example of problem solving with patterns

Fun teaching resources & tips to help you teach math with confidence

Math Strategies: Problem Solving by Finding a Pattern

One important math concept that children begin to learn and apply in elementary school is reading and using a table. This is essential knowledge, because we encounter tables of data all the time in our everyday lives! But it’s not just important that kids can read and answer questions based on information in a table, it’s also important that they know how to create their own table and then use it to solve problems, find patterns, graph equations, and so on. And while some may think of these as two different things, I think problem solving by making a table and finding a pattern go hand in hand!

This is such a useful math problem solving strategy! Kids can get so overwhelmed by math word problems, but helping them organize the information in a table and then find a pattern can make things easier!

–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

Finding Patterns in Math Problems:

So when should kids use problem solving by finding a pattern ? Well, when the problem gives a set of data, or a pattern that is continuing and can be arranged in a table, it’s good to consider looking for the pattern and determining the “rule” of the pattern.

As I mentioned when I discussed problem solving by making a list , finding a pattern can be immensely helpful and save a lot of time when working on a word problem. Sometimes, however, a student may not recognize the pattern right away, or may get bogged down with all the details of the question.

Setting up a table and filling in the information given in the question is a great way to organize things and provide a visual so that the “rule” of the pattern can be determined. The “rule” can then be used to find the answer to the question. This removes the tedious work of completing a table, which is especially nice if a lot of computation is involved.

But a table is also great for kids who struggle with math, because it gives them a way to get to the solution even if they have a hard time finding the pattern, or aren’t confident that they are using the “rule” correctly.

Because even though using a known pattern can save you time, and eliminate the need to fill out the entire table, it’s not necessary. A student who is unsure could simply continue filling out their table until they reach the solution they’re looking for.

Helping students learn how to set up a table is also helpful because they can use it to organize information (much like making a list) even if there isn’t a pattern to be found, because it can be done in a systematic way, ensuring that nothing is left out.

If your students are just learning how to read and create tables, I would suggest having them circle their answer in the table to show that they understood the question and knew where in the table to find the answer.

If you have older students, encourage them to find a pattern in the table and explain it in words , and then also with mathematical symbols and/or an equation. This will help them form connections and increase number sense. It will also help them see how to use their “rule” or equation to solve the given question as well as make predictions about the data.

It’s also important for students to consider whether or not their pattern will continue predictably . In some instances, the pattern may look one way for the first few entries, then change, so this is important to consider as the problems get more challenging.

There are tons of examples of problems where creating a table and finding a pattern is a useful strategy, but here’s just one example for you:

Ben decides to prepare for a marathon by running ten minutes a day, six days a week. Each week, he increases his time running by two minutes per day. How many minutes will he run in week 8?

Included in the table is the week number (we’re looking at weeks 1-8), as well as the number of minutes per day and the total minutes for the week. The first step is to fill in the first couple of weeks by calculating the total time.

Making a table and finding a pattern is an excellent math problem solving strategy! This is a great example!

Once you’ve found weeks 1-3, you may see a pattern and be able to calculate the total minutes for week 8. For example, in this case, the total number of minutes increases by 12 each week, meaning in week 8 he will run for 144 minutes.

If not, however, simply continue with the table until you get to week 8, and then you will have your answer.

I think it is especially important to make it clear to students that it is perfectly acceptable to complete the entire table (or continue a given table) if they don’t see or don’t know how to use the pattern to solve the problem.

I was working with a student once and she was given a table, but was then asked a question about information not included in that table . She was able to tell me the pattern she saw, but wasn’t able to correctly use the “rule” to find the answer. I insisted that she simply extend the table until she found what she needed. Then I showed her how to use the “rule” of the pattern to get the same answer.

I hope you find this helpful! Looking for and finding patterns is such an essential part of mathematics education! If you’re looking for more ideas for exploring patterns with younger kids, check out this post for making patterns with Skittles candy .

And of course, don’t miss the other posts in this Math Problem Solving Series:

Problem Solving by Solving an Easier Problem
Problem Solving by Drawing a Picture
Problem Solving by Working Backwards
Problem Solving by Making a List

One Comment

I had so much trouble spotting patterns when I was in school. Fortunately for her, my daughter rocks at it! This technique will be helpful for her when she’s a bit older! #ThoughtfulSpot

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Problem-Solving Strategies: Finding Patterns

In this lesson, we will learn different problem solving strategies to find patterns.

The following are some examples of problem solving strategies.

Explore it/Act it/Try it (EAT) method (Basic) Explore it/Act it/Try it (EAT) method (Intermediate ) Explore it/Act it/Try it (EAT) method (Advanced) Finding a Pattern (Basic) Finding a Pattern (Intermediate) Finding a Pattern (Advanced)

Find A Pattern (Advanced)

Here we will look at some advanced examples of “Find a Pattern” method of problem solving strategy.

Example: Each hexagon below is surrounded by 12 dots. a) Find the number of dots for a pattern with 6 hexagons in the first column. b) Find the pattern of hexagons with 229 dots.

a) The number of dots for a pattern with 6 hexagons in the first column is 142.

b) If there are 229 dots then the pattern has 8 hexagons in the first column.

Example: Each member of a club shook hands with every other member who came for a meeting. There were a total of 45 handshakes. How many members were present at the meeting?

Solution: Total = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 handshakes

There were 10 members

Example: In the figure, a pinball is released at A.

How many paths are there for it to drop from A to E?

Solution: from A to B: 2 B to C: 6 A to C: 2 × 6 = 12 C to D: 70 A to D: 12 × 70 = 840 D to E: 2 A to E: 2 × 840 = 1680

There are 1680 paths from A to E

Example: A group of businessmen were at a networking meeting. Each businessman exchanged his business card with every other businessman who was present.

a) If there were 16 businessmen, how many business cards were exchanged?

b) If there was a total of 380 business cards exchanged, how many businessmen were at the meeting?

Solution: a) 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 120 exchanges

120 × 2 = 240 business cards

If there were 16 businessmen, 240 business cards were exchanged.

b) 380 ÷ 2 = 190

190 = (19 × 20) ÷ 2 = 19 + 18 + 17 + … + 3 + 2 + 1

If there was a total of 380 business cards exchanged, there were 20 businessmen at the meeting.

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Patterns Questions

Patterns questions and answers are given here to help students understand how to solve patterns questions using simple techniques. Solving different pattern questions will be beneficial for a quick understanding of the logic in patterns. In this article, you will learn how to solve patterns in maths with detailed explanations.

What are Patterns in Mathematics?

In mathematics, a pattern is a sequence of numbers that are formed in a particular way. Every pattern contains a specific rule. For example, the sequence of even numbers is a pattern since each number is obtained by adding 2 to the previous number.

i.e., 2, 4, 6, 8, 10, 12, 14,….

Here, 2 + 2 = 4

8 + 2 = 10 and so on.

Also, check:

Number patterns in Whole numbers
Algebra as pattern

A pattern can be of numbers or figures, which means we can also observe patterns in a sequence of similar figures. Let’s have a look at the solved problems on the various number and figure patterns.

Patterns Questions and Answers

1. Identify the pattern for the following sequence and find the next number.

2, 3, 5, 8, 12, 17, 23, ____.

2, 3, 5, 8, 12, 17, 23, ____

The pattern involved in the given sequence is:

12 + 5 = 17

17 + 6 = 23

23 + 7 = 30

Therefore, the next number of the given sequence is 30.

2. Observe the below figure and identify the missing part.

Consider the question figure, where the design in each part will be obtained by rotating the previous design by 90 degrees in the clockwise direction.

So, the missing part will be option (b).

Hence, the complete figure is:

3. Write the next three numbers of the following sequence.

173, 155, 137, 119, 101

The pattern in the given sequence is:

173 – 18 = 155

155 – 18 = 137

137 – 18 = 119

119 – 18 = 101

So, the next three numbers can be written as:

101 – 18 = 83

83 – 18 = 65

65 – 18 = 47

Thus, the sequence is 173, 155, 137, 119, 101, 83, 65, 47.

4. Observe the following figure and choose the correct option.

Each part of the square box contains a triangle inscribed in the circle in the given figure. Also, the triangle in the next circle is the vertical image of the previous one.

Similarly, in the second row, the triangle in the missing part will be an image of the previous one.

Thus, the missing part is option (a).

5. What is the formula for the pattern for this sequence?

11, 21, 31, 41, 51, 61, 71

The numbers in this sequence are written as:

11 + 10 = 21, 21 + 10 = 31, 31 + 10 = 41, and so on.

This can also be expressed as:

11 = 10 + 1 = 10 × 1 + 1

21 = 20 + 1 = 10 × 2 + 1

31 = 30 + 1 = 10 × 3 + 1

41 = 40 + 1 = 10 × 4 + 1 and so on.

From this, we can write the formula for the above pattern as: 10n + 1, where n = 1, 2, 3, etc.

6. Find the pattern in the sequence and write the next two numbers.

10, 17, 36, 73, 134,…

Given sequence is:

10 = 1 3 + 9

17 = 2 3 + 9

36 = 3 3 + 9

73 = 4 3 + 9

134 = 5 3 + 9

So, the next number = 6 3 + 9 = 216 + 9 = 225

Again, the next number = 7 3 + 9 = 343 + 9 = 352

Therefore, the sequence is:

10, 17, 36, 73, 134, 225, 352.

7. Observe the pattern given below. Find the missing number.

In the given figure, we can observe that the sum of the four numbers is equal to the number written in the middle of the shape.

That means,

11 + 22 + 33 + 44 = 110

16 + 24 + 32 + 40 = 112

? + 23 + 34 + 12 = 114

? = 114 – 23 – 34 – 12 = 45

Therefore, the missing number is 45.

8. What is the next number of the following sequence?

20, 18, 21, 16, 23, 12, 25, 8, 27, 4, 33, ?

Given sequence:

20, 18, 21, 16, 23, 12, 25, 8, 27, 4, 33

Let us find the difference between two consecutive numbers of the sequence to identify the pattern.

18 – 20 = -2

21 – 18 = 3

16 – 21 = -5

23 – 16 = 7

12 – 23 = -11

25 – 12 = 13

8 – 25 = -17

27 – 8 = 19

4 – 27 = -23

33 – 4 = 29

Here, we can see that the differences are the prime numbers.

