PROBLEM SOLVING
Problem solving is what happens when many of this book’s skills align. Insight, creativity, practical intelligence, wisdom, perhaps expertise—all can contribute to a solution. That is, if you line them up right. That’s what this chapter’s about: how do you set up a problem and choose a solving strategy so that your other skills can knock it down? It’s trickier than it sounds.
For example, between 2010 and 2012 Metropolitan State College of Denver worked on the problem of a name change. See, the word “college” translates as “high school” in Spanish and connotes a technical school in French, and so partly to boost recruitment of students for whom English is a second language and partly to highlight its continuing education and graduate programs, the college wanted to change its name to university. The problem seemed easy: what’s the best wording for a new name that includes the word “university” instead of “college”?
There’s a way you go about a higher education name change. First, the administrators conducted a survey and found that the name Denver State University came out on top, favored by 36 percent of faculty, staff, and students. But the existing University of Denver didn’t appreciate the closeness of the new name, and the Colorado state legislature promised to veto the school’s attempt at becoming DSU. So MSCD went back to the drawing board and came up with the seemingly easy switcho-changeo of university for college, making Metropolitan State University of Denver.
They checked to see if MSUD.edu was available. It was. Then the college hired a firm to make sure the acronym didn’t duplicate that of any out-of-state institutions. Now moving smoothly, it turned out MSUD was unique in academia. The new name coasted through the Colorado state legislature and on April 18, 2012, Colorado governor John Hickenlooper signed bill SB12-148, making the name change official.
“But no one thought to Google it,” said university spokesperson Cathy Lucas, doing damage control on Colorado Public Radio. It was damage control because, unfortunately, when you Google “MSUD,” the first full page consists of information and parent forums for Maple Syrup Urinary Disease, a serious pediatric health condition in which the body can’t break down certain proteins and instead excretes them in urine. It can be fatal. And due to an extremely active parent community that posts about the disease, even a search engine optimization blitz by the college-now-university would be unlikely to unseat the insidious condition. Confounding its name with a pediatric disease perhaps wasn’t the path to prestige the university imagined.
Richard Mayer, professor at the University of California–Santa Barbara, author of the book Thinking, Problem Solving, Cognition, and one of the most prolific researchers in the field of educational psychology knows how you and I can avoid similar mistakes. “Sometimes it’s useful to see a problem in abstract—represent it as its salient conditions and then strip these conditions of false assumptions,” Mayer says. This generates what researchers call an initial state and helps you see a problem’s constraints. On the far side of a problem is the goal state. And to get there, you’ll have to perform some sort of workable operation. And in the language of problem solving, that’s it: initial state, constraints, operations, and goal state.
It’s tempting to focus on the operations needed to transform the initial state to the goal state. I mean, that’s the solution! But Mayer says that the most striking feature of people who successfully solve real-world problems is the time they spend studying the initial state and the constraints—the extra time they spend clarifying the problem.
Take the job of Agriculture Minister of the former Soviet Union. Way back in 1983—when the USSR still existed—researcher James Voss posed subjects the following problem: “Suppose you are the Minister of Agriculture for the Soviet Union. Crop productivity has been too low for the past several years. What would you do to increase crop production?” Description of the initial state says nothing about the weather or the Soviet political system. The constraints say nothing about the availability of arable land. Operations are open-ended. And how much must production increase in order to reach the goal state—1 percent? 25 percent? 200 percent? When Voss gave the problem to Soviet policy experts, political science undergrads, and chemistry professors, he saw something important in the way they went about solving it: The Soviet policy experts spent 24 percent of their solution time elaborating the initial state and constraints, whereas the poly-sci undergrads and chemistry professors spent only 1 percent of their time getting the facts straight.
In part, this was due to Soviet experts’ prior knowledge about the Soviet Union—they knew enough to ask about arable land and the Soviet political and social systems; they knew enough to poke and prod the problem’s constraints. Of course, the Soviet experts’ solutions were generally more reasonable than the other groups’ solutions. Understand the problem and a solution will follow.
