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### About Methods for Solving ProblemsCourse

In this course, we will explore different types of problems, the boundaries of what makes a problem solvable, and the different algorithms, rules, and heuristics we can implement to solve these problems.

### WHAT YOU WILL LEARN

• Ascribe human and computational problems to various definitions of difficulty and problem-solving techniques.
• Use problem-solving strategies to break down problems.
• Describe the effects of various sources of bias in judgment and decision-making.
• Analyze and apply heuristic models and probability modeling to a variety of decision-making scenarios.

Course Apply Link – Methods for Solving Problems

### Methods for Solving ProblemsQuiz Answers

#### Quiz 1: Recognizing and Solving Problems

Q1. In the chapter by Pretz et al., there is a discussion of “well-defined” versus “ill-defined” problems. Which of the following is true of ill-defined problems?

• Ill-defined problems generally don’t lend themselves to “problem-space” representation.
• Ill-defined problems always have a clear and correct solution.
• Ill-defined problems tend not to be important.
• Ill-defined problems can be solved by well-understood search algorithms.

Q2. The Chase-Simon “chess memory” experiment illustrates which of the following ideas?

• Experts in a domain like chess have far better memory for all sorts of things than do chess novices.
• There are no measurable differences of any kinds between chess experts and chess novices.
• Chess experts are likely to be better than chess novices at solving problems in other domains (like physics or math).
• Chess experts internally represent chess games differently than do novices, and thus have better memory for realistic chess board positions than do novices.

Q3. Suppose we have a problem space representation in which the (lone) goal state can be reached via a sequence of edges from some states but not others. Which of the following statements is true of this problem (and its graph)?

• This is an ill-defined problem.
• As long as our start state is among the states with paths to the goal state, then we can consider this problem solvable.
• As long as our start state is among the states with paths to the goal state, then we can consider this problem easy.
• The number of “good” start states will always be fewer than the number of “bad” (no available path) start states.

Q4. Which of the following best describes (on the basis of experimental evidence) people’s use of logical reasoning?

• People can and do use logic extensively in their everyday reasoning.
• People are capable of reasoning logically, but it takes effort and they do so inconsistently (depending on context) in everyday situations.
• People are hopelessly incapable of reasoning logically.
• People can reason logically, but only if they are urged to do so consciously.

#### Quiz 2: A Classic Puzzle

Q1. Here’s a famous old math puzzle: the “college student” puzzle. A student sends home an addition problem, written out in letters:

` S E N D`
• M O R E

M O N E Y

Each of the 8 distinct letters in the puzzle represents a distinct digit from 0 to 9. (For example, if “M” means “5”, then no other letter can mean “5”.)

Suppose you were to try an exhaustive search for solutions to this puzzle, trying out every possible assignment of digits for the eight letters. How many possibilities would you have to try?

Q2. Now actually solve the puzzle from the previous question (It’s not easy, but it’s not a killer, either):

• S E N D
• M O R E

M O N E Y

The numbers in the addition problem are assumed to be written in standard fashion, where the first digit is not “0”. Thus, you know right away that neither “S” nor “M” in our problem can mean “0”.

Submit your answer by copying and pasting the following, but replacing lowercase variables with the relevant digits:

S=x,​ E=y, N=z, D=w, M=v, O=t, R=p, Y=q

*​your answer may not be recognized if you don’t directly copy the above line of text and substitute digits

`Enter answer here`

### Week 2

#### Quiz 1: Computers and Logic

Q1. Among Polya’s problem-solving heuristics is the suggestion “Look for a related problem that you know.” Why might this be an interesting or challenging suggestion for a computational problem solving system.

• Most problems are one-of-a-kind, and aren’t, in fact, related to any other problems.
• Pursuing this heuristic would involve tackling the notion of similarities or analogies between various (superficially distinct) problems. This would be an interesting research project!
• People do not in fact use Polya’s heuristic for problem-solving.
• There is rarely enough computer memory available to create a “repository” of problems.

Q2. The “rotating-quarters” problem is difficult to approach via the same methods as (say) Rubik’s Cube. Which of the following reasons is most relevant to this difficulty?

• Finding a solution appears to involve elements of mental imagery or mental simulation.
• The behavior of coins on a table involves a great deal of “common-sense” knowledge.
• The problem is not well-defined enough to have a clear solution.
• There are many ways to solve the problem; it’s not clear which one to use.

