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Problem Solving

• Goals• Objects / givens• Operations / transformations• Constraints / obstacles

Types of Problems

• Well-defined• Ill-defined

2

Problem Solving Process

1. Form initialproblem

representation

2. Try to planpotentialsolution

4. Execute planand check

results

3. Try toreformulate

problem

Done!

Stuck!(return to step3 after a break)

then success

if plan fails

if fa

il

if success

if fail

Initial Representation

One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. A narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended at varying rates of speed, stopping many times along the way to rest and eat dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting and meditation he began his journey back along the same path, starting at sunrise and again walking at variable speeds, with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed. Show that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

2 3 4 51 6 87 9

Initial RepresentationEach of the digits 1 through 9 is written on a separate piece of paper. Two players draw digits alternately. As soon as either player gets any three digits that sum to 15, that player wins. If all nine digits are drawn without a win, then the game is a draw.

Player 1:0

Player 2:0

294

753

618

3

Types of Representations

PSY 502

PSY 217

PSY 529,531Thurs.

PSY 216

PSY 101

PSY 522Wed.

PSY 502

PSY 217

PSY 529,531Tues.

PSY 216

PSY 101

Fri.

PSY 216

PSY 101

PSY 522Mon.

SJ 003SJ 231SJ 325

Plan a Solution

• Algorithm vs. heuristic

• Some heuristics:– Generate-test method– Simple search (e.g. hill climbing)– Means-ends analysis– Working backward

Plan a Solution:Remove Unnecessary Constraints

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Plan a Solution:Applying an Analogy

Blocks that Hinder Problem Solving

25376288183391572034923631459205216429451043183

99251631421003127211

CBAAmount to get

Capacity of Jars

Blocks that Hinder Problem Solving

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Blocks that Hinder Problem Solving

Analyzing Problem-Solving Behavior: Verbal Protocols

• Solve:CROSS+ROADS------DANGER

1. Experimenter: R is 6:C60SS+6OADS------DANGE6

2. Since R is 6 and the two S’s are equal…

3. S must be equal to 3, or 8.4. And D must be equal to 1…5. Because C plus R can’t be

greater than 19…6. Or greater than 16 in this

case.7. So D must be a 1.

8. Would you make D a 1?9. Experimenter: D is 1.10. And seeing that two S’s are

equal…11. They must be either 3 or 8…12. If they are 8…13. Then E would be 0.14. If they are 3…15. Then E would be 4.16. So let’s try the S’s as 8.17. Could you make the S an 8?18. Experimenter: S is 8.19. That would make E a 0.20. Would you make E a 0?21. Experimenter: E is 0.

Analyzing Problem-Solving Behavior: Verbal Protocols

1 R = 6

2 2 S’s are =

3 S = 3 or 8

4 D = 1

10 11 S = 3 or 8

12 If S=8, E=0

14 If S=3, E=4

15 16 Set S = 8

19 Set E = 0

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Analyzing Problem-Solving Behavior: Computer Simulations

• GPS• Explicit about process• Internal consistency• Sufficiency

• Does not demonstrate whether program solves the problem in the same way as humans

Analyzing Problem-Solving Behavior: Intermediate Products