OK, so how exactly do you go about clarifying those initial states and constraints?
For a classic example of this process in action, Richard Mayer points to a problem first posed by Karl Duncker in 1945. “The classic tumor problem works like this,” Mayer says, “you’re a doctor and your patient has an inoperable malignant tumor in the middle of his abdomen. Radiation will kill the tumor, but radiation strong enough to do the job also destroys any healthy tissue it passes through on the way to the tumor. Without operating, how can you use radiation to kill the tumor without killing any healthy tissue?”
Take a minute. Or two. Can you solve this problem? When I chatted with Mayer, I certainly couldn’t. Like many problems, the solution depends on correctly conceptualizing it from the start. First explore inside the problem and try to isolate the smallest possible point where you seem hopelessly stuck—work to abstract the initial state and the constraints.
In the tumor problem, Mayer says that people usually pass through a couple of common ideas before narrowing in on the real crux. First, they think to toughen up the healthy tissue and then blast radiation through the whole thing at an intensity that kills the tumor but not the strengthened healthy tissue. “But this violates one of the problem’s constraints,” says Mayer, namely the fact that radiation strong enough to kill the tumor necessarily kills healthy tissue it touches. “There are a few other versions of that,” Mayer says, “and then people generally start to think about how to avoid contact with the tissue—maybe they think to remove the tumor and then zap it, or insert a tube and shoot the radiation through it.” But both require an operation. “And eventually problem solvers come to understand that the radiation has to be weak in the healthy tissue and stronger in the tumor,” Mayer says.
Bingo! This is the small box in which the real problem rests: how can the radiation be weak in the tissue and strong in the tumor? Restating the problem in this way focuses problem-solving power on the most difficult part. Assuming a solution exists, it will exist in this box. This is the problem’s true initial state.
Now that you’ve got the initial state nailed down, it’s time to figure out the problem’s true constraints. Are you making any false assumptions that are keeping you from finding a solution? Now that you’ve drawn a small box, widen it to make sure you include all possibilities.
Mayer suggests separating constraints from false assumptions by trying to “wiggle” your visualization. In your visualization of the tumor problem, do the constraints allow you to wiggle from a male to a female patient? Sure, but it doesn’t really matter, so move on to the next. Could the patient be sitting or standing? Of course, but that doesn’t seem relevant either. Do you visualize a single, gun-like ray delivering the radiation? If so, maybe it’s worth releasing this assumption. Wiggle your visualization to imagine other possibilities. “Examining their actual knowledge about a situation and comparing it to their assumptions tends to lead people to recognize that you could have many rays, all firing at varying intensities,” says Mayer.
With this initial state and constraints, the operation that leads to the goal state is fairly simple: low-intensity radiation must be delivered by many coordinated rays all firing into the tumor from different angles. No one ray will be strong enough to kill the healthy tissue it passes through, but the head-on collision of all the radiation at the point of the tumor will be enough to fry it.
And that brings us back to the MSMetropolitan State University of Denver. Their administrators isolated the initial state—how could they replace “college” with “university” in a way that made the new name unique? But they failed to separate constraints from false assumptions, namely failing to recognize and revise the assumption that the new name would compete only with acronyms within higher education.
Now you have the tools to handle the first two steps in solving life’s ill-defined problems: examining initial states and constraints. But if even after elucidating the problem’s conditions the solution doesn’t smack you in the head, you’ll need to move on to thinking about which solving strategies can help you arrive at the answer—the domain of operations. The following exercises will help you do it.
PROBLEM-SOLVING OPERATIONS: RANDOM, DEPTH-FIRST, BREADTH-FIRST, AND MEANS-ENDS ANALYSIS SEARCH
There are a number of ways humans go about solving problems on paper and in the real world, and knowing the names of these techniques and how they work can help you match the problems of your life with the best solving strategy. So let’s look at random, depth-first, breadth-first, and means-ends analysis searches.
This sounds a little tricky and technical until you imagine problem solving as a journey through a maze—the initial state is your starting point, the goal state is the finish, and the constraints are the rules: you can’t jump walls. Simple! And keeping these three conditions simple allows us to work on the fourth: the operation you use to draw the path.
