complexity, and when it’s meaningful because there are consequences. Your son’s climbing is a perfect example. There are consequences, and it’s complicated, and there’s uncertainty—can he make that grab? I think we can apply that same frame almost at any level. You can think about questions like “What town would I like to live in?” or “Should I put solar panels on my roof?”

You might think that’s a funny thing to apply problem solving to, but in my mind it’s not fundamentally different from business problem solving, which answers the question “What should my strategy be?” Or problem solving at the policy level: “How do we combat climate change?” “Should I support the local school bond?” I think these are all part and parcel of the same type of question, “What should I do?”

I’m a big fan of structured problem solving. By following steps, we can more clearly understand what problem it is we’re solving, what are the components of the problem that we’re solving, which components are the most important ones for us to pay attention to, which analytic techniques we should apply to those, and how we can synthesize what we’ve learned back into a compelling story. That’s all it is, at its heart.

I think sometimes when people think about seven steps, they assume that there’s a rigidity to this. That’s not it at all. It’s actually to give you the scope for creativity, which often doesn’t exist when your problem solving is muddled.

Simon London: You were just talking about the seven-step process. That’s what’s written down in the book, but it’s a very McKinsey process as well. Without getting too deep into the weeds, let’s go through the steps, one by one. You were just talking about problem definition as being a particularly important thing to get right first. That’s the first step. Hugo, tell us about that.

Hugo Sarrazin: It is surprising how often people jump past this step and make a bunch of assumptions. The most powerful thing is to step back and ask the basic questions—“What are we trying to solve? What are the constraints that exist? What are the dependencies?” Let’s make those explicit and really push the thinking and defining. At McKinsey, we spend an enormous amount of time in writing that little statement, and the statement, if you’re a logic purist, is great. You debate. “Is it an ‘or’? Is it an ‘and’? What’s the action verb?” Because all these specific words help you get to the heart of what matters.

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Simon London: So this is a concise problem statement.

Hugo Sarrazin: Yeah. It’s not like “Can we grow in Japan?” That’s interesting, but it is “What, specifically, are we trying to uncover in the growth of a product in Japan? Or a segment in Japan? Or a channel in Japan?” When you spend an enormous amount of time, in the first meeting of the different stakeholders, debating this and having different people put forward what they think the problem definition is, you realize that people have completely different views of why they’re here. That, to me, is the most important step.

Charles Conn: I would agree with that. For me, the problem context is critical. When we understand “What are the forces acting upon your decision maker? How quickly is the answer needed? With what precision is the answer needed? Are there areas that are off limits or areas where we would particularly like to find our solution? Is the decision maker open to exploring other areas?” then you not only become more efficient, and move toward what we call the critical path in problem solving, but you also make it so much more likely that you’re not going to waste your time or your decision maker’s time.

How often do especially bright young people run off with half of the idea about what the problem is and start collecting data and start building models—only to discover that they’ve really gone off half-cocked.

Hugo Sarrazin: Yeah.

Charles Conn: And in the wrong direction.

Simon London: OK. So step one—and there is a real art and a structure to it—is define the problem. Step two, Charles?

Charles Conn: My favorite step is step two, which is to use logic trees to disaggregate the problem. Every problem we’re solving has some complexity and some uncertainty in it. The only way that we can really get our team working on the problem is to take the problem apart into logical pieces.

What we find, of course, is that the way to disaggregate the problem often gives you an insight into the answer to the problem quite quickly. I love to do two or three different cuts at it, each one giving a bit of a different insight into what might be going wrong. By doing sensible disaggregations, using logic trees, we can figure out which parts of the problem we should be looking at, and we can assign those different parts to team members.

Simon London: What’s a good example of a logic tree on a sort of ratable problem?

Charles Conn: Maybe the easiest one is the classic profit tree. Almost in every business that I would take a look at, I would start with a profit or return-on-assets tree. In its simplest form, you have the components of revenue, which are price and quantity, and the components of cost, which are cost and quantity. Each of those can be broken out. Cost can be broken into variable cost and fixed cost. The components of price can be broken into what your pricing scheme is. That simple tree often provides insight into what’s going on in a business or what the difference is between that business and the competitors.

If we add the leg, which is “What’s the asset base or investment element?”—so profit divided by assets—then we can ask the question “Is the business using its investments sensibly?” whether that’s in stores or in manufacturing or in transportation assets. I hope we can see just how simple this is, even though we’re describing it in words.

When I went to work with Gordon Moore at the Moore Foundation, the problem that he asked us to look at was “How can we save Pacific salmon?” Now, that sounds like an impossible question, but it was amenable to precisely the same type of disaggregation and allowed us to organize what became a 15-year effort to improve the likelihood of good outcomes for Pacific salmon.

Simon London: Now, is there a danger that your logic tree can be impossibly large? This, I think, brings us onto the third step in the process, which is that you have to prioritize.

Charles Conn: Absolutely. The third step, which we also emphasize, along with good problem definition, is rigorous prioritization—we ask the questions “How important is this lever or this branch of the tree in the overall outcome that we seek to achieve? How much can I move that lever?” Obviously, we try and focus our efforts on ones that have a big impact on the problem and the ones that we have the ability to change. With salmon, ocean conditions turned out to be a big lever, but not one that we could adjust. We focused our attention on fish habitats and fish-harvesting practices, which were big levers that we could affect.

People spend a lot of time arguing about branches that are either not important or that none of us can change. We see it in the public square. When we deal with questions at the policy level—“Should you support the death penalty?” “How do we affect climate change?” “How can we uncover the causes and address homelessness?”—it’s even more important that we’re focusing on levers that are big and movable.

Simon London: Let’s move swiftly on to step four. You’ve defined your problem, you disaggregate it, you prioritize where you want to analyze—what you want to really look at hard. Then you got to the work plan. Now, what does that mean in practice?

Hugo Sarrazin: Depending on what you’ve prioritized, there are many things you could do. It could be breaking the work among the team members so that people have a clear piece of the work to do. It could be defining the specific analyses that need to get done and executed, and being clear on time lines. There’s always a level-one answer, there’s a level-two answer, there’s a level-three answer. Without being too flippant, I can solve any problem during a good dinner with wine. It won’t have a whole lot of backing.

Simon London: Not going to have a lot of depth to it.

Hugo Sarrazin: No, but it may be useful as a starting point. If the stakes are not that high, that could be OK. If it’s really high stakes, you may need level three and have the whole model validated in three different ways. You need to find a work plan that reflects the level of precision, the time frame you have, and the stakeholders you need to bring along in the exercise.

Charles Conn: I love the way you’ve described that, because, again, some people think of problem solving as a linear thing, but of course what’s critical is that it’s iterative. As you say, you can solve the problem in one day or even one hour.

