Podcast
Questions and Answers
Who is known as the father of modern problem solving and created a famous four-step process?
Who is known as the father of modern problem solving and created a famous four-step process?
George Polya
What is the first step in Polya's four-step problem-solving strategy?
What is the first step in Polya's four-step problem-solving strategy?
Understand the Problem
According to Polya's strategy, what should you do after understanding the problem?
According to Polya's strategy, what should you do after understanding the problem?
Devise a Plan
What is the third step in Polya's problem-solving process?
What is the third step in Polya's problem-solving process?
What is the final step in Polya's four-step strategy?
What is the final step in Polya's four-step strategy?
A baseball team won two out of their last four games. In how many different orders could they have achieved two wins (W) and two losses (L) in these four games?
A baseball team won two out of their last four games. In how many different orders could they have achieved two wins (W) and two losses (L) in these four games?
The product of the ages, in years, of three different teenagers (ages 13-19) is 4590. What are the ages of the teenagers?
The product of the ages, in years, of three different teenagers (ages 13-19) is 4590. What are the ages of the teenagers?
Place the digits 8, 9, 10, 12, & 13 in the circles (one central, four surrounding) so that the sum of the three numbers vertically and the sum of the three numbers horizontally both equal 31.
Place the digits 8, 9, 10, 12, & 13 in the circles (one central, four surrounding) so that the sum of the three numbers vertically and the sum of the three numbers horizontally both equal 31.
Using only a 5L can and an 11L can, how can you measure out exactly 7L of water?
Using only a 5L can and an 11L can, how can you measure out exactly 7L of water?
Mary is thinking of a number. If you double it and subtract 7, you obtain 11. What is the number?
Mary is thinking of a number. If you double it and subtract 7, you obtain 11. What is the number?
Fill the missing number in the sequence: 1, 4, 9, 16, _____, 36, 49, 64, 81, 100, ...
Fill the missing number in the sequence: 1, 4, 9, 16, _____, 36, 49, 64, 81, 100, ...
Let F(n) denote the nth term in the Fibonacci Sequence where F(1)=1, F(2)=1, F(3)=2, etc. Find the least value of n such that F(n) > 500.
Let F(n) denote the nth term in the Fibonacci Sequence where F(1)=1, F(2)=1, F(3)=2, etc. Find the least value of n such that F(n) > 500.
The sum of the two digits of a 2-digit number is 11. Reversing the digits increases the number by 45. What is the number?
The sum of the two digits of a 2-digit number is 11. Reversing the digits increases the number by 45. What is the number?
What type of reasoning forms a conclusion based on the examination of specific examples or observations?
What type of reasoning forms a conclusion based on the examination of specific examples or observations?
Use inductive reasoning to predict the next number in the sequence: 5, 10, 15, 20, ?
Use inductive reasoning to predict the next number in the sequence: 5, 10, 15, 20, ?
A conclusion formed using inductive reasoning, which may or may not be correct, is often called a what?
A conclusion formed using inductive reasoning, which may or may not be correct, is often called a what?
The period of a pendulum (time for one full swing) seems to be related to its length. If periods (in heartbeats) for lengths (in units) are (1,1), (4,2), (9,3), (16,4), (25,5), (36,6), what period would you conjecture for a pendulum with a length of 49 units?
The period of a pendulum (time for one full swing) seems to be related to its length. If periods (in heartbeats) for lengths (in units) are (1,1), (4,2), (9,3), (16,4), (25,5), (36,6), what period would you conjecture for a pendulum with a length of 49 units?
Based on the pendulum data (lengths 1, 4, 16 producing periods 1, 2, 4), what appears to happen to the period when the length of the pendulum is quadrupled?
Based on the pendulum data (lengths 1, 4, 16 producing periods 1, 2, 4), what appears to happen to the period when the length of the pendulum is quadrupled?
What is the term for a specific case that shows a general statement is false?
What is the term for a specific case that shows a general statement is false?
Find a counterexample to show the statement 'For all numbers x, $|x| > 0$' is false.
Find a counterexample to show the statement 'For all numbers x, $|x| > 0$' is false.
Find a counterexample to show the statement 'For all numbers x, $x^2 > x$' is false.
Find a counterexample to show the statement 'For all numbers x, $x^2 > x$' is false.
What type of reasoning reaches a conclusion by applying general principles, procedures, or facts?
What type of reasoning reaches a conclusion by applying general principles, procedures, or facts?
Use deductive reasoning: If the procedure is 'Pick a number (x), multiply by 9, add 6, divide by 3, subtract 2', what is the resulting expression in terms of x?
Use deductive reasoning: If the procedure is 'Pick a number (x), multiply by 9, add 6, divide by 3, subtract 2', what is the resulting expression in terms of x?
Determine the occupation (editor, banker, chef, dentist) of each neighbor (Kristan, Michael, Luis, Francis) using these clues:
- Michael gets home from work after the banker but before the dentist.
- Luis, who is the last to get home, is not the editor.
- The dentist and Luis leave for work at the same time.
- The banker lives next door to Francis.
Determine the occupation (editor, banker, chef, dentist) of each neighbor (Kristan, Michael, Luis, Francis) using these clues:
- Michael gets home from work after the banker but before the dentist.
- Luis, who is the last to get home, is not the editor.