On the first maze that follows, try a random search. Really: don’t think. Don’t look ahead. Don’t look back. Put your pencil at the start and just go. Hit a dead end? Try it again from the start, randomly. OK, OK, this obviously doesn’t work very well and in this case is more a thought experiment than a solution strategy.
But there are situations in which random search is, in fact, the best operation! The ideal operation depends on what researchers call a problem’s search space. With a simple search space—if there were very few turns in this maze—it might actually be easier to solve it randomly than to donate brainpower to the planning and execution of a more nuanced operation. Likewise, if it took nanoseconds to test each path, it might be faster to randomly try them all rather than thinking about how to try them. So: Random search is best when trying paths is quick, easy, painless, and likely to lead to a solution more quickly than extensive planning. For example, imagine the power is out in your house at night—if your goal state is the bathroom across the hall from your bedroom, random search should do it. But if your goal state is the leftover carrot cake in the fridge downstairs, the more complex search space may require more than random search: it may require a flashlight. (This extra layer is referred to as “subgoaling,” and we’ll look at it more later.)
On the next maze, do a depth-first search. Still without looking ahead, follow one path to its absolute dead end, and if that end is not the goal state, back up to the most recent junction and try the path not previously chosen, following it to its end. If that isn’t the goal state either, continue retreating to each previous junction and trying the next path. On a paper maze, your pencil shows which branches of the search space you’ve exhausted, and possibilities are likely few enough to remain manageable. But in life and in many other puzzles, like chess, a depth-first search quickly requires massive memory. So: a depth-first search is usually best when possibilities are few or when you don’t have enough information to nix unlikely branches of the decision space. (In the language of problem solving, a rule that allows you to chop out a portion of the search space is called a heuristic; adding heuristics to depth-first searches can be a powerful strategy.)
Try again, this time with a breadth-first search. Go to the first junction. Look down the left path. Does it eventually branch again? If so, it remains a possibility: draw a small arrow pointing down that left path. Now go back to check the right path. Does it eventually branch? If so, it also remains a possibility: draw a similar arrow pointing down this right path. Now go back and check each of those branches in turn to see if it branches again. If not, make an X instead of an arrow—this path is closed. A depth-first search allows you to get lucky, to serendipitously find a path that snakes all the way from initial state to goal state. But is it the best path? In a maze it doesn’t really matter. But in life, a breadth-first search keeps all options in play until they are absolutely nixed. Instead of following one thread to its lucky or unlucky end, a breadth-first search explores all paths at once a little at a time and so may lead to multiple answers, some better than others.
Finally, try a means-ends analysis. This solution strategy explores the differences between the initial state and the goal state and tries to chip away at each difference until none remain. It’s how water would trickle down through a jumble of rocks, always flowing toward its goal even if the flow is along a slope of only one degree. In a maze, the difference between the initial state and the goal state is the difference between x and y coordinates, so at every turn, take the branch that heads most toward the goal. Unfortunately, in mazes, as in life, it’s frequently necessary to move away from your goal so that you can eventually move toward it—without the ability to flow upward, water may get stuck in a cupped boulder and never reach the ground.
Here’s where we get to subgoals. Your subgoals may initially appear to be parallel to or even headed away from the goal state, but in combination they’ll get you there—and unlike the convoluted path from initial state to goal state, the path to each subgoal can be straightforward (or at least much less twisty!). Chimps set subgoals, too. In 1927 German psychologist Wolfgang Köhler placed his prize chimp, Sultan, below a bunch of bananas with two sticks just too short to reach them. Sultan quickly saw the subgoal—he needed a tool long enough to reach the bananas and so hooked the two short sticks together.
In a maze, subgoals may be bottlenecks through which the path obviously must pass. Or you can create subgoals by working backward—starting at the goal state and retreating to the first ambiguous turn provides a subgoal for use when switching to working the maze from the initial state.