So, the number number = 33 – 31 = 2

9. Estimate the next number of the following sequence.

1, 2, 6, 15, 31, ?

Let’s write the difference between consecutive numbers.

2 – 1 = 1

6 – 2 = 4

15 – 6 = 9

31 – 15 = 16

Here, 1 = 1 2 , 4 = 2 2 , 9 = 3 2 , 16 = 4 2 .

Thus, the next number of the sequence will be obtained by adding 5 2 , i.e. 25, to the previous number.

Therefore, 31 + 5 2 = 31 + 25 = 56.

10. What will be the next number of the given sequence?

1, 5, 12, 22, 35, ?

Let’s write the difference between consecutive numbers.

5 – 1 = 4

12 – 5 = 7

22 – 12 = 10

35 – 22 = 13

The difference between these numbers follows a pattern that 3 is odd to the previous difference.

So, the next number will be obtained by adding 13 + 3, i.e. 16 to 35.

Hence, the next number = 35 + 16 = 51.

Practice Problems on Patterns

Find the correct number to complete the pattern given below.

20, 21, 23, 26, 30, 35, 41, ___.

Find the next number in the sequence, 12, 21, 23, 32, 34, 43.
Write the missing numbers in the following.

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Pattern Problem Solving: Teach Students to Find a Pattern in Math Problems

Pattern analysis is a critical 21st century skill.

Need more tips and tricks for teaching math? You can find them in our math resources center .

Pattern problem solving

In this article, we’ll delve into the concept of pattern problem solving, a fundamental mathematical strategy that involves the identification of repeated sequences or elements to solve complex problems.

This method is instrumental in enhancing logical thinking and mathematical comprehension among students. Let's explore how this works, why it's important, and how to teach this critical skill effectively.

Math teacher teaches problem solving. pattern problem solving.

What is pattern problem solving?

Pattern problem solving is a mathematical strategy in which students look for patterns in data to solve a problem. To find a pattern, students search for repeated items, numbers, or series of events.

The following problem can be solved by finding the pattern:

There are 1000 lockers in a high school with 1000 students. The first student opens all 1000 lockers; next, the second student closes lockers 2, 4, 6, 8, 10, and so on up to locker 1000; the third student changes the state (opens lockers that are closed, closes lockers that are open) of lockers 3, 6, 9, 12, 15, and so on; the fourth student changes the state of lockers 4, 8, 12, 16, and so on. This continues until every student has had a turn. How many lockers will be open at the end?

For the answer, visit The Locker Problem from the Math Doctors

Why is pattern problem solving important?

Pattern problem solving is an important strategy for students as it encourages them to observe and understand patterns in data, which is a critical aspect of mathematical and logical thinking.

This strategy allows students to predict future data points or behaviors based on existing patterns. It helps students understand the inherent structure of data sets and mathematical problems, making them easier to solve.

Pattern recognition also aids in the understanding of multiplication facts, for example, recognizing that 4 x 7 is the same as 7 x 4. Overall, pattern problem solving fosters analytical thinking, problem-solving skills, and a deeper understanding of mathematics.

Boy solves math problem in class. pattern problem solving.

How to teach students to find the pattern in a math problem (using an example)

In the upcoming section, we will break down the steps on how to find a pattern in a math problem effectively. We will use a practical example to illustrate each step and provide helpful teaching tips throughout the process.

The goal is to offer a clear and comprehensible guide for educators teaching students about pattern problem solving in math.

Sample question: If you build a four-sided pyramid using basketballs and don't count the bottom as a side, how many balls will there be in a pyramid that has six layers?

Helpful teaching tip: Use cooperative learning groups to find solutions to the above problem. Cooperative learning groups help students verbalize their thinking, brainstorm ideas, discuss options, and justify their positions. After finding a solution, each group can present it to the class, explaining how they reached their solution and why they think it is correct. Or, students can explain their solutions in writing, and the teacher can display the solutions. Then students can circulate around the room to read each group's solution.

1. Ensure students understand the problem

Demonstrate that the first step to solving a problem is understanding it. This involves identifying the key pieces of information needed to find the answer. This may require students to read the problem several times or put the problem into their own words.

Sometimes you can solve a problem simply through pattern recognition, but more often you must extend the pattern to find the solution. Making a number table will help you see the pattern more clearly.

In this problem, students understand:

The top layer will have one basketball. I need to find how many balls there will be in each layer of a pyramid, from the first to the sixth. I need to find how many basketballs will be in the entire pyramid.

2. Choose a pattern problem solving strategy

To successfully find a pattern, you need to be sure that the pattern will continue.

Have students give reasons why they think the pattern is predictable and not based on probability. Problems that are solved most easily by finding a pattern include those that ask students to extend a sequence of numbers or to make a prediction based on data.

In this problem, students may also choose to make a table or draw a picture to organize and represent their thinking.

3. Solve the problem

Start with the top layer of the pyramid, one basketball. Determine how many balls must be under that ball to make the next layer or a pyramid. Let students use manipulatives if needed— they can use manipulatives of any kind, from coins to cubes to golf balls. Let students also draw pictures to help solve the problem, if needed.

If your students are in groups, you may want to have each group use a different manipulative and then compare their solutions. This will help you understand if different manipulatives affect the solution.

Helpful teaching tip: If students are younger, solve this problem with only three layers.

If it helps to visualize the pyramid, use manipulatives to create the third layer. Record the number and look for a pattern. The second layer adds 3 basketballs and the next adds 5 basketballs. Each time you add a new layer, the number of basketballs needed to create that layer increases by 2.

1 1 + 3 = 4 4 + 5 = 9

Continue until six layers are recorded. Once a pattern is found, students might not need to use manipulatives. 9 + 7 = 16

16 + 9 = 25 25 + 11 = 36

Then add the basketballs used to make all six layers.

The answer is 91 balls .

Look at the list to see if there is another pattern. The number of balls used in each level is the square of the layer number. So the 10th layer would have 10 x 10 = 100 balls .

4. Check your students' answers

Read the problem again to be sure the question was answered:

Yes, I found the total number of basketballs in the six-layer pyramid.

Also check the math to be sure it’s correct:

1 + 4 + 9 + 16 + 25 +36 = 91

Determine if the best strategy was chosen for this problem, or if there was another way to solve the problem:

Finding a pattern was a good way to solve this problem because the pattern was predictable.

5. Explain

Students should be able to explain the process they went through to find their answers. Students must be able to talk or write about their thinking. Demonstrate how to write a paragraph describing the steps they took and the decisions they made throughout.

I started with the first layer. I used blocks to make the pyramid and made a list of the number of blocks I used. Then I created a table to record the number of balls in each layer. I made four layers, then saw a pattern. I saw that or each layer, the number of balls used was the number of the layer multiplied by itself. I finished the pattern without the blocks, by multiplying the number of balls that would be in layers 5 and 6. Then I added up each layer. 1 + 4 + 9 + 16 + 25 +36 = 91. I got a total of 91 basketballs

How can you stretch this pattern problem solving strategy?

Find the pattern math problem solving. pattern problem solving.

Math problems can be simple, with few criteria needed to solve them, or they can be multidimensional, requiring charts or tables to organize students' thinking and to record patterns.

In using patterns, it is important for students to find out if the pattern will continue predictably. Have students determine if there is a reason for the pattern to continue, and be sure students use logic when finding patterns to solve problems.

For example, if it rains on Sunday, snows on Monday, rains on Tuesday, and snows on Wednesday, will it rain on Thursday?
Another example: If Lauren won the first and third game of chess, and Walter won the second and fourth game, who will win the fifth game?
Another example: If a plant grew 13 centimeters in the first week and 10 centimeters in the second week, how many centimeters will it grow in the third week?

Because these are questions of probability or nature, be sure students understand why patterns can't be used to find these answers.

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Course: 3rd grade > Unit 8

Finding patterns in numbers.

Recognizing number patterns
Math patterns
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Arithmetic patterns and problem solving: FAQ

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Video transcript

Patterns are all around us!

Finding and understanding patterns gives us great power. With patterns we can learn to predict the future, discover new things and better understand the world around us.

And playing with patterns is fun.

Simple Patterns

See how good you are at solving these simple patterns:

And you can create your own patterns with Shapes Freeplay .

Many patterns we can see around us have Symmetry .

Number Patterns

You can even make your own number patterns .

Look for patterns everywhere, it will make you a smarter person and give you a better understanding of the world.

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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

$Screen Shot 2018-08-30 at 4.43.05 PM.png$

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

$clipboard_e6298bbd7c7f66d9eb9affcd33892ef0d.png$

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

$example of problem solving with patterns$

Looking back: How would you find the nth term?

$example of problem solving with patterns$

Find the 10 th term of the above sequence.

Let L = the tenth term

$example of problem solving with patterns$

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

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Patterns and Sequences Short Problems

example of problem solving with patterns

Triangular Clock

Trinni rearanges numbers on a clock face so each adjacent pair add up to a triangle number... What number did she put where 6 would usually be?

Printing Error

Every third page number in this book has been omitted. Can you work out what number will be on the last page?

Sliding Robot

A robot moves along the number line. Where will it be after 2011 slides?

Many Matildas

MatildaMatildaMatil... What is the 1000th letter?

How Many Rectangles?

By drawing 5 horizontal and four vertical lines, one can form 12 rectangles. What is the greatest number of rectangles that can be formed by drawing 15 lines?

Night Watchmen

Grannie's watch gains 30 minutes every hour, whilst Grandpa's watch loses 30 minutes every hour. What is the correct time when their watches next agree?

Weekly Problem 51 - 2016 Pegs numbered 1 to 50 are placed in a row. Alternate pegs are knocked down, and this process is repeated. What is the number of the last peg to be knocked down?

Pattern Snake

This pattern repeats every 12 dots. Can you work out what a later piece will be?

What a Coincidence!

Consider two arithmetic sequences: 1998, 2005, 2012,... and 1996, 2005, 2014,... Which numbers will appear in both?

Suit Sequence

A pattern repeats every six symbols. What are the 100th and 101st symbols?

Knights and Knaves

Knights always tell the truth. Knaves always lie. Can you catch these knights and knaves out?

Doubly Consecutive Sums

How many numbers less than 2017 are both the sum of two consecutive integers and the sum of five consecutive integers?

