This chapter begins with a discussion of classes of problem complexity. It then focuses on hierarchical pyramids that have been accepted for the last three decades as computational models of the visual system of primates, including humans. Anatomical and behavioral results that support this claim are reviewed. First, the speed-accuracy tradeoff results in visual perception are reviewed. Second, mental size transformation is described and it is shown that they can be explained by a pyramid model. With these results related to visual perception in hand, it is shown how the pyramid model can be applied to the TSP. Pyramid models are characterized by self-similarity across space and scale. This self-similarity is a form of invariance, also known as symmetry. A pyramid model produces near-optimal TSP tours by using a global-to-local (coarse-to-fine) sequence of computational steps. Good performance of pyramid models on TSP comes with a price when the geometry of a TSP problem is perturbed by inserting obstacles. Finally, subjects are tested in a real-life application of the TSP, where they collect tennis balls on a tennis court. The fact that humans are capable of producing near-optimal TSP tours in almost “no time” suggests that they solve the task of minimizing a tour's length without ever measuring the length of the tour. Instead, they infer the concept of direction in the search space, and use the direction, rather than the distance, when they navigate within the problem spaces.

We are now halfway through our book. It may appear to be a coincidence that insight and creative thinking appear at the “center” of a book on problem solving, a coincidence that the gestalt psychologists would surely have liked. Contemplating how the young Gauss solved an arithmetic problem, and considering how a mutilated checkerboard problem is solved, while realizing that the solution of these problems are analogous to how snowflakes look, may come as a surprise. It should not because all three are based on symmetry and invariance. Before discussing how “ordinary” insight problems are solved, this chapter describes the insights that led Galileo, Archimedes, and Einstein to their scientific discoveries. Physicists have known for over a century that there would be no science based on the natural laws, and no natural laws in the first place, if there were no symmetry in nature. So what encouraged this to happen just over a 100 years ago? In 1918, Emmy Noether formulated and proved her mathematical theorems that revolutionized physics. Her theorems showed how the conservation laws can be derived from the symmetry of these laws by applying a least-action principle. The review of symmetry in scientific discovery presented in this chapter provides the stage for a new formalism of problem solving that may apply not only to the sophisticated areas of science, but also to “ordinary” brain teasers, as well as to the TSP and the 15-puzzle.

This chapter discusses the traveling salesman problem (TSP), which is the best-known problem with a large search space. TSPs with a moderate number of cities are solved very well by human subjects within a couple of minutes despite the fact that the number of possible tours can be equal to the number of seconds since the Big Bang. Randomly generated tours are much worse than the tours that humans produce. This means that human subjects must have an uncanny ability to navigate very intelligently through enormous spaces without considering too many, if any, alternatives. The absence of search in human problem solving is arguably the biggest difference between artificial intelligence, as it is designed today, and human intelligence, which is obviously the natural Intelligence. Human performance is described and compared to several heuristic algorithms such as the nearest neighbor, convex hull, and hierarchical pyramids. The chapter ends with a description of the performance of young children who solve TSP nearly as well as their parents.

This chapter elaborates what you learned in Chapter 7. It points out that despite the fact that many problems have an obvious visual representation, we need to be able to incorporate more abstract cognitive representations in our theory of problem solving. Traditionally, multidimensional scaling (MDS) has been used to infer Euclidean representations of concepts based on judged similarities. Here, after providing an example of how MDS has been used in vision, some cautionary comments are made about what MDS can and cannot provide. Because MDS is usually used to represent clusters of concepts, a formal discussion of clustering is included in this chapter. The chapter continues with two examples: one that is related to clustering in long-term memory and the other related to clustering in short-term memory. In both cases, clustering is used to interpret the mental navigation of memory representations as being analogous to navigation in our physical environment, just as it was when we discussed the TSP. The last section of this chapter illustrates how MDS can be used to explain how TSP tours are produced in the presence of obstacles where obstacles change the pairwise distances and make the distances not Euclidean. MDS can use the pairwise distances around obstacles to produce a Euclidean approximation. Preliminary experimental evidence suggests that this is what the human mind does.

The 15-puzzle is another well-known and well-studied problem with a large search space that is quickly solved by humans. This chapter defines the concept called the parity of a state of the 15-puzzle that is invariant for legal moves. From here on, the reader will see the terms invariance and symmetry showing up frequently in our discussion of problem solving. This should be rather intriguing because this concept has been completely absent from all prior research on human and computer problem solving. Next, the formal concept of a heuristic is introduced as an estimated distance to a goal. The second half of the chapter describes one of the few, perhaps even the only study of human performance in solving the 15-puzzle. Humans transform the start state to the goal state quickly and avoid backtracking. This way of solving the 15-puzzle strongly suggests that humans use direction, rather than distance. These two concepts are well defined in metric spaces, such as the Euclidean space that we use to describe our familiar physical environment. Here, I point out that in spaces, other than Euclidean, direction generalizes into what is known as a geodesic. This is important because it provides suggestions for how human problem solving could, or should, be studied in the future. The chapter ends with a description of a pyramid model that emulates how humans solve the 15-puzzle by using direction.