In a maze, there’s really only one constraint: don’t jump the walls. But in life, subgoaling may look more like applying one constraint at a time. For example, when picking a baby name, you might narrow the problem space by boy names, and then by family names, and then by names that won’t get the poor child teased in middle school. Each subgoal is a bottleneck through which the solution must pass.
These are the most common human solution strategies—random search, depth-first search (likely with heuristics), breadth-first search, and means-ends analysis (likely with subgoals). If you have the time, it’s worth taking another pass through these strategies with new mazes (try the ones on KrazyDad.com). The idea here is not so much to solve the mazes, of course, as it is to practice putting these strategies into action and thus get them to stick in your head.
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MATCH: INITIAL STATE TO OPERATIONS
Now that you’ve previewed the most common human solution strategies on paper, it’s time to put them to use in the labyrinth of life. Following are examples of situations best solved by the four operations you learned in the previous exercise. Match the situation to the best solving strategy. Then in the space provided, add your own examples of problems best solved by each search strategy.
Random search: Without planning or reflecting, try for a quick solution.
Depth-first search: Once you start down a path, follow it to its end. Not the solution? Retreat and try again.
Breadth-first search: Dip a toe into each possible path to a solution, nixing only the ones that prove impossible.
Means-ends analysis: Chip away at every factor that divides initial state from goal state, perhaps pointing at subgoals along the way.
Situations:
1. You are a man dressing formally and need to put on underwear, pants, undershirt, shirt, socks, belt, cufflinks, tie, jacket, etc.
2. Looking at a map, there are many routes you can use to commute from home to work. Which is the best?
3. You need a pair of black socks in a hurry. There are many white along with equally many black socks in a drawer.
4. You are at work. You are hungry. There’s a cash-only vending machine. You have a card but no cash. There’s an ATM downstairs.
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FALSE ASSUMPTIONS
As we’ve seen, arriving at a clear-eyed understanding of a problem’s initial state can be the key to reaching a solution. And researchers Afia Ahmed and John Patrick of Cardiff University know how to do it. In 2006, they presented subjects with two problems, including the following, known as the Unseen Walker Problem:
On a busy Friday afternoon, a man walked several miles across London from Westminster to Knightsbridge without seeing anybody or being seen by anybody. The day was clear and bright. He had perfect eyesight and he looked where he was going. He did not travel by any method of transport other than by foot. London was thronged with people yet not one of them saw him. How?
Then Ahmed and Patrick asked subjects to talk through their answers. Only 42 percent of subjects solved the problem, and in unsuccessful solvers the researchers found something in common—what they called a “constraint that could block participants from reaching the correct answer.” We’ll call it a false assumption. What do you incorrectly assume about the Unseen Walker Problem? Well, if you assume the walker is traveling aboveground, as did the unsuccessful solvers, you were almost certainly stumped. But what if he travels belowground in the sewers? Aha!
Then Ahmed and Patrick ran the experiment again, only this time they taught their participants to be on the lookout for these false assumptions. The training had two parts: first, they used an example to make participants aware of the blocking effects of assumptions. You’ve had an example already in the Unseen Walker Problem. And the second part of their training helped participants identify and overcome these assumptions in new problems.
The trick is learning to compare a problem to your interpretation of the problem (in these pages or in life!). For example, compare the following problem and interpretation from the ingenious book Lateral Thinking Puzzles by Paul Sloan and used by Ahmed and Patrick:
Problem: Archie and Ben were professional golfers and keen rivals. One day during a game, they had each scored 30 when Ben hit a bad shot. Archie immediately added 10 to his own score. Archie then hit a good shot and he had won the game. Why?
Interpretation: Two friends were playing golf, they were both on 30 points, then one reached 40 points and won.
Now ask yourself how the interpretation fails to match the problem—what does the interpretation assume that the puzzle doesn’t necessarily provide? Must they be playing golf? Of course not! And releasing that assumption should allow an insightful solution: they’re playing tennis, in which Archie’s score goes from 30 to 40 with Ben’s bad shot and then Archie wins the game on his next good shot.