Charles Conn: We encourage our teams everywhere to do that. We call it the one-day answer or the one-hour answer. In work planning, we’re always iterating. Every time you see a 50-page work plan that stretches out to three months, you know it’s wrong. It will be outmoded very quickly by that learning process that you described. Iterative problem solving is a critical part of this. Sometimes, people think work planning sounds dull, but it isn’t. It’s how we know what’s expected of us and when we need to deliver it and how we’re progressing toward the answer. It’s also the place where we can deal with biases. Bias is a feature of every human decision-making process. If we design our team interactions intelligently, we can avoid the worst sort of biases.

Simon London: Here we’re talking about cognitive biases primarily, right? It’s not that I’m biased against you because of your accent or something. These are the cognitive biases that behavioral sciences have shown we all carry around, things like anchoring, overoptimism—these kinds of things.

Both: Yeah.

Charles Conn: Availability bias is the one that I’m always alert to. You think you’ve seen the problem before, and therefore what’s available is your previous conception of it—and we have to be most careful about that. In any human setting, we also have to be careful about biases that are based on hierarchies, sometimes called sunflower bias. I’m sure, Hugo, with your teams, you make sure that the youngest team members speak first. Not the oldest team members, because it’s easy for people to look at who’s senior and alter their own creative approaches.

Hugo Sarrazin: It’s helpful, at that moment—if someone is asserting a point of view—to ask the question “This was true in what context?” You’re trying to apply something that worked in one context to a different one. That can be deadly if the context has changed, and that’s why organizations struggle to change. You promote all these people because they did something that worked well in the past, and then there’s a disruption in the industry, and they keep doing what got them promoted even though the context has changed.

Simon London: Right. Right.

Hugo Sarrazin: So it’s the same thing in problem solving.

Charles Conn: And it’s why diversity in our teams is so important. It’s one of the best things about the world that we’re in now. We’re likely to have people from different socioeconomic, ethnic, and national backgrounds, each of whom sees problems from a slightly different perspective. It is therefore much more likely that the team will uncover a truly creative and clever approach to problem solving.

Simon London: Let’s move on to step five. You’ve done your work plan. Now you’ve actually got to do the analysis. The thing that strikes me here is that the range of tools that we have at our disposal now, of course, is just huge, particularly with advances in computation, advanced analytics. There’s so many things that you can apply here. Just talk about the analysis stage. How do you pick the right tools?

Charles Conn: For me, the most important thing is that we start with simple heuristics and explanatory statistics before we go off and use the big-gun tools. We need to understand the shape and scope of our problem before we start applying these massive and complex analytical approaches.

Simon London: Would you agree with that?

Hugo Sarrazin: I agree. I think there are so many wonderful heuristics. You need to start there before you go deep into the modeling exercise. There’s an interesting dynamic that’s happening, though. In some cases, for some types of problems, it is even better to set yourself up to maximize your learning. Your problem-solving methodology is test and learn, test and learn, test and learn, and iterate. That is a heuristic in itself, the A/B testing that is used in many parts of the world. So that’s a problem-solving methodology. It’s nothing different. It just uses technology and feedback loops in a fast way. The other one is exploratory data analysis. When you’re dealing with a large-scale problem, and there’s so much data, I can get to the heuristics that Charles was talking about through very clever visualization of data.

You test with your data. You need to set up an environment to do so, but don’t get caught up in neural-network modeling immediately. You’re testing, you’re checking—“Is the data right? Is it sound? Does it make sense?”—before you launch too far.

Simon London: You do hear these ideas—that if you have a big enough data set and enough algorithms, they’re going to find things that you just wouldn’t have spotted, find solutions that maybe you wouldn’t have thought of. Does machine learning sort of revolutionize the problem-solving process? Or are these actually just other tools in the toolbox for structured problem solving?

Charles Conn: It can be revolutionary. There are some areas in which the pattern recognition of large data sets and good algorithms can help us see things that we otherwise couldn’t see. But I do think it’s terribly important we don’t think that this particular technique is a substitute for superb problem solving, starting with good problem definition. Many people use machine learning without understanding algorithms that themselves can have biases built into them. Just as 20 years ago, when we were doing statistical analysis, we knew that we needed good model definition, we still need a good understanding of our algorithms and really good problem definition before we launch off into big data sets and unknown algorithms.

Simon London: Step six. You’ve done your analysis.

Charles Conn: I take six and seven together, and this is the place where young problem solvers often make a mistake. They’ve got their analysis, and they assume that’s the answer, and of course it isn’t the answer. The ability to synthesize the pieces that came out of the analysis and begin to weave those into a story that helps people answer the question “What should I do?” This is back to where we started. If we can’t synthesize, and we can’t tell a story, then our decision maker can’t find the answer to “What should I do?”

Simon London: But, again, these final steps are about motivating people to action, right?

Charles Conn: Yeah.

Simon London: I am slightly torn about the nomenclature of problem solving because it’s on paper, right? Until you motivate people to action, you actually haven’t solved anything.

Charles Conn: I love this question because I think decision-making theory, without a bias to action, is a waste of time. Everything in how I approach this is to help people take action that makes the world better.

Simon London: Hence, these are absolutely critical steps. If you don’t do this well, you’ve just got a bunch of analysis.

Charles Conn: We end up in exactly the same place where we started, which is people speaking across each other, past each other in the public square, rather than actually working together, shoulder to shoulder, to crack these important problems.

Simon London: In the real world, we have a lot of uncertainty—arguably, increasing uncertainty. How do good problem solvers deal with that?

Hugo Sarrazin: At every step of the process. In the problem definition, when you’re defining the context, you need to understand those sources of uncertainty and whether they’re important or not important. It becomes important in the definition of the tree.

You need to think carefully about the branches of the tree that are more certain and less certain as you define them. They don’t have equal weight just because they’ve got equal space on the page. Then, when you’re prioritizing, your prioritization approach may put more emphasis on things that have low probability but huge impact—or, vice versa, may put a lot of priority on things that are very likely and, hopefully, have a reasonable impact. You can introduce that along the way. When you come back to the synthesis, you just need to be nuanced about what you’re understanding, the likelihood.

Often, people lack humility in the way they make their recommendations: “This is the answer.” They’re very precise, and I think we would all be well-served to say, “This is a likely answer under the following sets of conditions” and then make the level of uncertainty clearer, if that is appropriate. It doesn’t mean you’re always in the gray zone; it doesn’t mean you don’t have a point of view. It just means that you can be explicit about the certainty of your answer when you make that recommendation.

