Problem Solving

Shortcuts through the Problem Space

Problem Solving

• Problem = a situation in which one is trying to reach a goal

• Problem solving = finding a means for arriving at a goal

Stages of Problem Solving

1. Define the problem -- Problem identification and representation

2. Select a strategy -- plan a solution

3. Carry out the strategy -- execute the plan

4. Evaluate the plan and the solution -- determine whether it worked

Defining the Problem

1. Identify the initial state and the goal state– Well-defined Problem: clear definition of problem

and goal states (example: locked out of your house)

– Ill-defined Problem: the problem or goal state is not clearly defined (example: increase crop production in USSR -- Voss, Grene, Post, & Penner, 1983)

2. Decide how to represent the initial and goal states

Problem Representation

• How difficult it is to solve a problem often depends on how you choose to represent it.

• Some examples:– "number scrabble" (Newell & Simon, 1972) – "Bird and Trains" math problem

(from Posner 1973, Cognition: An introduction) – The monk problem (Duncker, 1945)

Selecting a Strategy:Algorithms vs. Heuristics

• Algorithm– a procedure that is guaranteed to produce a

solution to the problem – Examples:

• solving the anagram "xbo“ by enumerating all possible combinations: xbo, xob, oxb, obx, bxo, box

• What about "ntraoc"? There are 6! (or 72) possible combinations.

Heuristics in Problem Solving

• Heuristic = a rule of thumb, or "mental shortcut" for solving a problem

• not guaranteed to give the right answer• usually much more efficient than an algorithm• Heuristics for solving anagrams:

– “xbo”o vowel in the middle o assume “x” is not word-initial

– “ntraoc”o start with likely groupings of letters: "ant, car, tan, tar, ton"

Problem Space and Computational Complexity

• Problem space = all the possible states of affairs that could be produced from transformations of the initial problem state.

• Problem solving consists of searching the problem space for a state that matches your goal state.

• Algorithms search the entire space; heuristics search only part

• If the problem space is too large, an algorithmic approach is impossible. Example: Chess.

Useful Problem-Solving Strategies (Heuristics)

• Simple Search (Hill Climbing)• Means-end Analysis

– Break the problem into subgoals

– Used in the General Problem Solver (Newell & Simon, 1963; Newell, Simon, & Shaw, 1958)

• Working Backwards. Useful when:

o There is only one goal state and it is clearly specified

o There are a number of possible ways to represent the problem state

Try Out Your Problem Solving Skills• The “Calvin & Hobbes” Game • The Water Jar Problem (Luchins, 1942)

• The “9 Dots” Problem – Can you connect all 9 of these dots by drawing 4 straight

lines, without lifting your pencil from the page?– Give up? Here is a solution.

“Set” in Problem Solving

• "set" = state of mind a person brings to a problem solving situation

• An inappropriate "set" can keep you from representing the problem in the most productive way, or from choosing the best solution strategy.

“Set” interfering with problem representation

• The nine dots problem -- including an unnecessary boundary in your representation of the problem

• Functional Fixedness: failing to see a new use for an object

– Duncker (1945) -- mount a candle on the wall– The two string problem

“Set” interfering with choosing an effective solution strategy

• "Persistence of set" in the water jar problem (Luchins, 1942)