After Ahmed and Patrick’s training, subjects’ solution rate for the Unseen Walker Problem jumped from 42 percent to just over 80 percent. On a second problem in which an airplane’s bomb doors open but no bombs fall, initially only 8 percent saw the solution—but after training, 50 percent of subjects saw that the airplane must be flying upside down. In both cases, the researchers verified that whenever participants remained unsuccessful, a pesky, unidentified assumption was to blame.
Now that you’ve received this training, let’s see how you do. Solving each of the following problems requires releasing a common false assumption. As in the examples earlier in the exercise, practice comparing your interpretation to the problem, looking for that sneaky assumption that blocks the solution. If needed, write or draw your interpretation (or both!) and then go through the problem, clause-by-clause, checking to see which elements are defined by the problem and which are figments of your interpretation. As Ahmed and Patrick’s work demonstrates, as you learn to eliminate these false assumptions your solution rate should rise.
There are four feisty kindergarteners on a standard elementary school playground. How can a playground aide arrange them so that they are all exactly the same distance apart?
Move 4 stars to form 5 straight rows of 4 “ships” each.
It’s a dark and stormy night and you’re driving down the street when you notice three people at a bus stop: an old woman who needs a doctor ASAP, your best friend, and the date of your dreams. You can only fit one other person in your car. What should you do?
There are 114 million households in the United States. The average household has 1.8 pets. If you multiply together the number of pets in each household, approximately what number do you get?
In 1942, Boston’s Coconut Grove nightclub caught fire. Patrons rushed the doors but despite the fact they were unlocked, the doors couldn’t be opened and 492 people were killed. The incident led to one important safety modification. What was this modification?
A prisoner is locked in a tower with a window high above the ground. He has a rope long enough to reach halfway down. He cuts the rope in half, ties the two ends together and lowers himself safely to the ground. How is this possible?
How can you cut a hole in a standard playing card large enough to put your head through?
White socks and black socks are in a drawer, with five white socks for every four black socks. How many socks must you pull from the drawer in order to ensure a pair of the same color?
Jane and Janet are sisters born five minutes apart to the same parents and yet they are not twins. How is this so?
How can you move one matchstick to make the following equation true?
V – VIII=III
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SLIDING TILE PUZZLES
You’ve seen sliding tile puzzles: with one open square, you must slide the other square tiles to put numbers in order or complete a picture. Psychologists use sliding tile puzzles to test and train three components of problem solving: subgoaling, chunking, and planning. First, subgoals provide manageable waypoints on the path to a seemingly insurmountable solution. For example, completing the final row and column of a 4×4 puzzle effectively makes it into a 3×3 puzzle. Second, “chunking” moves into learned sequences can help you reach these subgoals, for example automating the moves needed to switch a tile with one directly above it. Finally and most important, solving a sliding tile puzzle requires planning—because there are 181,440 possible board states in the 3×3 puzzle and about 653 billion states in the 4×4 puzzle, it’s pretty unlikely you’ll stumble blindly to an answer. Instead, you have to plan how you will sometimes move away from your goal so that you can move toward it. Researchers at the University of Nottingham showed that when solving the 3×3 puzzle, subjects who immediately started pushing tiles used an average 49.8 more moves than people who planned. And the preplanning group saved an average of 150 seconds per puzzle.
These skills of subgoaling, chunking, and planning ahead are major components of solving many real-world problems. Here you’ll practice these skills by making your own sliding tile puzzles. After looking at the following example, copy and cut out the number tiles and arrange numbers 1 through 8 into a randomized 3×3 grid with a tile missing (as in the example). Now your goal is to slide these tiles within the grid to put the numbers in order (again, as in the example). Spend some time planning: what’s a reasonable subgoal? What useful “chunked” sequences of moves can you memorize? Once you get comfortable with the 3×3 puzzle, try 4×4 or even—gasp!—5×5.
Note that these sliding tile puzzles are either “even” or “odd”—half of the starting positions allow you to put all numbers in order and half of the starting positions will end with the final two numbers frustratingly reversed. Consider either position an answer!
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