2. Problem-Solving

3. Water Jugs Problem You are given two jugs, a 4-gallon jug & a 3-gallon jug. Neither has any measuring marks on it. There is a pump that can be used to fill the jugs with water. And you can dump water out on the ground. How can you get exactly 2 gallons of water in the 4-gallon jug?

4. Anatomy of a Problem Initial state Goal state Problem space  All the possible solution paths from the initial state to the goal state  “What is a problem? … The problem lies in the gap which separates where you are from where you want to be.” Hayes (1978) States Operators 0 4 0 3 2 4 0 3 Fill Pour Empty Water Jug Problem?

5. 0 4 0 3 4 4 0 3 4 4 3 3 0 4 3 3 4 4 3 3 1 4 3 3 4 4 3 3 Operators: Fill Empty Pour 0 4 0 3 0 4 3 3 1 4 0 3 4 4 0 3 4 4 0 3 1 4 3 3 0 4 0 3 0 4 1 3 4 4 1 3 0 4 3 3 0 4 0 3 1 4 0 3 4 4 3 3 0 4 1 3 4 4 0 3 2 4 3 3 Problem State

6. 0 4 0 3 Operators: Fill Empty Pour

7. 0 4 0 3 4 4 0 3 4 4 3 3 0 4 3 3 Operators: Fill Empty Pour

8. 0 4 0 3 4 4 0 3 4 4 3 3 0 4 3 3 1 4 3 3 4 4 3 3 Operators: Fill Empty Pour 0 4 0 3

9. 0 4 0 3 4 4 0 3 4 4 3 3 0 4 3 3 4 4 3 3 1 4 3 3 4 4 3 3 Operators: Fill Empty Pour 0 4 0 3 0 4 3 3 1 4 0 3 4 4 0 3

10. 0 4 0 3 4 4 0 3 4 4 3 3 0 4 3 3 4 4 3 3 1 4 3 3 4 4 3 3 Operators: Fill Empty Pour 0 4 0 3 0 4 3 3 1 4 0 3 4 4 0 3 4 4 0 3 1 4 3 3 0 4 0 3 0 4 1 3

11. 0 4 0 3 4 4 0 3 4 4 3 3 0 4 3 3 4 4 3 3 1 4 3 3 4 4 3 3 Operators: Fill Empty Pour 0 4 0 3 0 4 3 3 1 4 0 3 4 4 0 3 4 4 0 3 1 4 3 3 0 4 0 3 0 4 1 3 4 4 1 3 0 4 3 3 0 4 0 3 1 4 0 3

12. 0 4 0 3 4 4 0 3 4 4 3 3 0 4 3 3 4 4 3 3 1 4 3 3 4 4 3 3 Operators: Fill Empty Pour 0 4 0 3 0 4 3 3 1 4 0 3 4 4 0 3 4 4 0 3 1 4 3 3 0 4 0 3 0 4 1 3 4 4 1 3 0 4 3 3 0 4 0 3 1 4 0 3 4 4 3 3 0 4 1 3 4 4 0 3 2 4 3 3 WOOT!

13. Haiku Problem Haiku Poems  5 syllable line  7 syllable line  5 syllable line Write a brilliant & moving haiku about a water jug. 

14. Well/Ill-Defined Problems Well-Defined problems  clear initial state & goal  clear problem space Ill-Defined Problems  No clear initial state  No clear goal state oWhen do you know you have succeeded? oWhat are the criteria?  Complex (un-mappable) problem-space School is filled with examples of well-defined problems, while everyday life is filled with ill-defined ones.