Expanding Pattern

How many squares are needed to continue this pattern?

Fibonacci Deduction

Leonard writes down a sequence of numbers. Can you find a formula to predict the seventh number in his sequence?

Hexagon Line

How many hexagons are required for the perimeter of the whole shape to have length 1002cm?

Street Lamps

Walking up a steep hill, I pass 10 equally spaced street lamps. How long do I take to walk from the first lamp to the last?

Fruit Line-up

This grocer wants to arrange his fruit in a particular order, can you help him?

Alternating Sum

Given that the number 2008 is the correct answer to a sum, can you find n?

Newspaper Sheets

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

If a number is even, halve it; if odd, treble it and add 1. If a sequence starts at 13, what will be the value of the 2008th term?

Repeat a pattern of numbers to form a larger number. Can you find the sum of all the digits?

Collatz-ish

A sequence is generated using these rules. For which values of n is the nth term equal to n?

Can you work out which number will appear directly below 400 in this pattern?

Trolley Park

In a supermarket, there are two lines of tightly packed trolleys. What is the length of one trolley?

SOLVING PROBLEMS WITH PATTERNS

Example 1 :

For the following matchstick pattern, find the number of matches M required to make the

(a) 8 th figure

(b) n th figure

By observing the figures

Number of matchstick in the 1 st figure = 4.

Number of matchstick in the 2 nd figure = 10.

Number of matchstick in the 3 rd figure = 16.

Number of matchsticks in every figure is 2 less than the multiple of 6.

So, creating formula for this

number of matchstick in n th figure = 6n - 2

(a) using the formula

Number of match sticks in 8th figure :

(b) Number of matchstick in n th figure = 6n - 2.

Example 2 :

Consider the pattern :

S 1 = 1/(1×2)

S 2 = 1/(1×2) + 1/(2×3)

S 3 = 1/(1×2) + 1/(2×3) + 1/(3×4)

.....................

a) Find the values of S 1 , S 2 , S 3 , and S 4

b) write down the value of :

(i) S 10 (ii) S n

S 1 = 1/(1 x 2) = 1/2

S 2 = 1/(1 x 2) + 1/(2 x 3) = 1/2 + 1/6 = 4/6 = 2/3

S 3 = 1/(1x2) + 1/(2x3) + 1/(3x4)

= 1/2 + 1/6 + 1/12

= (6 + 2 + 1)/12

Observing the results it is in the form n/(n + 1)

S n = n/(n + 1)

(where n is natural number)

(a)

(b) (i) If n = 10,

S 10 = 10/(10 + 1)

S 10 = 10/11

(ii) S n = n/(n + 1)

Example 3 :

S 1 = 1 2

S 2 = 1 2 + 2 2

S 2 = 1 2 + 2 2 + 3 2 , …………

a) Check that the formula

S n = [n(n + 1)(2n + 1)]/6

is correct for n = 1, 2, 3 and 4.

b) Assuming the formula in a is always true, find the sum of

1 2 + 2 2 + 3 2 + 4 2 + 5 2 + ………… + 100 2

which is the sum of the squares of the first one hundred integers.

(a) W hen n = 1,

S 1 = 1(1 + 1)(2 + 1)]/6

S 1 = 1

It can be written as 1 2 .

When n = 2,

S 2 = [2(2 + 1)(4 + 1)]/6

S 2 = 5

It can be written as 1 2 + 2 2 .

When n = 3,

S 3 = [3(3 + 1)(6 + 1)]/6

S 2 = 84

It can be written as 1 2 + 2 2 + 3 2 .

Hence its verified.

(ii) Given :

1 2 + 2 2 + 3 2 + 4 2 + 5 2 + ………… + 100 2

Here, n = 100.

S 100 = [100(100 + 1)(200 + 1)]/6

S 100 = 338350

Example 4 :

N 1 = 1 3

N 2 = 1 3 + 2 3

N 3 = 1 3 + 2 3 + 3 3 , …………

a) Verify that the formula

N n = [n 2 (n + 1) 2 ] /4

is correct for n = 1, 2, 3 and 4

b) Use the above formula to find the sum of

1 3 + 2 3 + 3 3 + 4 3 + ………… + 50 3

c) Find the sum : 2 3 + 4 3 + 6 3 + 8 3 + ………… + 100 3

(a) W hen n = 1,

N n = [n 2 (n + 1) 2 ]/4

N 1 = [1 2 (1 + 1) 2 ]/4

N 1 = 1

It can be written as 1 3 .

N 2 = [2 2 (2 + 1) 2 ]/4

N 2 = 9

It can be written as 1 3 + 2 3 .

N 3 = [3 2 (3 + 1) 2 ] /4

N 3 = 36

It can be written as 1 3 + 2 3 + 3 3 .

Hence, its verified.

(b) 1 3 + 2 3 + 3 3 + 4 3 + ………… + 50 3

here, n = 50.

S n = n 2 (n + 1) 2 ] /4

S 50 = (50) 2 (50 + 1) 2 ] /4

S 50 = 1625625

c) 2 3 + 4 3 + 6 3 + 8 3 + ………… + 100 3

Factoring 2 3 from the series.

= 2 3 (1 3 + 2 3 + 3 3 + 4 3 + ………… + 50 3 )

= 2 3 (1625625)

= 8(1625625)

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Patterns to Solve Problems

Problem solving “ no problem “ just follow the pattern.

I like puzzles because they tend to follow a pattern. I can see colors, shapes, number, which helps me to come to a solution. When I was in school I don't recall teachers or professors discussing patterns and looking for patterns to solve problems. Personally for me, I probably would have understood math a lot better if someone had taught me how to identify patterns in math. According to a research paper in the Mathematics Education Research Journal by Joanne Mulligan & Michael Mitchelmore The power of mathematics lies in relations and transformations which give rise to patterns and generalizations. Abstracting patterns is the basis of structural knowledge, the goal of mathematics learning

Starting Early with Repeating Patterns

Kindergarten students often work on finding a pattern.

They have images like this, which are called repeating patterns, following a rule like, ABAB or ABCABC or ABBC, ABBC. There are many variations and having students start to recognize these patterns is important. According to the above article, students that have good visualization skills, which involves recognition of pattern and structure, have increased mathematical achievement and analogical reasoning. These repetitive pattern problems are great ways to get those young children to start the mental development of critical thinking and analytical sense. I have found that with young children when presented with numerous objects of color they will first notice the color, then shape and then number. This is helpful in upper grades when teachers work with students on algebraic equations. A great game to work on this age level on pattern recognition is a game called BLINK . This game has students quickly noticing color, shape and number. It is a fun and easy game to play.

Branch out into Growing Patterns

From repeating patterns students can move on to growing patterns.

I like using a number grid. This activity Number Neighbors and Number Neighbors II from our 1st and 2nd grade math kits seem simple but it can be challenging as you start to apply the numbers. The students start with an unmarked grid, the center number can be a random number selected by the teacher, 44. Students are then to start filling in the blank squares like it is a section of the number grid. Some of the squares have circles in them, this is to help focus the students on filling out the easiest to solve numbers first (an example of how shapes can focus students visualization). What number comes directly before 44? What numbers comes directly after 44? What number would be directly above 44? How many less than 44 is that? This grid has students developing the patterning of one more (+1), one less (-1), ten more (+10), ten less (-10) as they fill in the numbers.

Patterns Keep on Growing

Another patterning game which I think is a great visualization activity is the game Jump Around . Having student starting on a certain number, 3. Now ask them to jump 5 and keep track of the numbers they land, 3, 8, 13, 18, 23â€¦ What pattern do the students notice, what will be the next number in the pattern? Can we determine what the 10 number in the pattern? Students may struggle with the circular pattern of this number line, if that is the case have them use a number grid. Color in the numbers as they land on them, so the patterning because more visual.

Stick-With-It Patterns will Develop

Growing patterns can be shown visually using sticks, squares, pattern blocks. Stick To It has students determining a pattern using sticks. Matchstick problems are very common in middle school.

First square has 4 sticks, second square has 8, third square has 12. Can student determine the number of sticks in the 10th square or the 100th square? What pattern is being followed.

Algebra and Patterns

The final pattern game I will show is developed from those math tricks you may have seen before. Tricks like these . Choose a number Multiply by 3 Add 6 Divide by 3 Subtract 1st number I can tell you what the answer is. How will I always know, no matter what number you start with? Use squares and circles to see how this works.

Square represents the number

Multiply by 3

Divide by 3

Subtract first number Patterns can help students who find math difficult and uninspiring into understandable and visual. Helping them see how patterning can help them become better mathematicians. MANGO Math wants to help. Learn more.

MANGO Math Blog

The ultimate guide to solving pattern recognition problems

Introduction.

Pattern recognition is a fundamental aspect of human cognition that allows us to make sense of the world around us. It involves the identification and interpretation of recurring structures or sequences in data or information. Pattern recognition plays a crucial role in various fields, including computer science, artificial intelligence, biology, psychology, and finance.

In computer science, pattern recognition is used in image and speech recognition, natural language processing, and data mining. In biology, it helps in understanding the genetic code and deciphering patterns in biological systems. In psychology, it aids in studying human behavior and understanding cognitive processes. In finance, it enables the identification of market trends and making predictions based on historical patterns.

The Ultimate Guide to Solving Pattern Recognition Problems aims to provide a comprehensive overview of pattern recognition and guide you through the steps involved in solving pattern recognition problems effectively. Whether you’re a student, a professional, or simply curious about pattern recognition, this guide will equip you with the essential knowledge and tools to tackle pattern recognition challenges.

Throughout this guide, we will explore the concept of patterns, their characteristics, and the different types of patterns that exist in various domains. We will delve into the intricacies of problem identification and the crucial process of data collection. Additionally, we will discuss the importance of data preprocessing and feature extraction in preparing the data for analysis.

Next, we will navigate the selection of appropriate pattern recognition algorithms and explore techniques for model training and evaluation. We will also delve into the optimization of the trained models to improve their performance. The guide will provide practical insights into real-world applications of pattern recognition and present examples and case studies from diverse domains.

Furthermore, we will address common challenges encountered in pattern recognition and discuss potential advancements and future directions in the field. Finally, we will conclude the guide with a recap of the main points and emphasize the importance of applying pattern recognition techniques in problem-solving.