Simon London: So it sounds like there is an underlying principle: “Acknowledge and embrace the uncertainty. Don’t pretend that it isn’t there. Be very clear about what the uncertainties are up front, and then build that into every step of the process.”

Hugo Sarrazin: Every step of the process.

Simon London: Yeah. We have just walked through a particular structured methodology for problem solving. But, of course, this is not the only structured methodology for problem solving. One that is also very well-known is design thinking, which comes at things very differently. So, Hugo, I know you have worked with a lot of designers. Just give us a very quick summary. Design thinking—what is it, and how does it relate?

Hugo Sarrazin: It starts with an incredible amount of empathy for the user and uses that to define the problem. It does pause and go out in the wild and spend an enormous amount of time seeing how people interact with objects, seeing the experience they’re getting, seeing the pain points or joy—and uses that to infer and define the problem.

Simon London: Problem definition, but out in the world.

Hugo Sarrazin: With an enormous amount of empathy. There’s a huge emphasis on empathy. Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don’t know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there’s a lot of divergent thinking initially. That’s slightly different, versus the prioritization, but not for long. Eventually, you need to kind of say, “OK, I’m going to converge again.” Then you go and you bring things back to the customer and get feedback and iterate. Then you rinse and repeat, rinse and repeat. There’s a lot of tactile building, along the way, of prototypes and things like that. It’s very iterative.

Simon London: So, Charles, are these complements or are these alternatives?

Charles Conn: I think they’re entirely complementary, and I think Hugo’s description is perfect. When we do problem definition well in classic problem solving, we are demonstrating the kind of empathy, at the very beginning of our problem, that design thinking asks us to approach. When we ideate—and that’s very similar to the disaggregation, prioritization, and work-planning steps—we do precisely the same thing, and often we use contrasting teams, so that we do have divergent thinking. The best teams allow divergent thinking to bump them off whatever their initial biases in problem solving are. For me, design thinking gives us a constant reminder of creativity, empathy, and the tactile nature of problem solving, but it’s absolutely complementary, not alternative.

Simon London: I think, in a world of cross-functional teams, an interesting question is do people with design-thinking backgrounds really work well together with classical problem solvers? How do you make that chemistry happen?

Hugo Sarrazin: Yeah, it is not easy when people have spent an enormous amount of time seeped in design thinking or user-centric design, whichever word you want to use. If the person who’s applying classic problem-solving methodology is very rigid and mechanical in the way they’re doing it, there could be an enormous amount of tension. If there’s not clarity in the role and not clarity in the process, I think having the two together can be, sometimes, problematic.

The second thing that happens often is that the artifacts the two methodologies try to gravitate toward can be different. Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they’ll bring an example, a thing, and that feels different. Then you spend your time differently to achieve those two end products, so that’s another source of friction.

Now, I still think it can be an incredibly powerful thing to have the two—if there are the right people with the right mind-set, if there is a team that is explicit about the roles, if we’re clear about the kind of outcomes we are attempting to bring forward. There’s an enormous amount of collaborativeness and respect.

Simon London: But they have to respect each other’s methodology and be prepared to flex, maybe, a little bit, in how this process is going to work.

Hugo Sarrazin: Absolutely.

Simon London: The other area where, it strikes me, there could be a little bit of a different sort of friction is this whole concept of the day-one answer, which is what we were just talking about in classical problem solving. Now, you know that this is probably not going to be your final answer, but that’s how you begin to structure the problem. Whereas I would imagine your design thinkers—no, they’re going off to do their ethnographic research and get out into the field, potentially for a long time, before they come back with at least an initial hypothesis.

## Want better strategies? Become a bulletproof problem solver

Hugo Sarrazin: That is a great callout, and that’s another difference. Designers typically will like to soak into the situation and avoid converging too quickly. There’s optionality and exploring different options. There’s a strong belief that keeps the solution space wide enough that you can come up with more radical ideas. If there’s a large design team or many designers on the team, and you come on Friday and say, “What’s our week-one answer?” they’re going to struggle. They’re not going to be comfortable, naturally, to give that answer. It doesn’t mean they don’t have an answer; it’s just not where they are in their thinking process.

Simon London: I think we are, sadly, out of time for today. But Charles and Hugo, thank you so much.

Charles Conn: It was a pleasure to be here, Simon.

Hugo Sarrazin: It was a pleasure. Thank you.

Simon London: And thanks, as always, to you, our listeners, for tuning into this episode of the McKinsey Podcast . If you want to learn more about problem solving, you can find the book, Bulletproof Problem Solving: The One Skill That Changes Everything , online or order it through your local bookstore. To learn more about McKinsey, you can of course find us at McKinsey.com.

Charles Conn is CEO of Oxford Sciences Innovation and an alumnus of McKinsey’s Sydney office. Hugo Sarrazin is a senior partner in the Silicon Valley office, where Simon London, a member of McKinsey Publishing, is also based.

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## Critical Thinking: Problem-Solving

Problem-solving is a central business skill, and yet it's the one many people struggle with most. This course will show you how to apply critical thinking techniques to common business examples, avoid misunderstandings, and get at the root of any problem.

Logical thinking is the most valuable asset any business professional can have. That's why logic trees are such a valuable tool—they can help you identify a problem, break it down, and build it back up to a solution.

## MECE Principle

Using the MECE principle can help ensure you categorize without gaps or overlaps. Check out this course from GLOBIS Unlimited for a practical demonstration of how it works!

If you work in business—any aspect of a business, from R&D to sales to back-office data entry—you’ve probably experienced an unpleasant surprise or two. Not every marketing campaign boosts sales the way we want. Not every event has the turnout we hope for. Not every promotion we expect comes our way.

Particularly with problems that have fiscal casualties, you’ll want to do a little analysis and find out what happened. And one of the absolute best tools to apply to your analysis is critical thinking.

Critical thinking applies logic to solve problems systematically. According to the World Economic Forum’s “Future of Jobs Report” in 2020, “The top skills . . . which employers see as rising in prominence in the lead up to 2025 include . . . critical thinking and analysis, as well as problem-solving.”

And 2020 wasn’t the first year critical thinking made the list.

When it comes to problem-solving, logic trees are a go-to critical thinking framework. Done right, they’ll get you to the root of a problem (pun intended) and even help you find solutions.

## What is a logic tree?

A logic tree takes a complex problem and breaks it up systematically, drilling down into smaller, more manageable components. If you’ve seen an image like this, you’ve seen a logic tree:

Looks pretty simple, right? It is! But there are some important rules to follow to make sure your logic tree grows up big and strong—and, more importantly, leads you to the answers you seek.

## The Logic Tree and the Case of the Missing Sandwich

Logic trees are often used for complex issues, which is why they’re also called “issue trees” or “decision trees.” But consider a simple (though still frustrating) problem faced by many office workers in the days before COVID-19 . . .

