paper folding problem solving

Fold it right there: The mathematical art of paper folding

Malgosia Ip , Mathematics & Statistics editor

“I really don’t think it’s possible,” I say again, unfolding the rumpled sheet of paper. I have been trying to solve one of Erik Demaine’s folding puzzles for a few hours now. Some of the creases have been folded so many times that the paper is starting to tear.

Demaine, a Canadian math prodigy and now Professor at MIT, has been inventing and solving puzzles with his father since he was six years old. When Demaine’s research took him down the path of computational origami, the puzzles followed suit, becoming paper-based folding puzzles. Now Erik and his dad’s folding puzzles have become a tradition at the MIT Computer Science and Artificial Intelligence Laboratory (CSAIL) annual meeting.

The neat thing about these puzzles is that each one illustrates an unsolved mathematical problem. The one I’m working on is from the 2007 meeting, and it’s one of Demaine’s favourites. (See image, above: The 2007 CSAIL puzzle created by Erik and Martin Demaine. Reproduced with permission .)

“It relates to a mathematical problem called ‘ turning inside out ’,” Demaine tells me. “What surfaces can be turned inside-out (reversing their inside and outside) by a continuous folding motion using [a finite number of] creases?” Knowing the problem that the puzzle represents is actually a clue to its solution.

Download the puzzle and join me on a trip down the mathematical origami rabbit-hole.

Fold the paper into a square…

The instructions are simple: fold the paper into a square so that both sides form perfect 8 x 8 checkerboards. No problem. I begin haphazardly by folding the paper into 4 x 4 grids and trying my luck. It almost seems like the first attempt works, but this puzzle plays tricks on the eyes. I soon realize that two black squares are touching in several places. I begin again, trying to do something different this time, but the result is very similar – some of the squares seem to work, but not all of it is correct. After several more attempts, I start to feel like I’m doing the same thing over and over. There must be a better way.

My first try: I had started by folding along all the 4 x 4 grids and thought I had it almost right away… but it was only half right.

Paper folding, or origami, has been around for fifteen centuries, and most of the iconic designs (paper crane, anyone?) arose from treating it like an art form, not a science. Origami masters developed a knowledge of the basic folding patterns and how to put them together to achieve certain shapes – it took much practice to develop a new pattern that looked nice.

It wasn’t until much later that mathematics was added to the origami toolkit.

Origami meets math

Beginning in the late-1980s, NASA physicist Robert Lang and Japanese biochemist Toshiyuki Meguro developed a systematic design method that could be used to build arbitrarily complicated origami structures. They called it the circle-river method because of the way the desired design is encoded – a circle for each flap of the final figure and a line or “river” for each connecting piece.

Combining this algorithm with the power of the computer, Lang was able to create new origami designs that were well beyond what a person with pencil and paper could do. This was the birth of computational origami, which brought people like Demaine onto the origami bandwagon.

“Computational origami is (in some sense) a subfield of computational geometry,” explains Demaine. He was looking for something to research for his PhD and had recently seen a paper by Lang published in a prominent computational geometry journal.

“[The field] was just getting started at that point (there were just two papers on the topic), and it sounded like a fun direction to explore. I’d never folded paper at that point.”

Since that seminal work, the power of mathematics has completely transformed the art of origami. In fact, Demaine recently published a paper describing a universal algorithm for folding any 3D structure that guarantees a minimum number of seams.

I’m sure a computer algorithm would be able to solve this puzzle with ease, but I am determined to get it myself. I decide to try and make some rules that can help me solve the folding puzzle. After all, there are a limited number of ways that this puzzle can be folded into an 8 x 8 square and, if I can eliminate some of them, I can start to home in on the solution. One thing that jumps out are the two squares with the instructions – those must be hidden in the finished product. I try to keep this in mind when I begin folding again.

When I realized I didn’t have to fold along the grid, things got a little crazy.

Theory into practice

The work is not just theoretical – various practical applications of these mathematical folding methods have emerged. Many devices that must start small and expand, like airbags, heart stents , and satellites , have all benefited from origami folding techniques. In fact, NASA has just put out a call for origami designs for a radiation shield.

