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)Bird and Trains –The monk problem (Duncker, 1945)monk problem

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” ovowel in the middle oassume “x” is not word-initial –“ntraoc” ostart 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: oThere is only one goal state and it is clearly specified oThere are a number of possible ways to represent the problem state

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