It’s lunchtime at your workplace, so you head to the office fridge—only to find your lunch is gone. You stare in disbelief at the empty space where your sandwich should be. A mocking smudge of condensation stares back.

As you are a logical person (not prone to throwing tantrums), you decide to approach this problem with critical thinking . You march back to your desk, grab a pen and paper, and write down four words: “Who took my sandwich?”

## Decide what you really need to know.

Before you jump into the branches, remind yourself that logic trees stem from problems—but knee-jerk responses often misidentify problems. So once you’ve got your initial question down, take a step back. Is there a branch that should have come before the first one you made?

Setting aside the sandwich problem for just a moment, consider you’re exploring a different issue: “How can I get promoted?” Think about the reason you’re asking that question. What’s the actual problem you need to solve?

Maybe you should have asked, “How can I earn more money?” That opens far more possibilities than just getting promoted—you could look for another job, start a side business, or invest in your buddy’s startup. “How can I get promoted?” becomes just one branch on a bigger tree.

This is how logic trees (and critical thinking in general) not only help you identify solutions, but think outside the box —innovate.

## Beware emotional bias in the branches.

You decide to start your logic tree with “Who took my sandwich?” From that root problem, you might break up the question into its seemingly logical components: “someone inside the office ate my sandwich” (Mary Anne, maybe? Or Phil?) and “someone came in off the street and stole it” (a hungry ninja, perhaps?).

The fact that you’ve defined the first two branches of your logic tree as “culprit here or there” means you’re sure there’s a thief in your midst. You’ve (perhaps subconsciously) ruled out the possibility that you forgot your sandwich at home or left it on the bus. The logic tree you’ve started will also not remind you if you didn’t make a lunch at all today because you’ll be eating out with Mary Anne and Phil.

Logic trees will only tell you what you ask them to tell you. They can only answer the questions you lay down. Don’t let your emotions limit the possibilities. Try to be aware of the assumptions you’re baking in.

Next Article

## Apply MECE to the branches of your logic tree.

Now that you’ve got your first two branches, you set to work breaking them down further. This is a good time to remember to follow the MECE principle . MECE stands for “mutually exclusive, collectively exhaustive.” In other words, it means you want to build the branches of your logic tree without gaps or duplicates.

Remember, logic trees are a critical thinking tool, and critical thinking is about systematic problem-solving. The MECE component of logic trees helps keep the system clean by eliminating possibilities, which increases efficiency toward finding an answer.

For example, if one branch says, “Someone in the office took my sandwich” and another says, “Someone on this floor took my sandwich,” you’re setting yourself up for some overlap. (Surely, this floor is in the office, no?)

It seems unlikely that someone came in off the street and took your sandwich, so you focus on the other possibility: an inside job. That leads to two new MECE-friendly branches: someone from your team, or someone from another team? Then more branches under those: Someone who decided your lunch looked way more delicious than their own, or someone who innocently mistook your lunch for theirs?

Aha! Looking at the breakdown, the answer strikes you. You bought that sandwich from Sandwich Heaven—the same place your teammate Rick sometimes buys his lunch. You check the fridge again, and sure enough, there’s another sandwich almost identical to yours (except this one has tomato, gross).

## Don’t expect logic trees to end with “the answer.”

While you may be tempted to rush off and have a chat with Rick about stealing your sandwich, it’ll serve you much better to reflect on why he took your sandwich in the first place. You and Rick bought sandwiches from the same shop, which makes it easy to mix them up. That may be the answer—but is it a solution ?

At the end of your logic tree, pose some further questions: Should you and Rick decide to put your sandwiches on different shelves in the fridge? Should you make it a point to label yours with your name? Or should you stop going to Sandwich Heaven every day?

This follow-up is important, even for logic trees targeting more serious issues. If you’re trying to determine why sales are down, perhaps your logic tree will reveal you’re targeting the wrong customers. In that case, what steps can you take to reset your targets?

## The Payoff of Learning Logic Trees

While they might sound like a lot of work (and yes, they can be a bit overwhelming at first), logic trees can actually save you a lot of time once you get the hang of them. Even better, you don’t have to slave away at mastering them on your own. There’s no reason a logic tree needs to be a solitary activity (even if you’re hunting down a sandwich thief).

In fact, if you’re working on a critical business issue, you really shouldn’t try to go it alone. Ask your boss, your team, a consultant, or another colleague to review your work. They don’t even have to understand the problem in depth—the point is to get a fresh pair of eyes on your logic tree and, by extension, your problem.

Finally, keep in mind that there’s no guarantee a logic tree will bring you to the perfect answer. What they will do is train your critical thinking skills and help widen your view of the problems you face.

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## How to think like a programmer — lessons in problem solving

by Richard Reis

If you’re interested in programming, you may well have seen this quote before:

“Everyone in this country should learn to program a computer, because it teaches you to think.” — Steve Jobs

You probably also wondered what does it mean, exactly, to think like a programmer? And how do you do it??

Essentially, it’s all about a more effective way for problem solving .

In this post, my goal is to teach you that way.

By the end of it, you’ll know exactly what steps to take to be a better problem-solver.

## Why is this important?

Problem solving is the meta-skill.

We all have problems. Big and small. How we deal with them is sometimes, well…pretty random.

Unless you have a system, this is probably how you “solve” problems (which is what I did when I started coding):

• Try a solution.
• If that doesn’t work, try another one.
• If that doesn’t work, repeat step 2 until you luck out.

Look, sometimes you luck out. But that is the worst way to solve problems! And it’s a huge, huge waste of time.

The best way involves a) having a framework and b) practicing it.

“Almost all employers prioritize problem-solving skills first.
Problem-solving skills are almost unanimously the most important qualification that employers look for….more than programming languages proficiency, debugging, and system design.
Demonstrating computational thinking or the ability to break down large, complex problems is just as valuable (if not more so) than the baseline technical skills required for a job.” — Hacker Rank ( 2018 Developer Skills Report )

## Have a framework

To find the right framework, I followed the advice in Tim Ferriss’ book on learning, “ The 4-Hour Chef ”.

It led me to interview two really impressive people: C. Jordan Ball (ranked 1st or 2nd out of 65,000+ users on Coderbyte ), and V. Anton Spraul (author of the book “ Think Like a Programmer: An Introduction to Creative Problem Solving ”).

I asked them the same questions, and guess what? Their answers were pretty similar!

Soon, you too will know them.

Sidenote: this doesn’t mean they did everything the same way. Everyone is different. You’ll be different. But if you start with principles we all agree are good, you’ll get a lot further a lot quicker.