“Computational origami turned out to be exciting both as a mathematical challenge and in terms of applications to engineering,” says Demaine. He’s currently most excited about folding nanoscale structures to transform 2D nanoprinting technology, like that used in computer chip manufacturing, into technology that can make any 3D shape with nanoscale resolution.

Still, the ability to design any origami structure doesn’t mean it’s easy to turn the flat folding pattern into the final shape. In fact, most complex patterns can’t be constructed one fold at a time and require the simultaneous folding of many creases at once – something called the collapse.

As Demaine tells me, patience is the only requirement to becoming a paper folding expert.

Paper proof

I did eventually solve the 2007 CSAIL puzzle by thinking about the original mathematical problem that it was meant to represent. Indeed, the final solution didn’t emerge fold by fold but came suddenly, as the paper collapsed in on itself from all sides at once. The feeling of satisfaction as I scanned the square and saw the perfect checkerboard is hard to describe. Never have I been so excited to show everyone a folded piece of paper.

Proof that I had managed to solve the 2017 CSAIL puzzle

I may have solved the puzzle, but the general “turning inside out problem”, which the puzzle represents, remains unsolved. Demaine and other mathematicians working in this field have no shortage of challenges remaining – and plenty of paper possibilities.

One thought on “ Fold it right there: The mathematical art of paper folding ”

That’s amazing. “The feeling of satisfaction as I scanned the square and saw the perfect checkerboard is hard to describe. Never have I been so excited to show everyone a folded piece of paper.”

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Purposeful Paper Folding

• Practical and purposeful collaborative activity is an essential part of a curriculum that seeks to develop problem solving and creativity. (Challenge 1 in both Paper Patchwork 1 and Folding Flowers 1 , for example, offer the chance to create and explore within given constraints.)
• That origami is an accessible endeavour that provides challenge in both these areas for a wide age range (Frigerio, 2002; Haga, 2002; Cornelius and Tubis, 2009; Pope and Lam, 2009; Hull 2006 all in Pope and Lam, 2011) and helps to develop fine motor skills in young children and refine them in older learners. See also Froebel's work on Kindergarten education (Heewart, 1992 in Pope and Lam, 2011).
• That encouraging collaboration and discussion amongst students of all ages can help to develop their confidence and understanding.
• That unfolding and re-folding origami offers learners the opportunity to work out how a model has been constructed and recreate those steps for themselves. 
• That looking at 2D designs to 'see' what shapes can be identified will develop creative learners, open to new ways of looking at things and dispel the mathematical myth that there always has to be one right answer.  (Challenge 2 in Paper Patchwork 2 and Regular Rings 2 provides photographic examples in the Teachers' Resources, of the sorts of designs that can be created and talked about, to 'see' new shapes.)
• That when children are encouraged to collaborate, promoting a 'those who can, help others' approach, this develops fluency of vocabulary in the describer and requires perseverance from both learners.

Teaching Math Through Paper Folding

Introduction: Teaching Math Through Paper Folding

Years and years ago, when I was still in college studying to become a math teacher, I had a professor who said we could attend the Detroil Area Council of Teachers of Mathematics conference instead of doing one of the projects for class. I do not remember much about that class or what we learned but I remember the conference. More specifically, I remember one of the sessions that I attended. The session was held in the library at Lake View (I think) High School in St. Clair Shores, Michigan. I remember that I sat in the back of the class. There was a morning session and an afternoon session--I attended both. I remember leaving with tons of ideas and have used many of them over the years. (The second session was called 'scrap paper math'.) The steps that follow this are from this session of the DACTM conference. I do not remember the instructor's name or I would give him the credit that he so richly deserves. I have made dozens of these over the years. They make very nice Christmas ornaments. Even made from old math papers they look pretty. Since it is totally hollow inside and can be disassembled without tearing the paper, I have used the 20 pointed star to wrap small gifts.

Step 1: Getting Started

In the class, the instructor modeled the procedure that he used with his students. I can speak from experience--it works. He was folding paper as he gave instruction but he was also talking and asking questions. He loved asking questions about fractions and the size of different angles. He kept using math vocabulary and encouraging us to use the vocabulary in our answers. The only way to really get comfortable with technical terms is to use them repeatedly. Start with a square sheet of paper--he used scrap paper. If there is anything that a teacher has a lot of, it is scrap paper. He had brought stacks of old worksheets, news letters, even student work that he had lying around. You do not have to go buy origami paper. All you need is a nice paper cutter--try the art room or the main office. You can use scissors but it takes longer and the cuts are not as straight. I usually mass produce a whole box of squares using whatever paper is available.

