Problem Solving and Reasoning

Flashcards covering inductive and deductive reasoning, logic puzzles, Polya's four-step problem-solving strategy, summation formulas, and sequence analysis using difference tables.

What is inductive reasoning?

The process of reaching a general conclusion by examining specific examples.

What is a conjecture?

A conclusion formed based on the examination of specific examples through inductive reasoning, which may or may not be correct.

Using inductive reasoning, what is the next number in the list $1, 3, 6, 10, 15, \dots$?

The next number is $21$. The differences between successive terms increase by $1$ each time ($2, 3, 4, 5$), so the next difference is $6$, giving $15 + 6 = 21$.

What conjecture is formed by testing the procedure: pick a number, multiply by $8$, add $6$, divide by $2$, and subtract $3$?

The procedure produces a resulting number that is four ($4$) times the original number.

How did Galileo Galilei (1564–1642) measure the period of pendulums, and what unit was designated as $1\text{ unit}$ of length?

He measured the period of pendulums in "heart-beats," and designated a length of $10\text{ inches}$ as $1\text{ unit}$.

Based on Galileo's pendulum data, what is the period for a length of $49\text{ units}$, and what happens to the period when length is quadrupled?

A pendulum with a length of $49\text{ units}$ has a period of $7\text{ heartbeats}$. Quadrupling the length of a pendulum doubles its period.

What is deductive reasoning?

The process of reaching a conclusion by applying general assumptions, procedures, or principles.

How is deductive reasoning used to prove that the procedure 'multiply by $8$, add $6$, divide by $2$, and subtract $3$' always yields four times the original number?

Letting $n$ represent the original number: multiply by $8$ gives $8n$, add $6$ gives $8n + 6$, divide by $2$ gives $\frac{8n + 6}{2} = 4n + 3$, and subtract $3$ yields $4n + 3 - 3 = 4n$.

In the logic puzzle with neighbors Sean, Maria, Sarah, and Brian, what are each neighbor's determined occupations?

Sean is the banker, Maria is the editor, Sarah is the chef, and Brian is the dentist.

Why is the argument "All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost ₱35,000. Thus, my home improvement will cost more than ₱35,000" classified as deductive reasoning?

Because the conclusion is reached by applying a specific case to a general assumption.

What are the four steps in George Polya’s Problem Solving Strategy?

1. Understand the problem.
2. Devise a plan.
3. Carry out the plan.
4. Review the solution.

What questions should be considered during Polya's step 1, 'Understand the Problem'?

• Can you restate the problem in your own words? • Can you determine what is known about these types of problems? • Is there missing information that, if known, would allow you to solve the problem? • Is there extraneous information that is not needed to solve the problem? • What is the goal?

Name at least five procedures commonly used to 'Devise a Plan' in Polya’s framework.

Making a list of known or needed information, drawing a diagram, making an organized list of possibilities, making a table or chart, working backwards, trying a similar simpler problem, looking for a pattern, writing an equation, performing an experiment, or guessing and checking.

How did Karl Friedrich Gauss quickly calculate the sum of the first $100$ natural numbers?

He grouped the numbers into $50$ pairs that each sum to $101$ ($1 + 100$, $2 + 99$, $3 + 98$, etc.) and multiplied $50 \times 101$ to obtain $5050$.

What is the summation formula for the first $n$ natural numbers?

$$1 + 2 + 3 + \dots + (n - 2) + (n - 1) + n = \frac{n(n + 1)}{2}$$

In how many different orders could a baseball team have two wins ($\text{W}$) and two losses ($\text{L}$) in four games?

Six ($6$) different orders: WWLL, WLWL, WLLW, LWWL, LWLW, and LLWW.

What is a sequence, and what notation is customarily used to designate its $n\text{th}$ term?

An ordered list of numbers is called a sequence, and the subscript notation $a_n$ is used to designate its $n\text{th}$ term.

What is the formula to find the $n\text{th}$ term of an arithmetic sequence, and what does it yield for $1, 4, 7, 10, \dots$?

$\text{nth term} = d n + (a - d)$, where $d$ is the difference between terms and $a$ is the first term. For $1, 4, 7, 10, \dots$, it becomes $3n + (1 - 3) = 3n - 2$.

In a difference table, what are first differences and second differences?

First differences are the differences between successive terms of the original sequence. Second differences are the successive differences between the first differences.

Using a difference table, how is the next term of the sequence $5, 14, 27, 44, 65, \dots$ found?

The second differences are constant at $4$. Adding $4$ to the last first difference ($21$) gives the next first difference of $25$. Adding $25$ to the last term ($65$) predicts the next term is $90$.