“The biggest mistake I see new programmers make is focusing on learning syntax instead of learning how to solve problems.” — V. Anton Spraul

So, what should you do when you encounter a new problem?

Here are the steps:

## 1. Understand

Know exactly what is being asked. Most hard problems are hard because you don’t understand them (hence why this is the first step).

How to know when you understand a problem? When you can explain it in plain English.

Do you remember being stuck on a problem, you start explaining it, and you instantly see holes in the logic you didn’t see before?

Most programmers know this feeling.

This is why you should write down your problem, doodle a diagram, or tell someone else about it (or thing… some people use a rubber duck ).

“If you can’t explain something in simple terms, you don’t understand it.” — Richard Feynman

Don’t dive right into solving without a plan (and somehow hope you can muddle your way through). Plan your solution!

Nothing can help you if you can’t write down the exact steps.

In programming, this means don’t start hacking straight away. Give your brain time to analyze the problem and process the information.

To get a good plan, answer this question:

“Given input X, what are the steps necessary to return output Y?”

Sidenote: Programmers have a great tool to help them with this… Comments!

Pay attention. This is the most important step of all.

Do not try to solve one big problem. You will cry.

Instead, break it into sub-problems. These sub-problems are much easier to solve.

Then, solve each sub-problem one by one. Begin with the simplest. Simplest means you know the answer (or are closer to that answer).

After that, simplest means this sub-problem being solved doesn’t depend on others being solved.

Once you solved every sub-problem, connect the dots.

Connecting all your “sub-solutions” will give you the solution to the original problem. Congratulations!

This technique is a cornerstone of problem-solving. Remember it (read this step again, if you must).

“If I could teach every beginning programmer one problem-solving skill, it would be the ‘reduce the problem technique.’
For example, suppose you’re a new programmer and you’re asked to write a program that reads ten numbers and figures out which number is the third highest. For a brand-new programmer, that can be a tough assignment, even though it only requires basic programming syntax.
If you’re stuck, you should reduce the problem to something simpler. Instead of the third-highest number, what about finding the highest overall? Still too tough? What about finding the largest of just three numbers? Or the larger of two?
Reduce the problem to the point where you know how to solve it and write the solution. Then expand the problem slightly and rewrite the solution to match, and keep going until you are back where you started.” — V. Anton Spraul

By now, you’re probably sitting there thinking “Hey Richard... That’s cool and all, but what if I’m stuck and can’t even solve a sub-problem??”

First off, take a deep breath. Second, that’s fair.

Don’t worry though, friend. This happens to everyone!

The difference is the best programmers/problem-solvers are more curious about bugs/errors than irritated.

In fact, here are three things to try when facing a whammy:

• Debug: Go step by step through your solution trying to find where you went wrong. Programmers call this debugging (in fact, this is all a debugger does).
“The art of debugging is figuring out what you really told your program to do rather than what you thought you told it to do.”” — Andrew Singer
• Reassess: Take a step back. Look at the problem from another perspective. Is there anything that can be abstracted to a more general approach?
“Sometimes we get so lost in the details of a problem that we overlook general principles that would solve the problem at a more general level. […]
The classic example of this, of course, is the summation of a long list of consecutive integers, 1 + 2 + 3 + … + n, which a very young Gauss quickly recognized was simply n(n+1)/2, thus avoiding the effort of having to do the addition.” — C. Jordan Ball

Sidenote: Another way of reassessing is starting anew. Delete everything and begin again with fresh eyes. I’m serious. You’ll be dumbfounded at how effective this is.

• Research: Ahh, good ol’ Google. You read that right. No matter what problem you have, someone has probably solved it. Find that person/ solution. In fact, do this even if you solved the problem! (You can learn a lot from other people’s solutions).

Caveat: Don’t look for a solution to the big problem. Only look for solutions to sub-problems. Why? Because unless you struggle (even a little bit), you won’t learn anything. If you don’t learn anything, you wasted your time.

Don’t expect to be great after just one week. If you want to be a good problem-solver, solve a lot of problems!

Practice. Practice. Practice. It’ll only be a matter of time before you recognize that “this problem could easily be solved with <insert concept here>.”

How to practice? There are options out the wazoo!

Chess puzzles, math problems, Sudoku, Go, Monopoly, video-games, cryptokitties, bla… bla… bla….

In fact, a common pattern amongst successful people is their habit of practicing “micro problem-solving.” For example, Peter Thiel plays chess, and Elon Musk plays video-games.

“Byron Reeves said ‘If you want to see what business leadership may look like in three to five years, look at what’s happening in online games.’
Fast-forward to today. Elon [Musk], Reid [Hoffman], Mark Zuckerberg and many others say that games have been foundational to their success in building their companies.” — Mary Meeker ( 2017 internet trends report )

Does this mean you should just play video-games? Not at all.

But what are video-games all about? That’s right, problem-solving!

So, what you should do is find an outlet to practice. Something that allows you to solve many micro-problems (ideally, something you enjoy).

For example, I enjoy coding challenges. Every day, I try to solve at least one challenge (usually on Coderbyte ).

Like I said, all problems share similar patterns.

That’s all folks!

Now, you know better what it means to “think like a programmer.”

You also know that problem-solving is an incredible skill to cultivate (the meta-skill).

As if that wasn’t enough, notice how you also know what to do to practice your problem-solving skills!

Phew… Pretty cool right?

Finally, I wish you encounter many problems.

You read that right. At least now you know how to solve them! (also, you’ll learn that with every solution, you improve).

“Just when you think you’ve successfully navigated one obstacle, another emerges. But that’s what keeps life interesting.[…]
Life is a process of breaking through these impediments — a series of fortified lines that we must break through.
Each time, you’ll learn something.
Each time, you’ll develop strength, wisdom, and perspective.
Each time, a little more of the competition falls away. Until all that is left is you: the best version of you.” — Ryan Holiday ( The Obstacle is the Way )

Now, go solve some problems!

And best of luck ?

Special thanks to C. Jordan Ball and V. Anton Spraul . All the good advice here came from them.

Thanks for reading! If you enjoyed it, test how many times can you hit in 5 seconds. It’s great cardio for your fingers AND will help other people see the story.

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## 10.3: Basic Arguments- Using Logic

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An argument requires a number of premises (facts or assumptions) which are followed by a conclusion (point of the argument). The premises are used as justification for a conclusion. A conclusion which is correctly supported by the premises is known as a valid argument , while a fallacy is a deceptive argument that can sound good but is not well supported by the premises.

We will look at examples where the first two statements are the premises, and the third statement is the conclusion.

Determine if the following argument is valid.

All men are mortal.

John Smith is a man.

John Smith is mortal.