Q3. The “Teddy Roosevelt” problem is difficult to approach via the same methods as Rubik’s Cube. Which of the following reasons is most relevant to this difficulty?

• The problem is given in words rather than mathematical equations.
• There is a potentially (extremely) wide range of real-world or common-sense knowledge involved in answering the question; so the problem is not self-contained as many puzzles are.
• The problem is not well-defined enough to have a clear solution.
• We can only answer the problem with a statement of probability; we can’t be certain of the answer.

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Q4. Consider the “10 coins in three cups” problem given at: https\://curiosity.com/topics/can-you-solve-the-3-cups-10-coins-logic-puzzle-curiosity

What might make this difficult to solve with a computational system?

• A “standard” search program assumes a particular representation of the problem, while this particular problem involves finding a creative reconsideration of the problem statement itself.
• There is a tremendous amount of common-sense knowledge about (e.g.) coins and cups required to solve this problem.
• Since the problem can’t be solved, a computer program is bound to fail at it.
• This problem is not well-defined enough to have a clear solution.

Q5. Consider the “monkey climbing a rope” problem given at:

https\://activityworkshop.net/puzzlesgames/monkey/index.html

What makes this problem difficult?

• The problem involves an unavoidable degree of probability or uncertainty.
• The problem involves elements of physics knowledge and (most likely) mental simulation and imagery.
• The problem has an astronomically large problem-space representation.
• There is no solution to the problem.

Q6. Consider the “sand timers” problem (Problem 4) at the following website: https\://www.analyticsvidhya.com/blog/2016/07/20-challenging-job-interview-puzzles-which-every-analyst-solve-at-least

• This problem involves an unavoidable degree of probability or uncertainty in its solution.
• This problem requires way too much common-sense knowledge about the behavior of (e.g.) sand inside a bottle.
• This problem involves a high degree of mental imagery and thus would be difficult for a computer program to solve.
• This problem seems, in fact, amenable to a “standard” problem-space representation, and to solution via search. Occasionally our standard methods work, thank heavens.

Q7. Consider the following propositional logic statements:

• IF (P AND Q) THEN R
• P AND S
• Q AND V

Which of the following statements cannot be derived from these three statements?

• R
• P AND R
• S AND V
• (NOT P) OR (NOT Q)

Q8. Consider the hammerhead shark. Why do you think the animal evolved this sort of head shape?

• Presumably the distance between the two eyes helps with binocular vision (particularly for points in the region between the two eyes).
• It couldn’t have had anything to do with binocular vision.
• This is a likely instance of “sexual selection”, like the peacock’s tail: males evolved this head shape because females prefer it.
• The head shape is probably a good instance of predatory camouflage.

Q9. Why do we need to use a toroid shape in the Schelling model of neighborhoods as discussed in the lecture?

• A toroid allows all individuals to have exactly 8 neighbors, otherwise the corner location cannot possibly have 4 neighbors.
• A toroid is just more interesting – it doesn’t actually matter for our neighborhood problem.
• A toroid is a better representation of a neighborhood, because neighborhoods aren’t usually perfect squares
• A toroid is better suited to a multiple-agent model because there are edges.

Q10. What does this photograph suggest about the difficulty of the “vision” problem?

• Vision, because it involves all sorts of guesswork and heuristics in interpreting three dimensions from a two-dimensional projection, is capable of being confused or misled by certain images
• Vision is an easier problem for interpreting curved objects than interpreting objects with corners, like the “impossible triangle” in the photo.
• We can never get the vision problem “right” so it’s not even worth our while to attempt it.
• Vision is easy because we can always interpret retinal images unambiguously.

### Week 3

#### Quiz 1: Judgment and Decision Making

Q1. In the Kahneman and Tversky paper, which feature of human valuation is most relevant to the \$300 bonus scenario?

• People tend to be over-eager to engage in gambling scenarios.
• People place a higher value on small amounts of money than they do on large amounts of money.
• People are unpredictable when it comes to judgments about risk and reward.
• People’s valuation function is asymmetric, placing a greater relative weight on loss than on gain.

Q2. Which of these sentences, referring to the distinction between judgment and problem solving, is not true?

