Introduction to Problem Solving

1. Introduction to Problem Solving Psychology 355: Cognitive PsychologyInstructor: John Miyamoto05/27/2014: Lecture 09-1 This Powerpoint presentation may contain macros that were used to create the slides. The macros aren’t needed to view the slides. If necessary, you can disable the macros without any change to the presentation.

2. Outline • Definition of “problem” • Information processing versus Gestalt approach to problem solving. Algorithmic problems & insight problems • Tower of Hanoi – an example of an algorithmic problem • Insight problems • Problem representation • Problem restructuring • Problem isomorphs Definition of Problem Solving Psych 355, Miyamoto, Spr '14

3. Definition of Problem Solving • A problem exists when the present state differs from a goal state. The problem is to change the present state into the goal state. • Initial state • Goal state • Permissible "moves" – ways to change the problem state from the initial state towards the goal state. • Interesting problems are situations where it is not obvious how to change the initial state into the goal state. • Cognitive psychology of problem solving – how do people solve problems. Examples of Problem Solving Situations Psych 355, Miyamoto, Spr '14

4. Problem Solving - Examples • Math problems, physics problems, science problems generally. • Initial state: The given information in the problem. • Goal state: The “answer” or solution to the problem. • Practical problems, e.g., arranging furniture, building a mechanical device. • Winning strategies in games, business, public health, law & war. Key Ideas in Theory of Problem Solving Psych 355, Miyamoto, Spr '14

5. Key Ideas in the Psychology of Problem Solving • Problem representation – The mental representation of the problem that the problem solver manipulates while trying to solve the problem. • Initial state • Goal state • Moves or transformations. Constraints and rules. ------------------------- • Insight problems & algorithmic problems • Restructuring a problem representation ------------------------- • Set • Functional fixedness Algorithmic vs Insight Problems Psych 355, Miyamoto, Spr '14

6. Algorithmic Problems versus Insight Problems • Algorithmic problems: The initial problem state can be transformed to the goal state by a systematic procedure. • Example: The Tower of Hanoi • Example: Solving a long division problem • Insight problems require mental restructuring of the problem representation to get a solution. • Circle problem • Mutilated checkerboard problem • Algorithmic and insight problems require somewhat different psychological processes to solve them. Tower of Hanoi – Example of an Algorithmic Problem Psych 355, Miyamoto, Spr '14

7. The Tower of Hanoi (A Problem with an Algorithmic Solution) • Tower of Hanoi is an algorithmic problem – there is a logically adequate strategy that will always solve this problem. General Idea of an Insight Problem Psych 355, Miyamoto, Spr '14

8. General Idea of an Insight Problem The Problem Representation Mental Representation of a Problem = Restructuring the Problem Representation Finding a New Way to Represent a Problem = • The solution of insight problems usually depends on finding a new way to represent the problem. Ideas from Gestalt Psychology • The mind searches for structure in perception • The mind searches for structure in problem solving Solving the Circle Problem by Restructuring the Problem Representation Psych 355, Miyamoto, Spr '14

9. The Circle Problem: An Example of an Insight Problem b Given: • radius r = 1 • length of a = 0.9 • line b is perpendicular to line a Question: What is the length of x? Hint: Change the problem representation. Initial Representation Restructuring the Representation of the Circle Problem Psych 355, Miyamoto, Spr '14

10. Restructuring the Representation of the Circle Problem b If r = 1, a = 0.9, and a and b are perpendicular, what is the length of x? • Solution: Add dashed line that connects the opposite corners. • Alternative representation: The answer is obvious: x = r = 1. • Alternative problem representa-tionmakes the solution obvious. • Solutions to insight problems often depend on a “trick”. • Here the trick is to change the problem representation. Alternate Representationfor the Circle Problem Another Insight Problem – the Mutilated Checkerboard Problem Psych 355, Miyamoto, Spr '14