There are two premises (the first 2 sentences) and one conclusion (the last sentence). If we think of the premises as a and b, and the conclusion as c, then the argument in symbolic form is: $$a \land b) →c$$. In order for the argument to be valid, we need this conditional statement to always be true. If there is ever a time, even just one time , when this conditional statement is false, then it is an invalid argument . Another way to think of this is to say that the conclusion must follow from the premises. If the premises are true, then the conclusion must be true in order for the argument to be valid.

Look at the argument – if we assume that a and b are both true, then does the conclusion have to follow? YES! If all men are mortal, and if John Smith is a man, then John Smith must be mortal. This is valid.

All dogs are yellow.

Flippy is a dog.

Flippy is yellow.

All dogs are yellow is equivalent to “If it is a dog then it is yellow.” That is equivalent to “If it is not yellow, then it is not a dog” by the contrapositive. Assume the premises are true. Does the conclusion have to follow? YES! So this is valid!

In both of the examples above, the first statement of the premises could be written as an “if-then” statement. “All dogs are yellow” means the same thing as “If it is a dog, it is yellow."

The above examples are examples of Modus Ponens , which is always a valid argument.

Format of Modus Ponens (which is a valid logical argument)

Basically Modus Ponens states that if p implies q, and p is true, then q must also be true!

One could create a truth table to show Modus Tollens is true in all cases : [$$(p → q) \land p ] → q$$

Determine if the following argument is valid. (Hint: rewrite the “all” as “if-then”, then also write the contrapositive)

Chipper is yellow.

Chipper is a dog.

“All dogs are yellow” is equivalent to “If it is a dog then it is yellow.” or “If it is not yellow, then it is not a dog” by the contrapositive. Assume the premises are true. Does the conclusion have to follow? It states all dogs are yellow, but doesn’t say anything about yellow things, or that everything yellow is a dog. It is possible to have something yellow (like a lemon) that is not a dog; that means the conclusion isn’t necessarily true. This argument is invalid. Let p stand for “It is a dog.” Let q stand for “It is yellow.” The format of the above argument, shown below, is not Modus Ponens.

It is an example of Fallacy by Converse Error .

Remember the example where p is “You live in Vista” and q is “You live in California”? Consider

This is a valid logical statement because it is of the form Modus Ponens .

This is an invalid argument, and is an example of Fallacy by Converse Error .

This is also an invalid argument, and is an example of Fallacy by Inverse Error .

This is a valid argument, and is an example of Modus Tollens .

Modus Tollens is based on the contrapositive. Remember that p → q is logically equivalent to (~ q) → (~ p)

So the above argument could be written in four steps:

The last three statements LOOKS like Modus Ponens. But the original argument only had three lines. It wasn’t written as the contrapositive. So it’s not called Modus Ponens. One could create a truth table to show Modus Tollens is true in all cases: [(p → q) $$\land ~q] → ~p$$

Another reasoning argument is called the Chain Rule (transitivity) . Below is an example. The first two sentences are the premises, and the last is the conclusion. If the first two are true, the conclusion is true.

If I have a bus pass, I will go to school.

If I go to school, I will attend class.

If I have a bus pass, I will attend class.

So the idea is that if “if p, then q” and “if q, then r” are both true, then “if p, then r” is also true.

Symbolically, the chain rule is: [(p → q) $$\land (q → r)] → (p → r)$$

One could create a truth table to show the truth table is true in all cases, but it’s more complicated because there are 3 statements, hence 8 rows in the truth table.

The format for the Chain Rule where the first two lines are the premises and the third is the conclusion is: p → q

## Exercise 17

17. If the two statements below are premises, use the Chain Rule to state the conclusion.

If Mia doesn’t study, then Mia does not pass the final.

If Mia does not pass the final, then Mia does not pass the class.

What about a logic statement where all of the outcomes of a formula are true in every situation? When this happens, it is called a tautology . Modus Ponens, Modus Tollens, and the Chain Rule (transitivity) are tautologies. A truth table will show the statement true in each row of the column for that statement. A fallacy is when all the outcomes of a logic statement are false. An example of a fallacy in words is “I called Jim and I did not call Jim.” If p is “I called Jim,” the logic statement in symbols for this fallacy is $$p \land ~ p$$). A tautology would be “I called Jim or I did not call Jim,” which is written as $$p \lor ~ p$$)

You will create your own truth tables for Modus Ponens and Modus Tollens in the next exercises. Create intermediate columns so it is clear how you get the final column, which will show each is a tautology.

## Exercise 18

18. Make a Truth Table showing Modus Ponens is a valid argument. In other words, create and fill out a truth table where the last column is [(p → q) $$\land p] → q$$, and show that in all four situations, it is true, which means it is a tautology

## Exercise 19

19. Make a Truth Table showing Modus Tollens is a valid argument. In other words, create and fill out a truth table where the last column is [(p → q) $$\land ~ q] → ~ p$$, and show that in all four situations, it is true.

SUMMARY of arguments, where the first two statements are premises, and the third is the conclusion.

## Exercise 20

20. Determine if the following arguments are valid or not. If they are valid, write if it is by Modus Ponens, Modus Tollens, or the Chain Rule. If it is not valid, write if it is by Fallacy by Converse Error, or Fallacy by Inverse Error, or neither. If it looks like the chain rule, but has a false conclusion, write the correct conclusion.

## Exercise 21

21. Write a conclusion that would make each argument valid, and state if you used Modus Ponens or Modus Tollens.

## Exercise 22

22. Determine whether there is a problem with the person’s thinking. Explain your reasoning.

A) John’s mom told him “If you get home after 10pm, then you are grounded.” John got home at 9:30pm and was grounded. He was really ticked off because he said that she lied to him. Did she?

B) Marcia told her daughter: “If you get home before 10pm, then I will give back your cell phone.” Her daughter got home at 9:45pm, but her mom didn’t give back the cell phone. Did her mother lie?

## Exercise 23

23. Create a truth table for $$p \lor (~ p → q)$$

## Logic and Problem Solving Worksheets

• Most Popular
• Most Recent

Published: February 13, 2023

Biron Clark

Former Recruiter

Contributor

Dr. Kyle Elliott

Career Coach

Hayley Jukes

Editor-in-Chief

Biron Clark , Former Recruiter

Kyle Elliott , Career Coach

Hayley Jukes , Editor

As a recruiter , I know employers like to hire people who can solve problems and work well under pressure.

A job rarely goes 100% according to plan, so hiring managers are more likely to hire you if you seem like you can handle unexpected challenges while staying calm and logical.

But how do they measure this?

Hiring managers will ask you interview questions about your problem-solving skills, and they might also look for examples of problem-solving on your resume and cover letter.