- The dentist and Luis leave for work at the same time.
- The banker lives next door to Francis.
What is an ordered list of numbers called?
What is an ordered list of numbers called?
What are the individual numbers in a sequence called?
What are the individual numbers in a sequence called?
What is the nth term formula for the sequence of even numbers: 2, 4, 6, 8, 10, ...?
What is the nth term formula for the sequence of even numbers: 2, 4, 6, 8, 10, ...?
A sequence is defined by the formula $a_n = 3n^2 + n$. What is the 20th term ($a_{20}$) of this sequence?
A sequence is defined by the formula $a_n = 3n^2 + n$. What is the 20th term ($a_{20}$) of this sequence?
In a KenKen puzzle, a number cannot be repeated within a heavily outlined cage.
In a KenKen puzzle, a number cannot be repeated within a heavily outlined cage.
In a KenKen puzzle, the numbers in each row and column must be unique.
In a KenKen puzzle, the numbers in each row and column must be unique.
What is the minimum number of moves required to solve the Tower of Hanoi puzzle with n disks?
What is the minimum number of moves required to solve the Tower of Hanoi puzzle with n disks?
What is the minimum number of moves required to solve the Tower of Hanoi puzzle with 3 disks?
What is the minimum number of moves required to solve the Tower of Hanoi puzzle with 3 disks?
Flashcards
Polya's Four-Step Problem Solving Strategy
Polya's Four-Step Problem Solving Strategy
A four-step approach created by George Polya used to solve problems. Includes understanding the problem, devising a plan, carrying out the plan, and reviewing the solution.
Inductive Reasoning
Inductive Reasoning
Forming a conclusion based on examining specific examples; conclusions may not always be accurate.
Deductive Reasoning
Deductive Reasoning
It is the process of reaching a conclusion by applying general principles and procedures.
Counterexample
Counterexample
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Sequence
Sequence
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Terms of a sequence
Terms of a sequence
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an
an
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nth term formula
nth term formula
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KenKen Puzzle
KenKen Puzzle
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Tower of Hanoi
Tower of Hanoi
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Study Notes
- Problems are unavoidable, needing skills for solutions, this module studies approaches to problem solving, focused on pattern recognition, critical/logical thinking and creativity, to improve problem-solving skills and make it enjoyable.
Learning Outcomes
- Explain Polya's four-step problem-solving strategy.
- Apply Polya's strategy to solve problems.
- Distinguish between inductive and deductive reasoning.
- Give examples of inductive and deductive inference processes.
- Use inductive or deductive reasoning to solve practical problems.
- Solve problems using various approaches/strategies.
Polya's Strategy
- George Polya, considered the father of modern problem solving, developed a four-step process:
- Understand the Problem: Have a clear understanding
- Device a Plan: Enumerate steps, list information, use variables, translate to math, draw, etc.
- Carry Out the Plan: Carefully and accurately execute the plan, noting attempts.
- Review the Solution: Ensure consistency, interpret in context, review details, check validity.
Problem Solving Strategies
- Guess and test (trial and error) is a common method, trying answers and checking them.
- Picture/Diagram/Experiment is easier because of visualising the problem
- Working backwards involves starting from the final output and reversing steps to find the initial value.
- Looking for patterns involves identifying a repeating sequence or relationship in numbers.
- Listing/tabular method involves listing possibilities to look at it in an organised way.
- Algebraic equations are for complex problems requiring variables and equations.
Logical Reasoning
- Uses analysis to find solutions, applicable with other methods.
Inductive Reasoning
- Creates a conclusion from specific examples.
- The conclusion is called a conjecture, which may not always be correct.
- A counterexample proves a statement false if one case is untrue.
Deductive Reasoning
- Reaches a conclusion by applying general principles.
Mathematical Problems Involving Patterns
- A sequence is an ordered list of numbers (e.g., 6, 12, 18, 24).
- Terms refer to the numbers in a sequence(e.g, 6 is the first term).
- an is the nth term of a sequence, where a1, a2,..., an represents a sequence of n terms.
- To identify the next term, analyze the pattern in a sequence.
- The nth term formula generates sequence terms.
- For the sequence 2, 4, 6, 8, 10: The nth term formula is an=2n.
KenKen® Puzzles
- KenKen® is an arithmetic-based logic puzzle by Tetsuya Miyamoto in 2004, named after the words "knowledge" and "awareness".
- Rules include:
- For an n by n puzzle, each square contains the numbers 1 to n;
- Do not repeat a number in any row or column;
- Cages produce the target number;
- One-square cages are filled with the target number;
- Numbers cannot repeat within same cage in the same columns/ rows;
Basic Puzzle Solution Strategies
- Single-Square Cages: Fill with the target number.
- Cages with Two Squares: These usually have only two possible digits.
- Large or Small Target Numbers: Look for combinations.
- Duplicate Digit in a Cage: Digits can occur more than once but not in the same row/column.
Additional tips
- Remember rules: Each row/column must contain 1 to n; subtraction/division order is not important.
- Make a list of possible digits for each cage.
- Guess and check to test assumptions.
Tower of Hanoi
- The Tower of Hanoi is a puzzle where the objective is to transfer a tower of disks from one peg to another, subject to disk size restrictions.
- The minimal number of moves is 2n-1, where n is the number of disks.
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