15. Stages in Problem-Solving Familiarization Stage. Time spent understanding the nature of the problem & the goal (planning)  Production stage. Production of solution paths that define the problem space (Incubation stage) Occurs when no solution path is found; period spent not actively working on problem  Not working on problem @ ‘unconscious level’  Dissipate fatigue  Dissipate set effect (functional fixedness) Evaluation stage. Evaluation of the solution paths in order to select best one

16. Problem Planning IDEAL Model (Bransford & Stein, 1993) Identify the Problem Define & represent the problem* Explore possible strategies Act on the strategies Look back & evaluate the effects of your activities Tips on Representing Problems  Represent the goal in multiple ways  Write down all given (& inferred) information  Group related information  Leverage relevant prior knowledge  Represent the problem in multiple modalities e.g., graph, diagram, chart, models, text, etc.

17. Heuristics for Problem Representation External representations Problem type Graph Experimental results Patterns of numbers Diagrams Math problems Spatial problems Many complex interrelationships Hierarchical Trees Hierarchical information Matrices Givens can be grouped into categories for comparisons Models Movement/placement determine solution 

18. (Some) Problem-Solving Strategies Means-end analysis. Break problem down into sub-problems with sub-goals, solve in parts Working backwards. Begin with goal state. apply reverse operators to move toward initial state  (both)  (both) Simplification. Strip away details that complicate the problem, then later add them back in Generalization/Specialization. Consider the problem from both broad & narrow perspectives Analogies & Metaphors. Compare to similar problems that you have already solved   Trial & Error. If all else fails, use trial & error (best for well-defined problems) X

19. Haiku: Example Reach for water jug Pour us some marguaritas Waves attack my toes. ~ Magnifico, Alecia. (2004 

20. Why Representation Matters Mutilated Checkerboard Problem Suppose we have a checkerboard in which the two diagonally opposite corner squares have been cut out so that 62 squares remain. Now suppose that we have 31 dominoes, each of which covers exactly two squares of the board. Can you find some way of arranging these 31 dominoes on the boards so that they can cover all 62 squares? Cant be done! The trick is to include in your representation the fact that each domino must cover 1 black and 1 red square (not just any 2 squares). There’s just no other way. This means that with 31 dominoes we can cover 31 black & 31 red squares. But the mutilation removed two red squares, leaving 33 red & 32 black. So…its impossible. 

21. Tower of Hanoi  There are three pegs & three disks of differing sizes (A, B, C). The disks have holes in them so they can be stacked on the pegs. The disks can be moved from any peg to another peg. Only the top disk on a peg can be moved, and it can never be placed on a smaller disk. The disks start out on peg 1, but the goal is to move them all to peg 3, one disk at a time, by means of transferring disks among pegs. Initial State 123 A B C Goal State 123 A B C Production System

22. Analogy 1 Suppose you are a doctor faced with a patient who has a malignant tumor in his stomach. It is impossible to operate on the patient, but unless the tumor is destroyed the patient will die. There isa kind of ray that can be used to destroy the tumor. If the rays reach the tumor all at once at a sufficiently high intensity, the tumor will be destroyed. Unfortunately, at this intensity the healthy tissue that the rays pass through on the way to the tumor will also be destroyed. At lower intensities the rays are harmless to healthy tissue, but they will not affect the tumor either. What type of procedure might be used to destroy the tumor with the rays, and at the same time avoid destroying the healthy tissue? ~ Gentner (1983)

23. Analogy 2 A small country was ruled from a strong fortress by a dictator. The fortress was situated in the middle of the country, surrounded by farms and villages. Many roads led to the fortress through the countryside. A rebel general vowed to capture the fortress. The general knew that an attack by his entire army would capture the fortress. He gathered his army at the head of one of the roads, ready to launch a full-scale direct attack. However, the general then learned that the dictator had planted mines on each of the roads. The mines were set so that small bodies of men could pass over them safely, since the dictator needed to move his troops and workers to and from the fortress. However, any large force would detonate the mines. It therefore seemed impossible to capture the fortress. However the general devised a simple plan. He divided his army into small groups and dispatched each group to the head of a different road. When all was ready he gave the signal and each group marched down a different road. Each group continued down the road ot the fortress so that the entire army arrived together at the fortress at the same time. In this way, the general captured the fortress and overthrew the dictator. ~ Gentner (1983) 