In this article, I’m going to share a list of problem-solving examples and sample interview answers to questions like, “Give an example of a time you used logic to solve a problem?” and “Describe a time when you had to solve a problem without managerial input. How did you handle it, and what was the result?”

• Problem-solving involves identifying, prioritizing, analyzing, and solving problems using a variety of skills like critical thinking, creativity, decision making, and communication.
• Describe the Situation, Task, Action, and Result ( STAR method ) when discussing your problem-solving experiences.
• Provide numerical data or metrics to demonstrate the tangible impact of your problem-solving efforts.

## What are Problem Solving Skills?

Problem-solving is the ability to identify a problem, prioritize based on gravity and urgency, analyze the root cause, gather relevant information, develop and evaluate viable solutions, decide on the most effective and logical solution, and plan and execute implementation.

Problem-solving encompasses other skills that can be showcased in an interview response and your resume. Problem-solving skills examples include:

• Critical thinking
• Analytical skills
• Decision making
• Research skills
• Technical skills
• Communication skills

## Why is Problem Solving Important in the Workplace?

Problem-solving is essential in the workplace because it directly impacts productivity and efficiency. Whenever you encounter a problem, tackling it head-on prevents minor issues from escalating into bigger ones that could disrupt the entire workflow.

Beyond maintaining smooth operations, your ability to solve problems fosters innovation. It encourages you to think creatively, finding better ways to achieve goals, which keeps the business competitive and pushes the boundaries of what you can achieve.

Effective problem-solving also contributes to a healthier work environment; it reduces stress by providing clear strategies for overcoming obstacles and builds confidence within teams.

## Examples of Problem-Solving in the Workplace

• Correcting a mistake at work, whether it was made by you or someone else
• Overcoming a delay at work through problem solving and communication
• Resolving an issue with a difficult or upset customer
• Overcoming issues related to a limited budget, and still delivering good work through the use of creative problem solving
• Overcoming a scheduling/staffing shortage in the department to still deliver excellent work
• Troubleshooting and resolving technical issues
• Handling and resolving a conflict with a coworker
• Solving any problems related to money, customer billing, accounting and bookkeeping, etc.
• Taking initiative when another team member overlooked or missed something important
• Taking initiative to meet with your superior to discuss a problem before it became potentially worse
• Solving a safety issue at work or reporting the issue to those who could solve it
• Using problem solving abilities to reduce/eliminate a company expense
• Finding a way to make the company more profitable through new service or product offerings, new pricing ideas, promotion and sale ideas, etc.
• Changing how a process, team, or task is organized to make it more efficient
• Using creative thinking to come up with a solution that the company hasn’t used before
• Performing research to collect data and information to find a new solution to a problem
• Boosting a company or team’s performance by improving some aspect of communication among employees
• Finding a new piece of data that can guide a company’s decisions or strategy better in a certain area

## Problem-Solving Examples for Recent Grads/Entry-Level Job Seekers

• Coordinating work between team members in a class project
• Reassigning a missing team member’s work to other group members in a class project
• Speaking to your professor to get help when you were struggling or unsure about a project
• Asking classmates, peers, or professors for help in an area of struggle
• Researching solutions to an academic problem online, via Google or other methods
• Using problem solving and creative thinking to obtain an internship or other work opportunity during school after struggling at first

When you answer interview questions about problem-solving scenarios, or if you decide to demonstrate your problem-solving skills in a cover letter (which is a good idea any time the job description mentions problem-solving as a necessary skill), I recommend using the STAR method.

STAR stands for:

It’s a simple way of walking the listener or reader through the story in a way that will make sense to them.

Start by briefly describing the general situation and the task at hand. After this, describe the course of action you chose and why. Ideally, show that you evaluated all the information you could given the time you had, and made a decision based on logic and fact. Finally, describe the positive result you achieved.

Note: Our sample answers below are structured following the STAR formula. Be sure to check them out!

Dr. Kyle Elliott , MPA, CHES Tech & Interview Career Coach caffeinatedkyle.com

## How can I communicate complex problem-solving experiences clearly and succinctly?

Before answering any interview question, it’s important to understand why the interviewer is asking the question in the first place.

When it comes to questions about your complex problem-solving experiences, for example, the interviewer likely wants to know about your leadership acumen, collaboration abilities, and communication skills, not the problem itself.

Therefore, your answer should be focused on highlighting how you excelled in each of these areas, not diving into the weeds of the problem itself, which is a common mistake less-experienced interviewees often make.

## Tailoring Your Answer Based on the Skills Mentioned in the Job Description

As a recruiter, one of the top tips I can give you when responding to the prompt “Tell us about a problem you solved,” is to tailor your answer to the specific skills and qualifications outlined in the job description.

Once you’ve pinpointed the skills and key competencies the employer is seeking, craft your response to highlight experiences where you successfully utilized or developed those particular abilities.

For instance, if the job requires strong leadership skills, focus on a problem-solving scenario where you took charge and effectively guided a team toward resolution.

By aligning your answer with the desired skills outlined in the job description, you demonstrate your suitability for the role and show the employer that you understand their needs.

Amanda Augustine expands on this by saying:

“Showcase the specific skills you used to solve the problem. Did it require critical thinking, analytical abilities, or strong collaboration? Highlight the relevant skills the employer is seeking.”

Now, let’s look at some sample interview answers to, “Give me an example of a time you used logic to solve a problem,” or “Tell me about a time you solved a problem,” since you’re likely to hear different versions of this interview question in all sorts of industries.

The example interview responses are structured using the STAR method and are categorized into the top 5 key problem-solving skills recruiters look for in a candidate.

## 1. Analytical Thinking

Situation: In my previous role as a data analyst , our team encountered a significant drop in website traffic.

Task: I was tasked with identifying the root cause of the decrease.

Action: I conducted a thorough analysis of website metrics, including traffic sources, user demographics, and page performance. Through my analysis, I discovered a technical issue with our website’s loading speed, causing users to bounce.

Result: By optimizing server response time, compressing images, and minimizing redirects, we saw a 20% increase in traffic within two weeks.

## 2. Critical Thinking

Situation: During a project deadline crunch, our team encountered a major technical issue that threatened to derail our progress.

Task: My task was to assess the situation and devise a solution quickly.

Action: I immediately convened a meeting with the team to brainstorm potential solutions. Instead of panicking, I encouraged everyone to think outside the box and consider unconventional approaches. We analyzed the problem from different angles and weighed the pros and cons of each solution.

## 5. Persistence

Situation: During a challenging sales quarter, I encountered numerous rejections and setbacks while trying to close a major client deal.

Task: My task was to persistently pursue the client and overcome obstacles to secure the deal.