• Problem-solving is an evolutionarily more recent task than judgment.
• Problem-solving tends to involve effortful conscious search, whereas judgment is more reliant on (possibly unconscious) intuition.
• People are in general exceptionally good at both problem-solving and judgment tasks.
• Judgment tasks tend to involve more open-ended or common-sense scenarios than puzzle-like problem-solving tasks.

#### Quiz 2: Heuristics and Biases

Q1. Let’s see if you can recognize another (famous) example of the “conjunction fallacy”. Here’s a description of Linda: Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations. Here are your choices – your job is to rank order them in terms of probability:

• Linda is a teacher in an elementary school.
• Linda works in a bookstore and takes yoga classes.
• Linda is active in the feminist movement.
• Linda is a psychiatric social worker.
• Linda is a member of the League of Women Voters.
• Linda is a bank teller.
• Linda is an insurance salesperson.
• Linda is a bank teller and is active in the feminist movement.

Which of these statements is true?​

• Linda is more likely to be a bank teller than a bank teller who is active in the feminist movement.
• Linda is more likely to be a bank teller active in the feminist movement than a bank teller.
• Problem-solving tends to involve effortful conscious search, whereas judgment is more reliant on (possibly unconscious) intuition.
• Linda is most likely to be a teacher.
• There is no conclusion to be drawn from these choices.

Q2. The “decouple-the-metal-rings” problem is difficult to approach via the same methods as (say) Rubik’s Cube. Which of the following reasons is most relevant to this difficulty?

• There are very few steps required to solve the rings problem.
• Solving the metal rings problem does not require conscious thought, the way that Rubik’s Cube does.
• It is hard to know, on inspection, what constitute the distinct “states” of the problem space.
• There are very few people who are capable of solving the metal rings problem.

Q3. In fuzzy logic, it is possible to speak of “degrees of truth” using real numbers between 0 and 1. (Here, 0 corresponds to “false” and 1 to “true”.) Thus, we might say that the statement “The 2018 Boston Red Sox are a good baseball team” is “0.98 true” (i.e., close to certain) and “The 2018 Baltimore Orioles are a good baseball team” is “0.02 true” (i.e., very close to certainly false). Which of these statements – none of which is entirely true or false — is most true?

• October is a cold month in Boulder.
• February is a cold month in Boulder.
• May is a cold month in Boulder.
• July is a cold month in Boulder.

### Week 4

#### Quiz 1: Regression to the Mean

Q1. Suppose Mike is playing a coin game as follows. First he throws three fair coins into the air (First Toss). For each “heads”, he wins a dollar. Then, he tosses the three coins again (Second Toss). For each “heads” on this second toss, he also wins a dollar. (So he can win anywhere from \$0 to \$6 in one play of the game.)

What is the probability of tossing three heads on the first toss? (To three decimal places)

`Enter answer here`

Q2. What is the probability of tossing fewer than 3 heads on the first toss? (To three decimal places)

`Enter answer here`

Q3. What is the probability of tossing 0 heads on the first toss? (To three decimal places)

`Enter answer here`

Q4. Suppose Mike tosses 0 heads on the first toss. What is the probability of doing better than that on the second toss? (To three decimal places)

`Enter answer here`

Q5. Suppose Mike tosses 0 heads on the first toss and then curses the coins (“You stupid coins!”) What is the probability of doing better on the second toss? (To three decimal places)

`=Enter answer here`

Q6. Suppose Mike tosses 3 heads on the first toss. What is the probability of doing worse than that on the second toss? (To three decimal places)

`Enter answer here`

Q7. Suppose Mike tosses 3 heads on the first toss, and then praises the coins (“Good coins!”) What is the probability of doing worse on the second toss? (To three decimal places)

`Enter answer here`

Q8. Explain why these answers might lead Mike to (mistakenly) believe that criticizing the coins helps change their behavior in the right way, but praising the coins isn’t a good idea.

`What do you think?Your answer cannot be more than 10000 characters.`

#### Quiz 2: Small Numbers Fallacy

Q1. Suppose you toss 4 coins. What is the probability of getting 0 heads? What is the probability of getting 0 tails?

`Enter answers here`

Q2. Now, suppose you toss 100 coins. What is the probability of getting 0 heads? Or 0 tails?

Preview will appear here…

### Conclusion  