A 4 inch square is big enough for most kids when they are starting out. If you go too much smaller, it can be hard for them to get their fingers where they need to be. I usually demonstrate with 8 inch squares--easier for the kids in the back to see what I am doing. The smallest that I have worked with is an inch and a half but it was a bit difficult to work on even with my small fingers. I recommend at least a 2 inch square.

Step 2: First Folds

Before we start, let me just say that neatness does count on this project. Fold carefully. Crease sharply. Fold one sheet of paper in half so that you end up with a rectangle. When you unfold the paper, you will see that you have 2 congruent rectangles and that you bisected 2 opposite sides of the square. Fold the 2 outside edges in toward the original fold. You should have quartered the sheet of paper. Each quarter is a congruent rectangle. You once again bisected a pair of segments (the edge of the paper).

Step 3: Paper Airplane Folds

These are the hardest folds to make in the whole process, and the hardest to explain. Some students 'know' what you are going to tell them to do so they just don't bother to pay close attention. Once you know who these kids are, position yourself so that you can keep a close eye on them. That way you can stop them before they go too far off track.

I call them paper airplane folds because that is what the instructor called them. And because, who hasn't made a paper airplane. With that said, everyone doesn't make paper airplanes the same way. Keep an eye on the couple of students who don't follow directions well. Start with the folds running vertically. Fold the top right edge of the square down to line up with the quarter fold. Some students use the original half fold instead of the quarter fold--'cause teenagers hate to be told what to do. Once you get them back on track, they listen better. Now is an appropriate time to ask about angle measures. You bisected a right angle so you now have a 45 degree angle. The tougher question to ask students is "What fraction of the whole square is this little isosceles right triangle?" It usually takes a few tries before someone gets it right. Teenagers these days are even worse at fractions than when I was a kid. I blame it on the availability of calculators. The second airplane fold brings the folded edge to the same quarter fold. This bisectes the 45 degree angle leaving you with a 22 1/2 degree angle. Believe it or not, some teenagers have never considered that there might be something smaller than a whole degree.

Step 4: Rotate the Piece.

Once everyone has the 2 paper airplane folds in the upper right hand corner, you have the class rotate the piece 180 degrees. Watch to make sure that you catch the student or 2 that flip the thing over. Repeat the 2 airplane folds in the new upper right hand corner. You should have folded the opposite corners of the square--not adjacent corners. If you have a student who is left handed, it may be more comfortable for them to do the airplane folds on the upper left corner. The whole project will work but the 'left handed' pieces will not fit together with the 'right handed' ones.

Step 5: Next Set of Folds

Now, you fold both the left and right quarters in towards the center crease. If you folded neatly, the paper will not fight you. If you went past the quarter crease on your airplane folds, you may have to work to get it to lay flat. Now your piece is only 1/2 the size of the original square. You need to fold the upper LEFT corner all the way over to lighten up with the right hand edge of the rectangle. What fraction of the whole original square is this isosceles right triangle--tough question for some kids because they are not looking at the whole square of paper at this point. Open up the paper airplane fold and close it over the isosceles right triangle. This will hold the triangle in place. The shape is now a right trapezoid. Rotate the whole piece 180 degrees and repeat the fold. Tuck this triangle under the airplane folds. What shape did you make? Most kids know 'parallelogram' by the time they get to me.

Step 6: Finishing the Piece

Your piece has 2 long edges and 2 short edges. Lay the piece on the table so that the long edges run horizontally. Flip the piece so that the tucks are face down. You are going to bisect the long edge by bringing the acute (45 degrees) vertex (corner) over to the obtuse (135 degrees) one. Crease it well--there are a lot of layers of paper. Use the back of your finger nail. Don't worry that it doesn't want to stay folded.

Do the same and bisect the other long edge of the piece.

You now have the square that you need to build your cube.

Step 7: Make More Pieces.

Have each student begin the whole process over again with a fresh square of paper. You should be ready to remind them of the strips when they get stuck. By the time they have completed 2 or 3 pieces, some of them will remember the steps--then they can remind each other of the steps.