Whether you’re trying to identify patterns in stock market data, recognize fraudulent transactions, or classify images, this guide will serve as your go-to resource, providing you with the necessary knowledge and tools to succeed in solving pattern recognition problems. So, let’s embark on this journey to unravel the mysteries of patterns and discover the power of pattern recognition.

Understanding Patterns

In order to solve pattern recognition problems effectively, it is important to have a clear understanding of what patterns are and their characteristics. A pattern can be defined as a recurring and recognizable sequence or arrangement of elements or events. Patterns exhibit certain characteristics that distinguish them from random or unrelated data. Some key characteristics of patterns include:

Consistency: Patterns are consistent and repeat in a predictable manner. They maintain a certain structure or sequence that can be observed over time or across different instances.

Repetition: Patterns involve the repetition of certain elements or features. This repetition can occur within a single instance of the pattern or across multiple instances.

Regularity: Patterns often follow a regular or systematic arrangement. They may have a specific order or sequence in which their elements occur.

Symmetry: Some patterns exhibit symmetry, meaning that they are balanced or have a mirror-like structure. The elements on one side of the pattern are mirrored or reversed on the other side.

Variation: While patterns have certain consistent elements, they may also exhibit variations or deviations. These variations can be intentional or unintentional and may add complexity or diversity to the pattern.

Predictability: One of the main purposes of pattern recognition is to identify and understand patterns in order to make predictions or draw conclusions. Patterns are predictable to some extent, as they follow certain rules or principles.

Patterns can be found in various domains and scenarios, ranging from simple everyday occurrences to complex scientific phenomena. Some examples of patterns in real-life scenarios include:

Weather patterns: The weather follows certain seasonal patterns, with variations in temperature, precipitation, and wind patterns occurring in a predictable manner.

Language patterns: Languages have patterns in terms of grammar, syntax, and vocabulary. These patterns enable us to understand and communicate effectively.

Financial market patterns: Financial markets exhibit patterns in terms of price movements, trading volume, and market behavior. Traders and analysts use pattern recognition to identify trends and make investment decisions.

Biological patterns: Biological systems, such as DNA sequences and protein structures, have patterns that help scientists understand genetic information and biological processes.

Traffic patterns: Traffic flow on roads and highways follows certain patterns, with peak hours and congestion occurring predictably. This information is used to optimize traffic management and transportation planning.

Music patterns: Musical compositions have patterns in terms of rhythm, melody, and harmony. These patterns create a sense of structure and organization in the music.

Understanding the different types of patterns and their characteristics is essential in solving pattern recognition problems effectively. It enables us to identify and interpret patterns, make predictions, and draw meaningful insights from the data.

Steps in Solving Pattern Recognition Problems

1. problem identification.

In this initial step, the first task is to recognize the need for pattern recognition in a particular problem or challenge. By understanding the problem at hand, one can identify the areas where pattern recognition techniques can be applied effectively. This involves analyzing the data and determining whether there are any underlying patterns that can be leveraged to solve the problem.

2. Data Collection

Once the problem has been identified, the next step is to gather relevant data and information. This can include collecting data from various sources such as databases, files, or even web scraping. It is crucial to ensure that the data collected is accurate and complete, as any inaccuracies or missing information can negatively affect the pattern recognition process.

3. Data Preprocessing

Before the data can be used for pattern recognition, it often needs to undergo preprocessing. This step involves cleaning the data by removing any noise, errors, or irrelevant information. Additionally, missing values and outliers need to be handled appropriately in order to avoid biases in the pattern recognition process. Various techniques such as imputation and outlier detection can be employed to handle these issues effectively.

4. Feature Extraction

Feature extraction is a critical step in pattern recognition, as it involves identifying the key features that represent the underlying patterns in the data. These features can be numerical or categorical, depending on the nature of the problem. Feature extraction techniques such as Principal Component Analysis (PCA) and Discrete Wavelet Transform (DWT) can be used to transform the data into a more suitable format for pattern recognition algorithms.

5. Algorithm Selection

Choosing the right pattern recognition algorithm is crucial for achieving accurate and reliable results. There are various algorithms available, each with its own strengths and weaknesses. The selection process involves reviewing different algorithms, considering their applicability to the problem at hand, and evaluating their performance metrics. It is important to select an algorithm that aligns with the type of patterns and data being analyzed.

6. Model Training

Once the algorithm has been selected, the next step is to prepare the data for training the pattern recognition model. This involves splitting the data into training and testing sets, and applying appropriate techniques such as cross-validation to ensure the model’s robustness. The training process involves feeding the data into the selected algorithm and allowing it to learn the patterns from the labeled data.

7. Model Evaluation

After the model has been trained, it is essential to evaluate its performance. This is done by assessing various metrics such as accuracy, precision, recall, and F1 score. These metrics help to determine how well the model is able to identify and classify patterns in the data. The evaluation results provide insights into the model’s effectiveness and can guide any necessary adjustments or improvements.

8. Model Optimization

In order to achieve better results, model optimization is often necessary. This involves fine-tuning the model parameters to improve its performance. Techniques such as grid search and random search can be employed to explore different parameter combinations and find the optimal settings. Optimization aims to enhance the model’s predictive capabilities and increase its accuracy in pattern recognition tasks.

9. Pattern Recognition Applications

The field of pattern recognition has extensive applications in various domains. This step involves exploring and discussing these applications, ranging from image and speech recognition to data mining and bioinformatics. Case studies are often used to demonstrate the practical use of pattern recognition techniques in solving real-world problems across different industries and research areas.

10. Challenges and Future Directions

Pattern recognition is not without its challenges. This final step addresses common challenges faced in pattern recognition, such as dealing with high-dimensional data, handling class imbalance, and interpreting complex patterns. Additionally, the section discusses potential advancements and future directions in pattern recognition, including the integration of deep learning techniques, the use of unsupervised learning for pattern discovery, and the exploration of new data representation methods.

By following these steps in solving pattern recognition problems, individuals and organizations can effectively analyze and interpret complex data patterns, leading to valuable insights and informed decision-making.

In conclusion, pattern recognition is a powerful tool that plays a crucial role in various fields. By understanding and recognizing patterns, we can uncover valuable insights, make informed decisions, and solve complex problems more effectively.

Throughout this guide, we have explored the fundamentals of pattern recognition, including the different types of patterns and their characteristics. We have also discussed the step-by-step process of solving pattern recognition problems, starting from problem identification all the way to model optimization.

We began by emphasizing the importance of recognizing the need for pattern recognition and identifying the specific problem or challenge at hand. Next, we delved into the critical stage of data collection, where we gather relevant and accurate data to train our pattern recognition model.

Data preprocessing is then discussed, highlighting the significance of cleaning and transforming the data, as well as dealing with missing values and outliers. Following that, we explored feature extraction, which involves identifying the key features of the patterns and selecting appropriate techniques to extract them.

Afterwards, we moved on to algorithm selection, where we reviewed different pattern recognition algorithms and chose the most suitable one for the specific problem. Model training, model evaluation, and model optimization were covered next, as these stages are crucial for ensuring the model performs well and meets the desired criteria.

Furthermore, we expanded our understanding by discussing various applications of pattern recognition across different domains. Through case studies, we explored how pattern recognition techniques have been applied in real-world scenarios, such as image recognition, speech recognition, and fraud detection.

As we wrap up this guide, it is important to acknowledge the challenges that can arise in the field of pattern recognition. These challenges include issues regarding data quality, algorithm complexity, and the interpretability of results. However, with advancements in technology and ongoing research, we can look forward to potential solutions and advancements in the field.

In conclusion, pattern recognition is a valuable tool that enables us to uncover hidden patterns, make accurate predictions, and gain deeper insights into complex systems. By applying the principles and techniques discussed in this guide, we can harness the power of pattern recognition to solve problems and drive innovation in a wide range of domains.

So, I encourage you to take what you have learned in this guide and apply it to your own problem-solving endeavors. Whether you are in the field of healthcare, finance, engineering, or any other discipline, pattern recognition can be a game-changer. Embrace the power of patterns and let them guide you towards success.

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From zero to hero: the journey of goal achievement, a humorous take on the importance of pattern recognition skills in creative industries, addressing the current challenges in pattern recognition and innovating new solutions, 10 powerful techniques for enhancing pattern recognition skills, 8 power words to boost your pattern recognition prowess.

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26 Expert-Backed Problem Solving Examples – Interview Answers

Published: February 13, 2023

Interview Questions and Answers

Actionable advice from real experts:

Biron Clark

Former Recruiter

Contributor

Dr. Kyle Elliott

Career Coach

Hayley Jukes

Editor-in-Chief

Biron Clark , Former Recruiter

Kyle Elliott , Career Coach

Hayley Jukes , Editor

As a recruiter , I know employers like to hire people who can solve problems and work well under pressure.

A job rarely goes 100% according to plan, so hiring managers are more likely to hire you if you seem like you can handle unexpected challenges while staying calm and logical.

But how do they measure this?

Hiring managers will ask you interview questions about your problem-solving skills, and they might also look for examples of problem-solving on your resume and cover letter.

In this article, I’m going to share a list of problem-solving examples and sample interview answers to questions like, “Give an example of a time you used logic to solve a problem?” and “Describe a time when you had to solve a problem without managerial input. How did you handle it, and what was the result?”

Problem-solving involves identifying, prioritizing, analyzing, and solving problems using a variety of skills like critical thinking, creativity, decision making, and communication.
Describe the Situation, Task, Action, and Result ( STAR method ) when discussing your problem-solving experiences.
Tailor your interview answer with the specific skills and qualifications outlined in the job description.
Provide numerical data or metrics to demonstrate the tangible impact of your problem-solving efforts.

What are Problem Solving Skills?

Problem-solving is the ability to identify a problem, prioritize based on gravity and urgency, analyze the root cause, gather relevant information, develop and evaluate viable solutions, decide on the most effective and logical solution, and plan and execute implementation.

Problem-solving encompasses other skills that can be showcased in an interview response and your resume. Problem-solving skills examples include:

Critical thinking
Analytical skills
Decision making
Research skills
Technical skills
Communication skills
Adaptability and flexibility

Why is Problem Solving Important in the Workplace?

