Learn Polya’s problem-solving process and strategies, such as divide and conquer and pattern identification, to enhance your analytical skills, tackle puzzles, and solve real-world problems.

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Problem-solving skills are vital for personal growth, academic success, and professional achievement. They enable resilience, innovation, and effective decision-making. This course equips you with a lifelong skill set by teaching Polya’s problem-solving process and its applications in tackling real-world problems, games, and puzzles.

You will explore strategies, including top-down and bottom-up approaches, divide and conquer, stepwise refinement, pattern identification, and logical reasoning through puzzle-solving and computing logic.

By the end of the course, you’ll master analytical skills and critical thinking to apply Polya’s method and other strategies to complex challenges. This comprehensive approach establishes a strong foundation for systematically solving intricate problems.

Problem Solving

# Solution

We’ve learned how to use Gauss’s method to calculate the sum of the first 100 consecutive numbers. Let’s apply this technique to assist little Jill in determining the number of blocks needed for her wall. In this case, we have only $50$ numbers and the corresponding illustration will look like this:
 


So we have $25$ pairs of numbers that add up to $51$.

The total number of blocks required to make Jill’s triangular wall is: 

$$\text{Number of blocks} = 25 \times 51 = 1275$$

## Arithmetic series

A sequence of numbers like the one presented in this chapter is called an **arithmetic sequence**—a sequence of numbers in which the difference between two consecutive terms is constant. The sum of the terms in such a sequence is called an **arithmetic series**. 

## The general solution to arithmetic series

We’ve used Gauss’s method to calculate the required number of blocks for Jill’s triangular wall. To utilize it for any problem related to arithmetic series, we will need a generalized version of this formula.

We multiply the number of pairs by the sum of each pair. We can write it as follows:

$$\text{Sum} = \text{Number of pairs}\times(\text{first term + last term})$$

And what are the number of pairs here?
The number of pairs is half of the total terms of the list, which can be calculated as shown below:

$$\text{Sum} = \frac{\text{Total numbers}}{2}\times(\text{first term + last term})$$

Let’s say we have a list of numbers following the arithmetic sequence, $a_1, a_2,..., a_n$, where $n$ is the total number of terms and $a_1$, and $a_n$ are the first and last terms. Now, we can solve our problem using this formula. The sum of the series will be:

$$\text{Sum} = \frac{\text{n}}{2}\times(a_1 + a_n)$$

Now we can solve our problem using this formula.

Polyas method

Bottom-Up Approach

Top-Down Approach

Wire Length Calculator

Stepwise refinement

Guessing Games

Minesweeper Game

Triangular Wall

Pattern Identification

Conclusion

Devising a Plan: Solution

Solution

Arithmetic series