Problem Space

Intimate Engagement:

• Recall 2 properties:

• 1. Motivation --> psychological

• 2. Visualization

Visualization: Externalize

• Like the mathematician -- use a pencil and paper

• Relieves the mind of having to organize all the information

• Designing web pages

Special Features of the Problem Space

• Think about problems which are difficult to solve

• When we ask for a hint or receive a hint (gamers sometimes download hints from the WWW), we can be led in new directions in the problem space

Information-Processing Approach to Problem-Solving

• Problem-Space Theory– solving a problem involves negotiating

alternative paths to a solution

– initial state is linked to goal state

– knowledge states are produced by the application of mental operators

– algorithms vs. heuristics

– limited processing resources

Maier’s 2-String Problem• In this problem, the subject is in a room with two

strings tied to the ceiling. Both strings are of equal length. The objective is to tie the ends of the two strings together.

• The problem is that while the strings are long enough to be tied together they are short enough that one is unable to just take hold of one string, walk over to the other string, and tie them together. On the floor is a pair of pliers.

So what’s the problem?

• The string is too short

• My arms are too short

• The end of one sting will not stay anchored in place while I get the other

• The string won't come to me

• if the only object in the room were a pair of pliers, then the solutions become much more limited, as this object cannot be used to resolve all the possible problem sources.

• About 60% of the participants in Maier's study failed to find a solution within a 10-minute time limit.

• These participants saw the pliers only as the traditional tool they are, not recognizing that the pliers could be used as a pendulum bob, swinging at the end of one of the two strings, thus resolving the "string won't come to me" problem source.

• As it is known 2 multiplied by 2 produces the same result as 2 added to 2. It is 4 in both cases - just as shown in the lower left corner of the illustration.

Can you think of another pair of numbers (a and b) which when multiplied together give the same result when they are added together? In other words it should be such a and b numbers that the two equations shown in the illustration are true, i.e. a + b = c anda x b = c.

• The formula which helps to find such pairs is derived from the simple equality that a+b = a x b. From this we have that b always equals a/(a-1). The formula shows there is an infinity number of such pairs existing.

Thus if a is 3 then according to that formula above b is 1.5. When they are multiplied the result is the same as when they are added together.

Find the digit represented by each letter:

SEND

+ MORE

MONEY

9 5 6 7

+ 1 0 8 5

1 0 6 5 2

Flagpole Problem

• Two flagpoles are standing, each 100 feet tall. A 150-foot rope is strung from the top of one of the flagpoles to the top of the other and hangs freely between them. The lowest point of the rope is 25 feet above the ground. How far apart are the flagpoles?

• Ans: The flagpoles are right next to each

other.

Gestalt Viewpoint

• Problem-solving is both reproductive and productive

• Reproductive PS involves re-use of previous experience (can be beneficial or detrimental)

• Productive problem-solving is characterized by restructuring and insight

• Insight accompanied by subjective “ah-ha”

L

+ W

_______

LA

Ans: L = 1, W = 9, and A = 0

• A person starts striking a bass drum every 2 seconds. From drum beat 1 to drum beat 60, how much time has elapsed?

Tower(s) of Hanoi

• It consists of three pegs, and a number of discs of different sizes which can slide onto any peg. The puzzle starts with the discs neatly stacked in order of size on one peg, smallest at the top, thus making a conical shape.

• The object of the game is to move the entire stack to another peg, obeying the following rules:

* only one disc may be moved at a time* no disc may be placed on top of a smaller disc

N

6

5

4

3

2

Minimum # of movesN=# of discs

2N-1N

636

315

154

73

32

Minimum # of movesN=# of discs

Recursive algorithm

* label the pegs A, B, C -- these labels may move at different steps

* let n be the total number of discs* number the discs from 1 (smallest, topmost) to n

(largest, bottommost)

To move n discs from peg A to peg B:

1. move n−1 discs from A to C. This leaves disc #n alone on peg A

2. move disc #n from A to B

3. move n−1 discs from C to B so they sit on disc #n

The Notched Checkerboard

• You are given a checkerboard and 32 dominoes. Each domino covers exactly two adjacent squares on the board. Thus, 32 dominoes can cover all 64 squares of the checkerboard. Now suppose two squares are cut off at diagonally opposite corners of the board. Is it possible to place 31 dominoes on the board so that all of the 62 remaining squares are covered? If possible, show how it can be done. If not possible, give a convincing reason for the conclusion.

Answer:

• Since a domino covers adjacent squares, these squares are of a different color. The two squares that have been removed are of the same color, hence it is impossible to have 31 dominoes covering 62 squares.

Exchanging Knights

• A three by three chessboard has two black knights on adjacent corners of the board. Two white knights occupy the other two corners. A knight can only move to an unoccupied square that lies one square in one direction and two squares in a direction at right angles to the first direction. (That is, the legal moves permitted for the knight in chess.) It is desired to interchange the two black knights with the two white knights. What is the least number of moves required to accomplish this?

B

B

W

W