Problem-solving is essential in the workplace because it directly impacts productivity and efficiency. Whenever you encounter a problem, tackling it head-on prevents minor issues from escalating into bigger ones that could disrupt the entire workflow.

Beyond maintaining smooth operations, your ability to solve problems fosters innovation. It encourages you to think creatively, finding better ways to achieve goals, which keeps the business competitive and pushes the boundaries of what you can achieve.

Effective problem-solving also contributes to a healthier work environment; it reduces stress by providing clear strategies for overcoming obstacles and builds confidence within teams.

Examples of Problem-Solving in the Workplace

Correcting a mistake at work, whether it was made by you or someone else
Overcoming a delay at work through problem solving and communication
Resolving an issue with a difficult or upset customer
Overcoming issues related to a limited budget, and still delivering good work through the use of creative problem solving
Overcoming a scheduling/staffing shortage in the department to still deliver excellent work
Troubleshooting and resolving technical issues
Handling and resolving a conflict with a coworker
Solving any problems related to money, customer billing, accounting and bookkeeping, etc.
Taking initiative when another team member overlooked or missed something important
Taking initiative to meet with your superior to discuss a problem before it became potentially worse
Solving a safety issue at work or reporting the issue to those who could solve it
Using problem solving abilities to reduce/eliminate a company expense
Finding a way to make the company more profitable through new service or product offerings, new pricing ideas, promotion and sale ideas, etc.
Changing how a process, team, or task is organized to make it more efficient
Using creative thinking to come up with a solution that the company hasn’t used before
Performing research to collect data and information to find a new solution to a problem
Boosting a company or team’s performance by improving some aspect of communication among employees
Finding a new piece of data that can guide a company’s decisions or strategy better in a certain area

Problem-Solving Examples for Recent Grads/Entry-Level Job Seekers

Coordinating work between team members in a class project
Reassigning a missing team member’s work to other group members in a class project
Adjusting your workflow on a project to accommodate a tight deadline
Speaking to your professor to get help when you were struggling or unsure about a project
Asking classmates, peers, or professors for help in an area of struggle
Talking to your academic advisor to brainstorm solutions to a problem you were facing
Researching solutions to an academic problem online, via Google or other methods
Using problem solving and creative thinking to obtain an internship or other work opportunity during school after struggling at first

How To Answer “Tell Us About a Problem You Solved”

When you answer interview questions about problem-solving scenarios, or if you decide to demonstrate your problem-solving skills in a cover letter (which is a good idea any time the job description mentions problem-solving as a necessary skill), I recommend using the STAR method.

STAR stands for:

It’s a simple way of walking the listener or reader through the story in a way that will make sense to them.

Start by briefly describing the general situation and the task at hand. After this, describe the course of action you chose and why. Ideally, show that you evaluated all the information you could given the time you had, and made a decision based on logic and fact. Finally, describe the positive result you achieved.

Note: Our sample answers below are structured following the STAR formula. Be sure to check them out!

EXPERT ADVICE

Dr. Kyle Elliott , MPA, CHES Tech & Interview Career Coach caffeinatedkyle.com

How can I communicate complex problem-solving experiences clearly and succinctly?

Before answering any interview question, it’s important to understand why the interviewer is asking the question in the first place.

When it comes to questions about your complex problem-solving experiences, for example, the interviewer likely wants to know about your leadership acumen, collaboration abilities, and communication skills, not the problem itself.

Therefore, your answer should be focused on highlighting how you excelled in each of these areas, not diving into the weeds of the problem itself, which is a common mistake less-experienced interviewees often make.

Tailoring Your Answer Based on the Skills Mentioned in the Job Description

As a recruiter, one of the top tips I can give you when responding to the prompt “Tell us about a problem you solved,” is to tailor your answer to the specific skills and qualifications outlined in the job description.

Once you’ve pinpointed the skills and key competencies the employer is seeking, craft your response to highlight experiences where you successfully utilized or developed those particular abilities.

For instance, if the job requires strong leadership skills, focus on a problem-solving scenario where you took charge and effectively guided a team toward resolution.

By aligning your answer with the desired skills outlined in the job description, you demonstrate your suitability for the role and show the employer that you understand their needs.

Amanda Augustine expands on this by saying:

“Showcase the specific skills you used to solve the problem. Did it require critical thinking, analytical abilities, or strong collaboration? Highlight the relevant skills the employer is seeking.”

Interview Answers to “Tell Me About a Time You Solved a Problem”

Now, let’s look at some sample interview answers to, “Give me an example of a time you used logic to solve a problem,” or “Tell me about a time you solved a problem,” since you’re likely to hear different versions of this interview question in all sorts of industries.

The example interview responses are structured using the STAR method and are categorized into the top 5 key problem-solving skills recruiters look for in a candidate.

1. Analytical Thinking

Situation: In my previous role as a data analyst , our team encountered a significant drop in website traffic.

Task: I was tasked with identifying the root cause of the decrease.

Action: I conducted a thorough analysis of website metrics, including traffic sources, user demographics, and page performance. Through my analysis, I discovered a technical issue with our website’s loading speed, causing users to bounce.

Result: By optimizing server response time, compressing images, and minimizing redirects, we saw a 20% increase in traffic within two weeks.

2. Critical Thinking

Situation: During a project deadline crunch, our team encountered a major technical issue that threatened to derail our progress.

Task: My task was to assess the situation and devise a solution quickly.

Action: I immediately convened a meeting with the team to brainstorm potential solutions. Instead of panicking, I encouraged everyone to think outside the box and consider unconventional approaches. We analyzed the problem from different angles and weighed the pros and cons of each solution.

Result: By devising a workaround solution, we were able to meet the project deadline, avoiding potential delays that could have cost the company $100,000 in penalties for missing contractual obligations.

3. Decision Making

Situation: As a project manager , I was faced with a dilemma when two key team members had conflicting opinions on the project direction.

Task: My task was to make a decisive choice that would align with the project goals and maintain team cohesion.

Action: I scheduled a meeting with both team members to understand their perspectives in detail. I listened actively, asked probing questions, and encouraged open dialogue. After carefully weighing the pros and cons of each approach, I made a decision that incorporated elements from both viewpoints.

Result: The decision I made not only resolved the immediate conflict but also led to a stronger sense of collaboration within the team. By valuing input from all team members and making a well-informed decision, we were able to achieve our project objectives efficiently.

4. Communication (Teamwork)

Situation: During a cross-functional project, miscommunication between departments was causing delays and misunderstandings.

Task: My task was to improve communication channels and foster better teamwork among team members.

Action: I initiated regular cross-departmental meetings to ensure that everyone was on the same page regarding project goals and timelines. I also implemented a centralized communication platform where team members could share updates, ask questions, and collaborate more effectively.

Result: Streamlining workflows and improving communication channels led to a 30% reduction in project completion time, saving the company $25,000 in operational costs.

5. Persistence

Situation: During a challenging sales quarter, I encountered numerous rejections and setbacks while trying to close a major client deal.

Task: My task was to persistently pursue the client and overcome obstacles to secure the deal.

Action: I maintained regular communication with the client, addressing their concerns and demonstrating the value proposition of our product. Despite facing multiple rejections, I remained persistent and resilient, adjusting my approach based on feedback and market dynamics.

Result: After months of perseverance, I successfully closed the deal with the client. By closing the major client deal, I exceeded quarterly sales targets by 25%, resulting in a revenue increase of $250,000 for the company.

Tips to Improve Your Problem-Solving Skills

Throughout your career, being able to showcase and effectively communicate your problem-solving skills gives you more leverage in achieving better jobs and earning more money .

So to improve your problem-solving skills, I recommend always analyzing a problem and situation before acting.

When discussing problem-solving with employers, you never want to sound like you rush or make impulsive decisions. They want to see fact-based or data-based decisions when you solve problems.

Don’t just say you’re good at solving problems. Show it with specifics. How much did you boost efficiency? Did you save the company money? Adding numbers can really make your achievements stand out.

To get better at solving problems, analyze the outcomes of past solutions you came up with. You can recognize what works and what doesn’t.

Think about how you can improve researching and analyzing a situation, how you can get better at communicating, and deciding on the right people in the organization to talk to and “pull in” to help you if needed, etc.

Finally, practice staying calm even in stressful situations. Take a few minutes to walk outside if needed. Step away from your phone and computer to clear your head. A work problem is rarely so urgent that you cannot take five minutes to think (with the possible exception of safety problems), and you’ll get better outcomes if you solve problems by acting logically instead of rushing to react in a panic.

You can use all of the ideas above to describe your problem-solving skills when asked interview questions about the topic. If you say that you do the things above, employers will be impressed when they assess your problem-solving ability.

More Interview Resources

3 Answers to “How Do You Handle Stress?”
How to Answer “How Do You Handle Conflict?” (Interview Question)
Sample Answers to “Tell Me About a Time You Failed”

About the Author

Biron Clark is a former executive recruiter who has worked individually with hundreds of job seekers, reviewed thousands of resumes and LinkedIn profiles, and recruited for top venture-backed startups and Fortune 500 companies. He has been advising job seekers since 2012 to think differently in their job search and land high-paying, competitive positions. Follow on Twitter and LinkedIn .

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Posted on Nov 2, 2023

20 Essential Coding Patterns to Ace Your Next Coding Interview

Navigating through coding interviews requires more than just a good grasp of algorithms and data structures; it demands a strategic approach and a keen eye for patterns. In todays competitive world of tech job interviews, understanding and mastering coding patterns can significantly enhance your problem-solving skills and boost your performance.

Coding patterns, or as we like to call them, are recurring techniques that provide a structured approach to solving complex problems . Think of them as the building blocks of algorithms, helping you to break down problems into more manageable parts.

In this blog, we will explore 20 essential coding patterns that are pivotal for acing coding interviews. We will delve into the pros and cons of each pattern, providing you with a balanced view to help you make informed decisions during your interviews. And to top it off, we will equip you with real problem examples from the Grokking the Coding Interview course. I'm the author of this course, feel free to reach out to me if you have questions.

Let's go through each pattern one by one.

1. Two Pointers

Description.

The Two Pointers technique is a clever strategy used in algorithm design, particularly when dealing with arrays or linked lists. Imagine you have two fingers, and you place each at different ends or positions of an array. These ‘fingers’ or pointers then traverse through the array, helping you to compare, search, or even manipulate the data efficiently.