Step 8: Assembly

Six pieces will assemble into one cube. I usually have 2 or 3 students pool their pieces to finish the cube. If you look at the square side of a piece, you will notice that there are 4 pockets--2 that are triangular, 2 that are quadrilaterals. We will only be using the triangles. I colored in the quadrilateral pockets in the picture above. These will be covered by another piece but you will not be using them as pockets. Take a second piece and insert the acute vertex into one of the triangular pockets. Take a 3rd piece and put one of its acute corners into the pocket of the 2nd piece. Before you grab another piece, tuck the corner of the first piece into the 3rd piece. You should have 3 sides of a cube. Many kids can figure out the rest without help. Some need a little more instruction. The biggest problem that most kids have is that they allow the acute corners to slip inside the box. When I help them assemble their pieces, my job usually involves holding the acute points outside of the box. Then I have the student tuck it into the triangle pocket. Sometime, students forget to find the triangle pocket. Nothing will fit right if you use the quadrilateral ones. If one piece is made 'left handed', it will never fit. All must be made the same 'handedness'.

Step 9: The Challenge

This is where the instructor stopped teaching us. He started telling us a story. It had been about 5 years at this point since he had learned what he had just taught us. He went on to tell us that he had made dozens of pieces and had taught people--teens and adults--to assemble the cube. When he was originally taught, his teacher gave him a completed 'stellated icosahedron' and he pulled it out of the paper grocery bag that he had with him. He had been trying for 5 years to reproduce the thing. While he continued to talk, he pulled out all sorts of miss-shaped attempts. One, I remember, looked like like a snake. He told us that the icosahedron used 30 pieces like the ones we had just made but he couldn't figure out how to assemble them. I went back to the conference the following year ('cause I wanted to, not to get out of work for a class) and went to the same teacher's class. I got there late so I had to sit in the back (again). While I was learning all the new things he was teaching, I used my spare time to fold 30 pieces and assemble the star. When I got the last piece in, he stopped class (he knew what I was working on), and told them that he no longer included this project in his seminars. He dropped it because he thought since he could not teach the assembly process, it was a waste of our time. I told him that not only did I figure it out, but I showed it to my high school students. I taught them what he had taught us. Then I left it as an open challenge. Anyone who could make one by the end of the term got extra credit. 4 of them got the extra credit. I have included one extra hint if you want it. It is on the next step. You can choose to accept the challenge of creating your own 20 pointed star now or read one last step.

Step 10: Hint

There was one more fold that each piece needs if you are assembling all 30 pieces. See the picture above.

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5 Reasons Why Origami Improves Students’ Skills

Origami, the ancient art of paper folding, has applications in the modern-day classroom for teaching geometry, thinking skills, fractions, problem solving, and fun science.

What do pizza boxes, paper bags, and fancy napkins have in common? Well, you might have guessed it -- origami.

Origami, the ancient art of paper folding, is making a comeback. While some of the oldest pieces of origami have been found in ancient China and its deepest roots are in ancient Japan, origami can make an impact in today's education too. This art form engages students and sneakily enhances their skills -- including improved spatial perception and logical and sequential thinking.

An Art Form for All Subjects

Don’t believe me? Researchers have found a number of ways that origami can make lessons enticing, while giving students skills they need. (Think of it as vegetables blended into spaghetti sauce.) Here are some ways that origami can be used in your classroom to improve a range of skills:

According to the National Center for Education Statistics in 2003, geometry was one area of weakness among American students. Origami has been found to strengthen an understanding of geometric concepts, formulas, and labels, making them come alive. By labeling an origami structure with length, width, and height, students learn key terms and ways to describe a shape. You can use origami to determine the area by applying a formula to a real-world structure.

Thinking Skills

Origami excites other modalities of learning. It has been shown to improve spatial visualization skills using hands-on learning. Such skills allow children to comprehend, characterize, and construct their own vernacular for the world around them. In your class, find origami or geometric shapes in nature and then describe them with geometric terms.

The concept of fractions is scary to lots of students. Folding paper can demonstrate the fractions in a tactile way. In your class, you can use origami to illustrate the concepts of one-half, one-third, or one-fourth by folding paper and asking how many folds students would need to make a certain shape. The act of folding the paper in half and in half again and so on can also be used to demonstrate the concept of infinity.