Action: I maintained regular communication with the client, addressing their concerns and demonstrating the value proposition of our product. Despite facing multiple rejections, I remained persistent and resilient, adjusting my approach based on feedback and market dynamics.

Result: After months of perseverance, I successfully closed the deal with the client. By closing the major client deal, I exceeded quarterly sales targets by 25%, resulting in a revenue increase of \$250,000 for the company.

## Tips to Improve Your Problem-Solving Skills

Throughout your career, being able to showcase and effectively communicate your problem-solving skills gives you more leverage in achieving better jobs and earning more money .

So to improve your problem-solving skills, I recommend always analyzing a problem and situation before acting.

When discussing problem-solving with employers, you never want to sound like you rush or make impulsive decisions. They want to see fact-based or data-based decisions when you solve problems.

Don’t just say you’re good at solving problems. Show it with specifics. How much did you boost efficiency? Did you save the company money? Adding numbers can really make your achievements stand out.

To get better at solving problems, analyze the outcomes of past solutions you came up with. You can recognize what works and what doesn’t.

Think about how you can improve researching and analyzing a situation, how you can get better at communicating, and deciding on the right people in the organization to talk to and “pull in” to help you if needed, etc.

Finally, practice staying calm even in stressful situations. Take a few minutes to walk outside if needed. Step away from your phone and computer to clear your head. A work problem is rarely so urgent that you cannot take five minutes to think (with the possible exception of safety problems), and you’ll get better outcomes if you solve problems by acting logically instead of rushing to react in a panic.

You can use all of the ideas above to describe your problem-solving skills when asked interview questions about the topic. If you say that you do the things above, employers will be impressed when they assess your problem-solving ability.

## More Interview Resources

• 3 Answers to “How Do You Handle Stress?”
• How to Answer “How Do You Handle Conflict?” (Interview Question)

Biron Clark is a former executive recruiter who has worked individually with hundreds of job seekers, reviewed thousands of resumes and LinkedIn profiles, and recruited for top venture-backed startups and Fortune 500 companies. He has been advising job seekers since 2012 to think differently in their job search and land high-paying, competitive positions. Follow on Twitter and LinkedIn .

Read more articles by Biron Clark

Kyle Elliott , career coach and mental health advocate, transforms his side hustle into a notable practice, aiding Silicon Valley professionals in maximizing potential. Follow Kyle on LinkedIn .

Hayley Jukes is the Editor-in-Chief at CareerSidekick with five years of experience creating engaging articles, books, and transcripts for diverse platforms and audiences.

## 12 Expert-Approved Responses to ‘What Makes You Unique?’ in Job Interviews

15 most common pharmacist interview questions and answers, 15 most common paralegal interview questions and answers, top 30+ funny interview questions and answers, 60 hardest interview questions and answers, 100+ best ice breaker questions to ask candidates, top 20 situational interview questions (& sample answers), 15 most common physical therapist interview questions and answers.

## Enhancing Spotted Hyena optimization with fuzzy logic for complex engineering optimization

• Original Article
• Published: 06 May 2024

• N. Kumaratharan 2

Recently, solving complex real-world challenges has become a significant and vital task, many of these challenges involve combinatorial issues where optimal solutions are desired. Traditional optimization strategies have proven to be efficient for small-scale problems. However, when it comes to larger problems, such as those in the finance or business domains, a meta-heuristic search algorithm, like the Spotted Hyena optimization (SHO), is implemented. However, the SHO algorithm faces a challenge in selecting the optimal cluster head, resulting in low efficiency. To address this issue, a new bio-inspired optimization strategy called a novel Fuzzy Based Spotted Hyena Optimizer (FSHO) is proposed. This novel algorithm aims to solve the problem of optimum clustering and provides a solution to real-world engineering challenges. In addition, the Spotted Hyena algorithm is combined with Fuzzy C-Means to enhance its exploration capability, making it well-suited for optimization tasks. The implementation of this strategy is carried out using MATLAB, and its performance is evaluated using 29 benchmark functions, as well as by addressing various complex and computationally expensive engineering issues. Comparing the cost analysis of existing techniques such as FFA, AVOA, MGO, and GTO, the proposed FSHO achieves significantly better results with a cost reduction of 1.2%, as opposed to 5%, 3%, 2.7%, and 1.5% achieved by the other techniques, respectively. This method not only enhances performance but also provides cost-effective analysis with a low convergence rate.

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## Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Not applicable.

## Abbreviations

Spotted Hyena optimization

Fuzzy-Based Spotted Hyena Optimizer

Genetic algorithms

Gravitational search algorithm

Central force optimization

Simulated annealing

Firefly algorithm

Gravitational local search

Artificial neural network

Multi-objective optimization differential evolution

Non-dominated sorting genetic algorithm

Swarm intelligence

Evolutionary algorithms

Constrained multi-objective optimization problem

Ant-colony optimization

Particle-Swarm Optimization

Artificial-Bee Colony

Cuckoo Search

Grey Wolf Optimization

Firefly Algorithm

Emperor Penguin Optimizer

Genetic algorithm

Fuzzy inference system

Social network analysis

Fuzzy c-mean

• Spotted Hyena

Design of experiment

Farmland Fertility Algorithm

African Vultures Optimization Algorithm

Mountain Gazelle Optimizer

Artificial Gorilla Troops Optimizer

Set of n data points

Dataset function

Cluster center

Partition matrix

Node membership function

Distance between $$p_{j}$$ and $$c_{k}$$

Weighting exponent

Euclidean distance

Euclidean norm

Distance range between SH and the prey

Current iteration process

Coefficient vectors

Position of the prey

Spotted hyena’s position vector

Exact value

Multiplication process

Maximum number of iterations

Random vectors

First best SH and other SH positions

Population size and dimension

Wire’s diameter

Mean of the coil diameter

Active coils

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Department of CSE, IFET College of Engineering, Villupuram, Tamilnadu, India

Department of ECE, Sri Venkatesawara College of Engineering, Chennai, Tamilnadu, India

N. Kumaratharan

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## Contributions

All authors agreed on the content of the study. NP and NK collected all the data for analysis. NP agreed on the methodology. NP and NK completed the analysis based on agreed steps. Results and conclusions are discussed and written together. All authors read and approved the final manuscript.

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Padmapriya, N., Kumaratharan, N. Enhancing Spotted Hyena optimization with fuzzy logic for complex engineering optimization. Int. J. Mach. Learn. & Cyber. (2024). https://doi.org/10.1007/s13042-024-02130-6

Accepted : 10 March 2024

Published : 06 May 2024

DOI : https://doi.org/10.1007/s13042-024-02130-6

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