Ordered Data Structures : This pattern shines when applied to ordered arrays or lists, allowing for intelligent, position-based decisions that can significantly optimize the algorithm.
Efficiency : By reducing the need for nested loops, the Two Pointers technique helps in achieving linear time complexity, making your algorithm faster and more efficient.

Pros and Cons

Efficiency : Achieves O(n) time complexity for problems that might otherwise require O(n^2).
Simplicity : Once mastered, it provides a straightforward and elegant solution.
Applicability : Mainly beneficial for problems involving sequences or intervals.
Initial Complexity : It might take some time to get the hang of this pattern and understand where and how to move the pointers.

Example Problems from Grokking the Coding Interview

Pair with Target Sum : Find a pair in an array that adds up to a specific target sum.
Squaring a Sorted Array : Given a sorted array, create a new array containing squares of all the numbers of the input array in the sorted order.
Triplet Sum to Zero : Given an array of unsorted numbers, find all unique triplets in it that add up to zero.

2. Island (Matrix Traversal) Pattern

The Island pattern, also known as Matrix Traversal, is a technique used to navigate through a 2D array or matrix. The primary goal is to identify and process contiguous groups of elements, often referred to as ‘islands’. This pattern is particularly useful when you need to explore and manipulate grid-based data.

Grid-Based Problems : Excelling in problems where you need to traverse a grid to find connected components or regions.
Contiguous Elements : Ideal for situations where you need to group together adjacent elements that share a common property.
Comprehensive : Provides a thorough way to explore all the elements in a grid.
Versatile : Can be used to solve a variety of problems related to 2D arrays.
Complexity : Can be more complex to implement compared to linear data structure traversal.
Space Overhead : May require additional space for recursion or queue/stack for breadth-first/depth-first traversal.
Number of Islands : Count the number of islands in a given 2D matrix.
Biggest Island : Find the largest island in terms of area or number of cells.
Flood Fill : Change the color of an image represented by a 2D array.

3. Fast & Slow Pointers

The Fast & Slow Pointers technique involves two pointers traversing through a data structure at different speeds. This ingenious approach is particularly useful in identifying cycles, finding middle elements, and solving various other problems related to linked lists and arrays.

Cycle Detection : Perfect for identifying cycles in a linked list or array, which is a common interview question.
Finding Middle Elements : Efficiently find the middle element of a linked list without knowing the length beforehand.
Problem-Specific Solutions : Solve specific problems like finding the start of a cycle in a linked list.
Space Efficiency : Achieves solutions without the need for extra space, adhering to O(1) space complexity.
Versatility : Applicable to a variety of problems, making it a versatile pattern to know.
Initial Complexity : Understanding how to move the pointers and at what speed can be tricky at first.
Specificity : While versatile, it is mostly beneficial for problems related to linked lists and certain array problems.
LinkedList Cycle : Determine if a linked list has a cycle.
Middle of the LinkedList : Find the middle node of a linked list.
Palindrome LinkedList : Check if a linked list is a palindrome.

4. Sliding Window

The Sliding Window pattern involves creating a ‘window’ over a portion of data and sliding it across to solve problems efficiently. This technique is particularly useful for array or list-based problems where you need to find or calculate something among all the contiguous subarrays or sublists of a given size.

Contiguous Subarrays : Ideal for problems that require you to deal with contiguous subarrays or sublists.
Variable Sized Window : Can be adapted for problems where the window size is not fixed and needs to be adjusted based on certain conditions.
Efficiency : Provides a way to reduce time complexity from O(n^2) to O(n) for specific problems.
Versatility : Can be used for a variety of problems, including maximum sum subarray, smallest subarray with a given sum, and longest substring with K distinct characters.
Initial Complexity : Understanding how to adjust the window size and when to slide the window can be challenging initially.
Specificity : Mainly beneficial for problems involving contiguous subarrays or sublists.
Maximum Sum Subarray of Size K : Given an array of positive numbers and a positive number ‘k’, find the maximum sum of any contiguous subarray of size ‘k’.
Fruits Into Baskets : Given an array of characters where each character represents a fruit tree, you are given two baskets, and your goal is to put maximum number of fruits in each basket.
Longest Substring with K Distinct Characters : Given a string, find the length of the longest substring in it with no more than K distinct characters.

5. Merge Intervals

The Merge Intervals pattern is a powerful technique used to deal with overlapping intervals or ranges. It involves sorting and then merging intervals based on specific conditions. This pattern is incredibly useful for time-based problems, scheduling, and range manipulation.

Overlapping Intervals : Perfect for problems where you need to merge overlapping intervals or find if an interval overlaps with any other.
Interval Scheduling : Useful for problems that involve scheduling based on time intervals.
Clarity : Provides a clear and systematic way to deal with overlapping intervals.
Efficiency : Helps in reducing the problem complexity and achieving optimal solutions.
Sorting Overhead : Requires the intervals to be sorted beforehand, which could add to the time complexity.
Specificity : Mainly beneficial for problems involving intervals and ranges.
Merge Intervals : Given a list of intervals, merge all the overlapping intervals to produce a list that has only mutually exclusive intervals.
Insert Interval : Given a list of non-overlapping intervals sorted by their start time, insert a given interval at the correct position and merge all necessary intervals to produce a list that has only mutually exclusive intervals.
Intervals Intersection : Given two lists of intervals, find the intersection of these two lists. Each list consists of disjoint intervals sorted on their start time.

6. Cyclic Sort

Cyclic Sort is a unique and intuitive sorting algorithm, particularly well-suited for problems where you are given a range of numbers and asked to sort them. The beauty of this pattern lies in its ability to sort the numbers in-place, utilizing the fact that the numbers are consecutive or have a specific range.

Consecutive Numbers : Ideal for scenarios where you have an array of numbers in a specific range, and you need to sort them or find missing/duplicate numbers.
In-Place Sorting : Provides a way to sort the numbers without using any extra space.
Space Efficiency : Achieves sorting without the need for additional space, adhering to O(1) space complexity.
Time Efficiency : Offers a linear time complexity solution for specific range-based sorting problems.
Limited Applicability : Best suited for problems involving numbers in a specific range and may not be applicable for other types of sorting problems.
Initial Learning Curve : Understanding the cyclic sort pattern and knowing when to apply it can take some time.
Find the Missing Number : Given an array containing n distinct numbers taken from 0, 1, 2, ..., n, find the one that is missing from the array.
Find all Duplicates : Find all the duplicate numbers (without using extra space and in O(n) runtime).
Duplicates In Array : Given an array of integers, 1 ≤ a[i] ≤ n (n = size of array), some elements appear twice and others appear once.

7. In-place Reversal of a Linked List

The In-place Reversal of a Linked List pattern is a technique used to reverse the elements of a linked list without using additional memory. This is achieved by manipulating the pointers of the nodes in the linked list to reverse their direction.

Memory Efficiency : Since no additional data structures are used, this pattern is memory efficient.
Reversing Sub-lists : Can be extended to reverse sub-lists within a linked list, providing versatility in solving more complex problems.
Space Efficiency : Achieves in-place reversal, ensuring O(1) space complexity.
Versatility : Can be used to solve various problems related to linked lists, including reversing sub-lists and finding palindromes.
Pointer Manipulation : Requires careful manipulation of pointers, which can be error-prone.
Initial Learning Curve : Understanding how to reverse the pointers without losing the rest of the linked list can be challenging initially.
Reverse a LinkedList : Given the head of a Singly LinkedList, reverse the LinkedList.
Reverse a Sub-list : Given the head of a LinkedList and two positions ‘p’ and ‘q’, reverse the LinkedList from position ‘p’ to ‘q’.
Reverse Every K-element Sub-list : Given the head of a LinkedList and a number ‘k’, reverse every ‘k’ sized sub-list starting from the head.

8. Tree Breadth First Search

The Tree Breadth First Search (BFS) pattern involves traversing a tree level by level, ensuring that you visit all the nodes at the current depth before moving on to the nodes at the next depth level. This is usually implemented using a queue.

Level Order Traversal : Ideal for problems that require you to traverse a tree in level order or when you need to perform operations on nodes at the same depth.
Minimum Depth : Useful for finding the minimum depth of a tree, as you can stop the traversal once you find the first leaf node.
Complete Traversal : Ensures that every node in the tree is visited.
Level Order Information : Provides information about the depth or level of each node.
Space Overhead : Requires additional space for the queue, which can be as large as the number of nodes at the largest level.
Not as Efficient for Depth-Related Queries : For problems that depend on depth information, a depth-first search might be more efficient.
Binary Tree Level Order Traversal : Traverse a tree in level order and return the values of the nodes at each level.
Reverse Level Order Traversal : Traverse a tree in reverse level order.
Zigzag Traversal : Traverse a tree in a zigzag order.

9. Tree Depth First Search

The Tree Depth First Search (DFS) pattern involves traversing a tree in a depth-first manner, meaning you go as deep as possible down one branch before backing up and exploring other branches. This is typically implemented using recursion or a stack.

Path Finding : Ideal for problems where you need to find a path or check the existence of a path with certain properties.
Complex Tree Traversals : Useful for more complex tree traversal problems where you need to maintain state or perform operations as you traverse.
Space Efficiency : For a balanced tree, DFS uses less space than BFS.
Simplicity : Recursive implementations can be more straightforward and concise.
Can Be Less Efficient for Wide Trees : For very wide trees, DFS can use more space than BFS.
May Not Find the Shortest Path : If you're looking for the shortest path in an unweighted tree, BFS is generally a better choice.
Binary Tree Path Sum : Given a binary tree and a number ‘S’, find if the tree has a path from root-to-leaf such that the sum of all the node values of that path equals ‘S’.
All Paths for a Sum : Find all root-to-leaf paths in a binary tree that have a sum equal to a given number.
Count Paths for a Sum : Find the number of paths in a tree that sum up to a given value.

10. Two Heaps

The Two Heaps pattern involves using two priority queues (heaps) to maintain a running balance or median of a set of numbers. One heap keeps track of the smaller half of the numbers, and the other keeps track of the larger half.