11. Another Insight Problem – Mutilated Checkerboard Problem Problem: Cover the mutilated checkerboard with domino pieces so that every domino covers two squares OR if this is impossible, explain why it is impossible. The domino pieces must always be perpendicular or parallel to the sides of the board - they cannot be placed in a diagonal position. Failed Attempt to Solve the Mutilated Checkerboard Problem Psych 355, Miyamoto, Spr '14

12. Failed Attempt at Solving the Mutilated Checkerboard Problem Problem: Cover the mutilated checkerboard with domino pieces so that every domino covers two squares OR if this is impossible, explain why it is impossible. • Failure! • This is not a solution! • In fact, it is impossible to cover the mutilated checkerboard with dominoes. • Why is it impossible? Solution to the Mutilated Checkerboard Problem Psych 355, Miyamoto, Spr '14

13. Solution to the Mutilated Checkerboard Problem • Problem: Cover the checkerboard with domino pieces so that every domino covers two squares OR if this is impossible, explain why it is impossible. • A Solution is Impossible! • A domino piece always covers one dark square and one light square. Therefore any solution covers an equal number of dark and light squares. • The mutilated checkerboard has 30 dark squares and 32 light squares so it is impossible to cover an equal number of dark and light squares. Easy Version of the Mutilated Checkerboard Problem – The Matchmaker Problem Psych 355, Miyamoto, Spr '14

14. Easy Version of the Mutilate Checkerboard ProblemThe Russian Marriage Problem (a.k.a. the Matchmaker Problem) Hayes, 1978: [wording slightly altered below] In a small Russian village, there were 32 bachelors and 32 unmarried women. A matchmaker arranges 32 highly satisfactory marriages. The village was happy and proud. One night, two bachelors got drunk and killed each other. Can the matchmaker come up with heterosexual marriages (one man, one woman) among the 62 survivors? There are 30 men and 32 women. Obviously there is no way to match them into a complete set of heterosexual couples. Mutilated Checkerboard Problem & Russian Marriage Problem Are Isomorphs Psych 355, Miyamoto, Spr '14

15. Mutilated Checkerboard Problem & Russian Marriage Problem Russian Marriage Problem Mutilated Checkerboard Problem • The multilated checkerboard problem and the Russian marriage problem are problemisomorphs. • Problem Isomorphs: Problems that differ superficially but have identical logical structure. Concept of Problem Isomorphs Psych 355, Miyamoto, Spr '14

16. Concept of Problem Isomorphs • Problem isomorphs – structurally identical versions of a problem. • Basic fact about problem isomorphs: Some versions of a problem are harder to solve than other versions of the problem. • What is the psychological difference between the mutilated checkerboard problem and the matchmaker problem? • Kaplan and Simon: It is easier to solve the Russian marriage problem than the mutilated checkerboard problem, presumably because the Russian marriage version makes the importance of pairing men with women obvious. (See next slide) Four Isomorphic Versions of the Mutilated Checkerboard Problem Psych 355, Miyamoto, Spr '14

17. Kaplan & Simon: Four Isomorphic Versionsof the Mutilated Checkerboard Problem • Blank board is hardest problem. • “Bread”/“Butter” word labels are easiest problem. • Colored & “Pink”/“Black” word labels are intermediate difficulty. • The salience of the pairing affects difficulty. Blank(hardest) Colored(intermediate) “Pink” & “Black” Word Labels(intermediate) “Bread” & “Butter”(easiest) Conclusions re Problem Representation Psych 355, Miyamoto, Spr '14

18. Conclusion re Problem Representation • Some problem representations make problem solving easier than other problem representations. • Solving an insight problem often depends on finding a problem representation that make it obvious how to find the solution. Examples that support these claims: • Mutilated checkerboard problem; Russian marriage problem;other isomorphic versions. • Circle problem. . Cheap Necklace Problem – An Example of a False Constraint Psych 355, Miyamoto, Spr '14

19. Tuesday, May 27, 2014: The Lecture Ended Here Psych 355, Miyamoto, Spr '14