Problem solving is a goal-directed activity. As such, it depends critically on abstract, mental representations of a problem, including the identification of the goal that needs to be reached and the operations that allow the problem solver to navigate within the problem space. Because of this, mental representations of the physical, cognitive, and social environments take center stage when problem solving is discussed. The role of mental representations explains why the origins of research on problem solving are so closely related to the origins of the modern approach to perception initiated 100 years ago by the gestalt psychologists. The gestalt psychologists were particularly interested in insight problem solving, where the term “insight” provides an intuitive definition of such problems. We all know that Archimedes had an insight when he shouted out “Eureka” when he discovered the principle of buoyancy. Chapter 1 sets the stage for the remainder of the book, by promising to provide a new formalism that may be able to explain not only insight, but also many other research problems, including problems in mathematics and physics, as well as in scientific discovery. This ambitious plan should keep the students eager to see how it plays out, and by the end of Chapter 11 it should be clear why launching, 70 years ago, a new field called cognitive psychology, was called a scientific revolution.

If mental representations are important in problem solving, we need to provide a computational theory of how such representations are formed. Visual perception is the best place to start, considering how advanced research on visual representations is, when compared to other cognitive representations. Our physical world is three-dimensional (3D) and we see it as such, despite the fact that the sensory data input is a pair of 2D images on the back of the eye. Even if you close one eye, you still see the environment as 3D. It follows that the 3D percept is some kind of an educated guess (an inference) about the missing depth dimension. Formally, perception is an ill-posed inverse problem, whose solution requires a priori knowledge about the environment. What kind of a priori knowledge is both needed, and effective? It turns out that the symmetry of natural objects is both necessary and sufficient for making successful visual inferences. Combining sensory data with symmetry constraints leads to the veridical percepts of objects and scenes. In order to solve this visual problem optimally, the visual system finds the minimum of a cost function. The way the human mind solves this problem is completely analogous to how a least-action principle operates in physics. This analogy becomes important later when we discuss intuitive physics as a form of problem solving.

Chapter 10 discusses physics and math problems. Both have received some attention in the problem solving and education communities. The chapter begins with physics problems and evaluates what is called "intuitive physics," specifically, how well humans can understand the laws of physics before taking any classes in physics. Intuitive physics was introduced in Chapters 1 and 2 when animal problem solving was discussed. By now you know how important symmetry is in physics and that all basic phenomena can be explained by a least-action principle. Surprisingly, both of these concepts have been absent from research done on intuitive physics as well as absent from most or all conventional high school education. One reason for this is that these concepts require more sophisticated math than the math used in Newtonian physics. This chapter reviews a few well-known studies on intuitive physics and goes on to studies of causality. In the second half of the chapter, Polya’s contributions to problem solving in mathematics are highlighted and reviewed. This review focuses on his treatment of optimization problems. Polya’s elementary exposition of optimization is of particular interest because it suggests that optimization can be taught before college. Polya is aware of the relationship between optimization problems and a least-action principle, so his examples contribute to problem solving in both math and physics. If optimization and symmetry can be included in high school, or, perhaps, even in elementary school, this would revolutionize education in the STEM subjects, that is, science, technology, engineering, and mathematics.

By now we all know how important mental representations are for any goal-directed actions. But, if I have my mental representations and you have yours, can I improve my actions by having a representation of your representation? If I have such a representation, I am able to acknowledge that you are like me, in the sense that you have your own mental representations and goal-directed actions. This dialogue is referred to as the Theory of Mind because I have a “theory” (a model or representation) of your mind. In fact, I may even have a representation of your representation of my representations. These kinds of first-order and second-order representations are important in evaluating false beliefs and deception. The simpler types of reasoning are already present in preschool children, as illustrated in the first study described in this chapter. More complex types of reasoning show up later in development, so adults can use them. Forming representations of others’ representations happens in visual perception, as well, where it is called “visual perspective taking.” Children are capable of reasoning about what other people can or cannot see and they can make these visual inferences not only by looking at a 3D scene, but also by looking at pictures of 3D scenes. The second half of this chapter describes in some detail a systematic, quantitative study of a matrix game that has been used to examine levels of recursive resoning.

An analogy is a relational similarity. Analogies are objects or events that are isomorphic, that is, they have structures in common. Two objects or events are not considered analogs of each other because they refer to the same thing, but because the relationships among their components are the same. In Chapter 3, we saw how former Federal Reserve Chairman Ben Bernanke justified bailing out the finance industry following its collapse in 2008 through use of an analogy: Even if an irresponsible neighbor set fire to his house, we need to put out the fire in order to prevent the fire from spreading to other houses in the neighborhood. A burning house has nothing physically in common with a failing bank. But at a deeper level of analysis, they describe similar relations among their components: A burning house is in danger of physical collapse, and poses a threat to neighboring houses. A failing bank is in danger of financial collapse, and poses a danger to other banks. More importantly, both require intervention if the collapse is to be prevented.

“Houston, we have a problem.” It’s 1970, and NASA has just launched another manned space expedition, Apollo 13. It is the seventh manned expedition, and the third scheduled to land on the moon. People have become fairly jaded about these expeditions. As a result, little air time is devoted to televising the mission – until an oxygen tank explosion occurs that cripples the spacecraft and strands its three astronauts 205,000 miles above the Earth. While the astronauts search for ways to survive in the limited time they have left, NASA’s heroic ground crew searches for ways to bring the craft safely back home using nothing more than the items available to the men on board. As we all know from history (and a really great Ron Howard movie starring Tom Hanks), they find a solution, and all ends well.

While more dramatic than the usual types of problems we encounter in everyday life, the Apollo 13 story contains all of the elements that comprise all problem-solving, from the little annoying problems (like finding your lost keys) to potentially life-changing ones (like finding a cure for a deadly disease.)

According to philosopher David Hume (1711–1776), causality is an “illusion.” Here’s why: A Treatise of Human Nature (1739–40), Hume pointed out that, in order for one event to cause another event, the following conditions must be satisfied.