Running Median : Perfect for problems where you need to find the median of a set of numbers as new numbers are added.
Balanced Partition : Useful for problems where you need to maintain a balanced partition of numbers.
Efficiency : Provides a way to efficiently find the median or maintain balance in O(log N) time.
Dynamic : Can handle dynamic datasets where numbers are added over time.
Complexity : Implementation can be more complex due to the need to balance the two heaps.
Space Overhead : Requires additional space to store the numbers in the heaps.
Find the Median of a Number Stream : Design a class to calculate the median of a number stream.
Sliding Window Median : Find the median of all subarrays of size ‘K’ in the array.
Maximize Capital : Given a set of investment projects with their respective profits, we need to find the most profitable projects. We are given an initial capital and are allowed to invest only in a fixed number of projects. Our goal is to choose projects that give us the maximum profit.

11. Subsets

The Subsets pattern involves dealing with problems that require generating all possible combinations or subsets of a set. This pattern is particularly useful when you need to explore all the different ways to combine elements, which is a common scenario in many coding problems.

Combinatorial Problems : Ideal for problems where you need to generate all possible combinations of elements.
Exhaustive Search : Useful when you need to perform an exhaustive search over all possible subsets of a set.
Comprehensive : Ensures that you consider all possible combinations of elements.
Versatile : Can be used to solve a variety of problems, including generating power sets, combinations, and permutations.
Time Complexity : Can lead to exponential time complexity, as the number of subsets of a set is 2^N.
Space Complexity : Requires additional space to store all the subsets.
Subsets : Given a set with distinct elements, find all of its distinct subsets.
Subsets With Duplicates : Given a set of numbers that might contain duplicates, find all of its distinct subsets.
Permutations : Given a set of distinct numbers, find all of its permutations.

12. Modified Binary Search

The Modified Binary Search pattern involves adapting the classic binary search algorithm to solve various problems, often related to searching in a sorted array or finding the boundary of a condition.

Sorted Arrays : Perfect for problems involving searching or making decisions based on sorted arrays.
Finding Boundaries : Useful for finding the start or end of a condition in a sorted array.
Efficiency : Provides a logarithmic time complexity solution for searching problems, making it highly efficient.
Versatility : Can be adapted to solve a wide range of problems beyond simple searching.
Applicability : Mainly beneficial for problems involving sorted arrays or conditions with clear boundaries.
Implementation Nuances : Requires careful implementation to handle edge cases and avoid infinite loops.
Order-agnostic Binary Search : Given a sorted array of numbers, find the index of a given number. The array could be sorted in ascending or descending order.
Ceiling of a Number : Given an array of numbers sorted in ascending order, find the ceiling of a given number. The ceiling of a number is the smallest number in the given array greater than or equal to the given number.
Next Letter : Given an array of lowercase letters sorted in ascending order, find the smallest letter in the given array greater than a given ‘key’.

13. Bitwise XOR

The Bitwise XOR pattern involves using the XOR bitwise operator to solve problems, often related to finding missing numbers or duplicate numbers in an array. XOR is a binary operator that returns 1 when the two bits are different and 0 when they are the same.

Finding Missing or Duplicate Numbers : Ideal for problems where you need to find a missing number or duplicate numbers in an array.
Bit Manipulation : Useful for problems that require manipulation of bits to achieve the desired result.
Efficiency : Provides a constant space solution for certain problems, making it highly efficient.
Simplicity : Once understood, the XOR operator can be used to create elegant and simple solutions.
Specificity : Mainly beneficial for problems involving finding missing or duplicate numbers.
Learning Curve : Understanding how the XOR operator works and when to use it can take some time.
Single Number : In a non-empty array of integers, every number appears twice except for one, find that single number.
Two Single Numbers : In a non-empty array of numbers, every number appears exactly twice except two numbers that appear only once. Find the two numbers that appear only once.
Complement of Base 10 Number : For a given positive number N in base-10, return the complement of its binary representation as a base-10 integer.

14. Top 'K' Elements

The Top 'K' Elements pattern involves finding the 'K' largest or smallest elements in an array or stream of data. This pattern is particularly useful when dealing with large datasets and you need to maintain a subset of the data based on certain criteria.

Priority Queue : Utilizes a min-heap or max-heap to efficiently keep track of the 'K' largest or smallest elements.
Streaming Data : Ideal for scenarios where the data is streaming in, and you need to maintain the 'K' largest or smallest elements at any given time.
Efficiency : Provides a way to find the 'K' largest or smallest elements in O(N log K) time.
Space Efficiency : Only requires O(K) space, regardless of the size of the dataset.
Limited to 'K' Elements : Only maintains information about the top 'K' elements, not the entire dataset.
Heap Maintenance : Requires careful maintenance of the heap to ensure efficiency.
Top 'K' Numbers : Given an unsorted array of numbers, find the ‘K’ largest numbers in it.
Kth Smallest Number : Given an unsorted array of numbers, find the Kth smallest number in it.
‘K’ Closest Points to the Origin : Given an array of points in the a 2D plane, find ‘K’ closest points to the origin.

15. K-way Merge

The K-way Merge pattern involves merging multiple sorted arrays or lists into a single sorted list. This pattern is highly useful in scenarios where you have multiple sorted datasets that you need to combine and maintain the sorted order.

Multiple Sorted Arrays : Ideal for merging multiple sorted arrays or lists.
External Sorting : Useful in external sorting, where the data to be sorted does not fit into memory and is stored in sorted chunks.
Efficiency : Provides a way to merge multiple sorted arrays in O(N log K) time, where ‘N’ is the total number of elements across all arrays, and ‘K’ is the number of arrays.
Space Efficiency : Only requires O(K) space for the priority queue.
Dependent on Sorting : The efficiency of this pattern depends on the arrays being sorted.
Priority Queue Overhead : Requires maintenance of a priority queue, which adds to the complexity.
Merge K Sorted Lists : Given an array of ‘K’ sorted LinkedLists, merge them into one sorted list.
Kth Smallest Number in M Sorted Lists : Given ‘M’ sorted arrays, find the K’th smallest number among all the arrays.
Find the Smallest Range Covering Elements from K Lists : Given ‘M’ sorted arrays, find the smallest range that includes at least one number from each of the ‘M’ lists.

16. Topological Sort

Topological Sort is a pattern used for linearly ordering the vertices of a directed graph in such a way that for every directed edge (U, V), vertex U comes before V. This pattern is particularly useful in scenarios where you have a set of tasks and some tasks depend on others.

Task Scheduling : Ideal for problems where tasks need to be scheduled in a specific order, respecting their dependencies.
Course Scheduling : Useful in scenarios like course scheduling where some courses have prerequisites.
Clarity : Provides a clear and systematic way to order tasks or vertices.
Detecting Cycles : Helps in detecting cycles in a directed graph, which is important for understanding if a valid ordering is possible.
Applicability : Mainly beneficial for problems involving directed graphs and ordering of vertices.
Complexity : Implementation can be complex, especially for beginners.
Topological Sort : Given a directed graph, find the topological ordering of its vertices.
Tasks Scheduling : Find if it is possible to schedule all the tasks.
Tasks Scheduling Order : Find the order of tasks we should pick to finish all tasks.

A Trie, also known as a prefix tree, is a tree-like data structure used to store a dynamic set of strings, where the keys are usually strings. It is particularly useful for retrieval of a key in a dataset of strings, which makes it highly efficient for solving word-based problems.

Autocomplete : Ideal for implementing autocomplete functionality in search engines or text editors.
Spell Checker : Useful for building spell checkers in word processors.
IP Routing : Used in IP routing to store and search routes.
Efficiency : Provides fast retrieval of strings and is more efficient than hash tables or sets when it comes to string keys.
Prefix Searching : Excellent for problems that require prefix searching or matching.
Space Overhead : Can use more space compared to other data structures when the dataset is sparse.
Complexity : Implementation can be complex, especially when handling deletion of words from the Trie.
Insert into and Search in a Trie : Implement insertion and search in a Trie.
Longest Common Prefix : Find the longest common prefix of a set of strings.
Word Search : Given a 2D board and a word, find if the word exists in the grid.

18. Backtracking

Backtracking is a general algorithmic technique that considers searching through all the possible configurations of a search space in order to solve computational problems. It is particularly useful for optimization problems and when a complete search of the solution space is required. The main idea is to explore each possibility until the solution is found or all possibilities have been exhausted.

Combinatorial Problems : Ideal for solving problems that require generating all possible configurations like permutations, combinations, and subsets.
Puzzle Solving : Useful for solving puzzles such as Sudoku, crossword puzzles, and the N-Queens problem.
Completeness : Ensures that the entire solution space is explored, guaranteeing that the optimal solution will be found if it exists.
Space Efficiency : Uses less memory as it only needs to store the current state and the decision stack.
Time Complexity : Can lead to exponential time complexity, as it explores all possible configurations.
Optimization Required : May require additional optimizations like pruning to be practical for larger instances.
Subsets : Given a set of numbers, find all of its subsets.
N-Queens : Place N queens on an N×N chessboard so that no two queens threaten each other.

19. Monotonic Stack

A Monotonic Stack is a specialized data structure that maintains elements in a sorted order while supporting stack operations. It is particularly useful for problems where you need to find the next greater or smaller element in an array or when you need to maintain a running maximum or minimum value efficiently.

Next Greater Element : Ideal for finding the next greater element for each element in an array.
Maximum Area Histogram : Useful for problems like finding the largest rectangular area under a histogram.
Efficiency : Provides a way to solve certain problems in linear time, making it highly efficient.
Simplicity : Once understood, the monotonic stack can lead to concise and elegant solutions.
Specificity : Mainly beneficial for problems involving finding the next greater or smaller element and related problems.
Learning Curve : Understanding how and when to use a monotonic stack can take some time.

Example Problems

Next Greater Element : Given an array, find the next greater element for each element in the array.
Maximum Area Histogram : Given a histogram, find the largest rectangular area under the histogram.
Largest Rectangle in Histogram : Given n non-negative integers representing the histogram's bar height where the width of each bar is 1, find the area of largest rectangle in the histogram.

20. 0/1 Knapsack (Dynamic Programming)

The 0/1 Knapsack problem is a classic optimization problem that falls under the category of Dynamic Programming. In this problem, you are given a set of items, each with a weight and a value, and a knapsack with a maximum capacity. The goal is to determine the maximum value that can be accommodated in the knapsack without exceeding its capacity. The "0/1" part of the name reflects the fact that you can't break an item, you either take it or leave it.

