- Three hobbits and three orcs are running for their lives. They are being chased by Smaug the Dragon, who drives them towards a river. Fortunately, there is a boat that can ferry one or two creatures across the river. Unfortunately, if the orcs outnumber the hobbits, then they will be eaten. Strangely enough though, the orcs can be trusted to return the boat to the other side of the river. How can the hobbits safely cross the river?
- You wake up in a cabin in the woods. On the table, you find a box of tacks, a candle, and a book of matches. Since it is getting dark, your goal is to attach the candle to the wall.

While you were working through each of the problems, did you notice anything different about the way you approached the problem? Was there something different about the way you arrived at the solution? Once you had the solution, was it obvious?

As you know, problems come in many different shapes and sizes. Some problems are easy, others are hard. Some need extensive domain-specific knowledge, while others require relatively little knowledge. Some problems are well-defined and have "right" or "wrong" answers, whereas others are ill-defined and have "better" or "worse" answers.

The distinction between the problems stated above is that the first is a

**routine problem**and the second is an

**insight problem**. What's the difference?

## Just a Regular Day at the Office

A routine problem has the following characteristics. First, it starts with a well-defined**initial state**and

**goal state**. In other words, we know what must be done, and we know when to stop once we have solved it. We can see that these two criteria hold up for the "hobbit and orcs" problem. The initial state is three hobbits and three orcs on one side of the river. The goal state is all three hobbits safely on the other side of the river.

The second characteristic of a routine problem is that there exists an

**operator**to move the problem from the initial state to the goal state. The operator in this case is extremely basic. One or two creatures get in the boat and move to the other side of the river. Once they have moved, then the problem is in a

**transitional**state (i.e., somewhere between the initial and goal state).

Given the first two characteristics of routine problems, you can completely map out a

**problem space**. The problem space for the "hobbits and orcs" problem is pretty small because there are only a few transitional states. But for more interesting problems, or games such as chess, the problem space can be

*enormous*.

If a problem can be characterized in this way, then solving the problem includes searching through the problem space for a path that connects the initial state to the goal state. With experience, people can become quite proficient at solving problems because they generate different methods that apply across many different types of problems. These methods are called

**problem-solving heuristics**because they help solve the problem, but they do not guarantee a solution.

## A Problem Less Ordinary

Insight problems may lack one (or more) of the defining characteristics of a routine problem. For example, it might not be entirely obvious what the operators are for an insight problem. Nor can you plausibly describe or articulate any of the transitional states. For example, if you're like me, you probably needed a sheet of paper to solve the "hobbit and orcs" problem. My sheet of paper looks like a filmstrip. Each frame of the film is a different state of the problem. With the candle problem, I don't really have any transitional states. I immediately leapt from the initial state to the solution via an*aha!*moment.

Because insight problems lack the same characteristics of routine problems, does that mean there aren't any heuristics for solving them? That's a tough question, and I don't have a really solid answer. But I would say that you can train yourself to ask:

*What assumptions am I making? Which of those assumptions aren't true?*This is the essence of "insight" because you have to question that which you cannot see [1].

Going back to the nine-dot problem from a previous post, notice that most of us initially make the assumption that you are required to stay within the bounds of the dots. Well, it turns out that was not a requirement of the problem. In other words, we have artificially constrained the problem. A problem

**constraint**might be defined as the "rules of the game" or what counts as a valid solution. In the "hobbits and orcs" problem, there is an implied constraint that the hobbits can neither call in reinforcements nor get a bigger boat. I believe you can get better at solving insight problems by searching for problem constraints that are self-imposed.

## The STEM Connection

Most math problems are pitched as routine problems. You have an initial and goal state, and the lesson for the day is to learn the "rules" or operators that help transition from the initial to goal state. Let's take the topic of summation as an example. You want your class to sum all of the integers between 1 and 100. The initial state is given, and the goal state is well-defined.To motivate the lesson, you might talk about what the students already know. They already know the operator, which is adding up numbers, like this: 1+ 2 + 3 + 4 + ... + 98 + 99 + 100. Then, once they are done with that task, they learn about the summation symbol, its index, and the lower and upper bounds.

Solving this as a routine problem is generally necessary to go onto more advanced topics (e.g., computing the average of a dataset). But what if, after the solution is known, the problem is reframed as an insight problem? Wouldn't that make the lesson more interesting? Assuming your kids know about the commutative property of addition, the problem could be recast as: Can you find a more efficient way to calculate the sum of all the integers between 1 and 100? [2]

I realize that not all routine problems can be easily recast as insight problems; however, I do believe that knowing the distinction between routine and insight problems is useful because it highlights the fact that different mental operations are needed. When solving routine problems, the question one should be asking is:

*What operator do I apply?*Alternatively, when solving insight problems, one needs to be asking:

*What constraints do I need to relax?*Armed with these questions, there aren't any problems left that we can't solve!

Share and Enjoy!

Dr. Bob

## For More Information

[2] Perhaps apocryphal, but one of my favorite stories of insight problem solving is little Karl Gauss, who, in primary school, summed the values of 1 through 100 in his head. How did he do it? Easy! By recognizing that 1+100=101, and that there are 50 pairs of these sums. I can almost guarantee he had an

*aha!*moment when he arrived at his solution.