Resource Allocation : Ideal for problems where you need to optimally allocate limited resources to maximize profit or minimize cost.
Budgeting : Useful for budgeting scenarios where you need to choose a subset of projects or investments to maximize return.
Optimality : Ensures that the optimal solution is found, provided that the problem satisfies the principle of optimality.
Generality : Can be adapted to solve a wide variety of optimization problems.
Time and Space Complexity : The naive implementation has a time and space complexity of O(nW), where n is the number of items and W is the capacity of the knapsack. This can be prohibitive for large inputs.
Requires Integer Weights and Values : The classic 0/1 Knapsack problem requires weights and values to be integers.
0/1 Knapsack : Given the weights and profits of ‘N’ items, put these items in a knapsack which has a capacity ‘C’. The goal is to get the maximum profit out of the items in the knapsack.
Equal Subset Sum Partition : Given a set of positive numbers, find if we can partition it into two subsets such that the sum of elements in both subsets is equal.
Subset Sum : Given a set of positive numbers, determine if there exists a subset in the set whose sum is equal to a given number ‘S’.

Mastering these patterns is definitely not about memorizing solutions; it’s about understanding the underlying principles and learning how to apply them to a wide array of problems. The versatility of these patterns ensures that you are well-equipped to handle different challenges, making you a formidable candidate in any coding interview.

Remember, practice is key. The more problems you solve using these patterns, the more proficient you will become in recognizing problem types and applying the appropriate patterns. So, keep practicing, stay persistent, and you will find yourself excelling in coding interviews, ready to tackle any problem that comes your way.

Want to read more about coding patterns:

Don’t Just LeetCode; Follow the Coding Patterns Instead
Top LeetCode Patterns for FAANG Coding Interviews
Grokking Dynamic Programming Patterns for Coding Interviews

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Design Patterns Cheat Sheet – When to Use Which Design Pattern?

In system design , selecting the right design pattern is related to choosing the right tool for the job. It’s essential for crafting scalable , maintainable , and efficient systems. Yet, among a lot of options, the decision can be difficult. This Design Patterns Cheat Sheet serves as a guide, helping you on the path toward optimal design pattern selection. Simplifying complex concepts into easy insights empowers engineers to navigate design patterns confidently.

Important Topics for Design Patterns Cheat Sheet

What are Design Patterns?

When to use which design pattern.

When to Choose Creational Design Patterns?

When to Choose Structural Design Patterns?

When to Choose Behavioral Design Patterns?

Importance of Choosing the Right Design Pattern

Design patterns are reusable solutions to common problems encountered during software design and development. They represent established best practices for structuring code to address specific challenges in a standardized and efficient manner.

By following design patterns, developers can streamline the development process, enhance code clarity, and facilitate communication among team members.
Examples of design patterns include Singleton, Factory, Observer, Strategy, and many others, each helps to address distinct sets of problems.

Before we select a pattern, we must identify and understand the problem we have encountered, and based on that we can choose which sub-category of design pattern we can go for, Let us see in this Design Patterns Cheat Sheet, that in which scenario to choose which design pattern, that can effectively solve our problem:

Design Patterns Cheat Sheet

When to Choose Creational Design Patterns ?

According to this Design Patterns Cheat Sheet, choose creational design patterns when object creation is complex, involves multiple steps, or requires specific initialization. They’re useful for promoting reusability, encapsulating creation logic, and decoupling client code from classes it instantiates.

Creational patterns enhance flexibility, making it easier to change or extend object creation methods at runtime.
Common patterns include Singleton, Factory Method, Abstract Factory, Builder, and Prototype. Use them to improve maintainability, readability, and scalability of your codebase.

We must choose Creational Design Patterns when the Problem is related to Object Creation .

When to Choose Creational Design Patterns

Singleton: Makes sure there is just one instance.
Factory Method: Assigns subclasses the task of instantiating objects.
Abstract Factory: Constructs related object families without defining their concrete classes.
Prototype: Clones objects to provide a template example.
Builder: Helps in building the complex objects step by step.

When to Choose Structural Design Patterns ?

According to this Design Patterns Cheat Sheet, choose structural design patterns when you need to compose objects and classes into larger structures while keeping them flexible and efficient. These patterns are useful for clarifying relationships between classes, managing object hierarchies, and altering interfaces without affecting clients.

Structural patterns promote code reuse, simplify system design, and enhance scalability.
They’re beneficial when dealing with complex systems, integration of new components, or refactoring existing codebases.

We must choose Structural Design Patterns when the Problem is related to Object Assembly .

When to Choose Structural Design Patterns?

Adapter: Acts as a bridge between two incompatible interfaces
Bridge: Separates the abstraction from the implementation.
Composite: Handles single and composite objects equally.
Decorator: Adds behaviors to objects dynamically.
Facade: Helps in Simplifying the complex system interfaces.
Flyweight: Shares common parts of state between multiple objects to reduce memory.
Proxy: Controls the access to an object.

When to Choose Behavioral Design Patterns ?

According to this Design Patterns Cheat Sheet, choose behavioral design patterns when you need to manage algorithms, communication, or responsibilities between objects. They’re useful for encapsulating behavior that varies and promoting loose coupling between objects.

Behavioral patterns facilitate code reuse, flexibility, and maintainability by defining how objects interact and communicate.
Use them to address scenarios like handling complex workflows, managing state transitions, or implementing communication between objects.

We must choose Behavioral Design Patterns when the Problem is related to Object Interactions.

When to Choose Behavioral Design Patterns?When to Choose Behavioral Design Patterns?

Observer: Observes and notifies changes in multiple objects.
Strategy: Encapsulates the interchangeable algorithms.
Command: Encapsulates requests as objects for decoupled execution.
State: It Changes the behavior of object with internal state.
Visitor: It separates algorithms from objects.
Memento: Pattern to manage object state and actions.
Iterator: It Sequentially accesses the elements of a collection.
Mediator: Central controller managing communication between objects.
Chain of Responsibility: Pass request through handlers until one handles it.
Template Method: Defines the skeleton of an algorithm.

According to this Design Patterns Cheat Sheet, choosing the right design pattern is crucial due to:

Scalability: Ensures the architecture can accommodate growth without excessive restructuring.
Flexibility: Enables easy adaptation to changing requirements and future enhancements.
Maintainability: Facilitates code readability and comprehension, easing maintenance tasks.
Reusability: Promotes reuse of proven solutions, saving development time and effort.
Performance: Optimizes code execution and resource utilization for efficient operation.
Reducing Errors: Helps avoid common pitfalls and design flaws, leading to fewer bugs and issues.

With the help of this Design Patterns Cheat Sheet, selecting the appropriate design pattern is crucial for effective problem-solving in software development. By understanding the characteristics of different design patterns and analyzing the specific requirements and constraints of the problem at hand, developers can make informed decisions that lead to more maintainable, scalable, and efficient solutions.

Moreover, the choice of design pattern can significantly impact the readability, flexibility, and extensibility of the codebase. It allows developers to leverage proven solutions to common design problems, reducing the chances of errors and promoting code reusability.

FAQs related to Design Patterns Cheat Sheet – When to Use Which Design Pattern?

Here are some frequently asked questions (FAQs) related to when to use various design patterns:

Q. When should I use design patterns?

Design patterns should be used when you encounter recurring design problems in your software development process. They help in making your code more maintainable, flexible, and easier to understand.

Q. How do I know which design pattern to use?

You should choose a design pattern based on the problem you’re trying to solve. Each design pattern addresses a specific concern, such as object creation, behavior management, or structural organization. Understanding the problem and the characteristics of each pattern will help you make the right choice.

Q. Are design patterns applicable to all programming languages?

Yes, design patterns are applicable to most programming languages. While some patterns may be more commonly used in certain languages or environments, the concepts behind them are language-agnostic.

Q. Can I create my own design patterns?

Yes, you can create your own design patterns if you encounter a problem that is not addressed by existing patterns. However, it’s important to ensure that your pattern is well-documented, widely applicable, and follows established design principles.

Q. Should I always use design patterns in my code?

While design patterns can be very helpful, they should not be applied blindly. It’s essential to understand the problem you’re solving and whether a design pattern is the appropriate solution. Overusing patterns can lead to overly complex and difficult-to-maintain code.

Q. How do design patterns relate to software architecture?

Design patterns are often used within the context of software architecture to address specific concerns at different levels of abstraction. They help in organizing and structuring the components of a system to promote flexibility, scalability, and maintainability.

Q. Are there any drawbacks to using design patterns?

While design patterns can offer many benefits, they can also introduce complexity to your codebase, especially if used inappropriately. Additionally, relying too heavily on patterns without fully understanding their implications can lead to over-engineering.

Q. Where can I learn more about design patterns?

There are numerous resources available for learning about design patterns, including books, online tutorials, and courses. Additionally, studying real-world examples and practicing applying patterns in your own projects can help deepen your understanding.

Q. Can design patterns be combined or used together?

Yes, design patterns can often be combined or used together to address complex design problems. Understanding how different patterns interact and complement each other can lead to more robust and flexible solutions.

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Pattern Problem Solving: Teach Students to Find a Pattern in Math Problems

Pattern problem solving

What is pattern problem solving?

Why is pattern problem solving important?

How to teach students to find the pattern in a math problem (using an example)

1. Ensure students understand the problem

2. Choose a pattern problem solving strategy

3. Solve the problem

4. Check your students' answers

5. Explain

How can you stretch this pattern problem solving strategy?

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Course: 3rd grade > Unit 8

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Video transcript

Simple Patterns

Number Patterns

Module 1: Problem Solving Strategies

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

For younger learners

Advanced mathematics

Patterns and Sequences Short Problems

Triangular Clock

Printing Error

Sliding Robot

Many Matildas

How Many Rectangles?

Night Watchmen

Pattern Snake

What a Coincidence!

Suit Sequence

Knights and Knaves

Doubly Consecutive Sums

Expanding Pattern

Fibonacci Deduction

Hexagon Line

Street Lamps

Fruit Line-up

Alternating Sum

Newspaper Sheets

Collatz-ish

Trolley Park

SOLVING PROBLEMS WITH PATTERNS

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