Problem Solving

Often in assignments, there is one set answer and one way to get there. Origami provides children an opportunity to solve something that isn't prescribed and gives them a chance to make friends with failure (i.e. trial and error). In your class, show a shape and ask students to come up with a way to make it. They may get the solution from various approaches. Remember, there is no wrong answer.

Fun Science

Origami is a fun way to explain physics concepts. A thin piece of paper is not very strong, but if you fold it like an accordion it will be. (Look at the side of a cardboard box for proof.) Bridges are based on this concept. Also, origami is a fun way to explain molecules. Many molecules have the shape of tetrahedrons and other polyhedra.

Bonus: Just Plain Fun!

I hope that I don't need to explain fun. Here are some activities ( with diagrams ) to keep those young hands and minds working.

No Papering Over Origami's Benefits

Children love origami as evidenced by how they are enamored with their first paper airplane, paper hat, or paper boat. And while we might not always think about it, origami surrounds us -- from envelopes, paper fans, and shirt folds to brochures and fancy towels. Origami envelops us (forgive the pun). Origami has been found to improve not only 3D perception and logical thinking (PDF), but also focus and concentration.

Researchers have found that students who use origami in math perform better. In some ways, it is an untapped resource for supplementing math instruction and can be used for geometric construction, determining geometric and algebraic formulas, and increasing manual dexterity along the way. In addition to math, origami is a great way to merge science, technology, engineering, art, and math all together: STEAM.

Origami is a STEAM Engine

While schools are still catching up to the idea of origami as a STEAM engine (the merging of these disciplines), origami is already being used to solve tough problems in technology. Artists have teamed up with engineers to find the right folds for an airbag to be stored in a small space, so that it can be deployed in a fraction of a second. Additionally, the National Science Foundation, one of the government's largest funding agencies, has supported a few programs that link engineers with artists to use origami in designs. The ideas range from medical forceps to foldable plastic solar panels.

And origami continues to amaze scientists with its presence in nature. Many beetles have wings that are bigger than their bodies. In fact they can be as much as two or three times as large. How are they able to do that? Their wings unfold in origami patterns. Insects are not alone. Leaf buds are folded in intricate ways that resemble origami art, too. Origami is all around us and can be a source of inspiration for children and adults alike.

So no matter how you fold it, origami is a way to get children engaged in math, could improve their skills, and makes them appreciate the world around them more. When it comes to making lessons exciting, origami is above the fold.

Math with Paper: Fold Some Math into Your Day!

by Sarah Eason , Michelle Hurst , Susan Levine , Amy Claessens , Madeleine Oswald , Kassie Kerr & Abrea Greene

Many fun math games for families can be done with materials you probably have at home already, such as scrap paper. Learn how to create origami shapes, be a paper math wizard, and support children’s learning with these activities.

There are lots of fun math activities for families to do together that don’t require special materials. In fact, with just a few sheets of paper, families can find fun ways to explore math ideas and problem solving!

Why Do Math with Paper?

Kids can learn a lot about math at home through hands-on, playful activities that inspire conversations about  numbers , shapes , and  spatial relations .

The activities outlined below introduce and reinforce math concepts while encouraging creativity. They are also easy to adapt for different ages and may be done with multiple kids at once. Because families can do them together, it’s a great opportunity to talk about math.

Many of these activities can be done with whatever paper is available—even scrap paper, newspapers, or magazine pages would work. Scissors, hole punchers, and pencils or markers will be helpful, too.

Math with Paper Activities

Origami is great for thinking about shapes and ideas of space and place. The same sheet of paper can look completely different depending on where and how it gets folded. 

A quick search online will lead you to lots of origami instructions. Here are some origami projects to get you started:

• Beginner:  Dog origami
• Intermediate:  Heart origami
• Advanced:  Swan origami

Paper Wizard 

Mental imagery—the ability to imagine what a shape looks like or to move objects around in your mind—is an important part of math. This activity helps children practice thinking about spatial information and imagining how space can be transformed without actually seeing it. 

• Get started:  Paper wizard
• Try another one:  Advanced Paper wizard  

Even More Ideas

There are lots of ways to explore math with paper! Check out our printable  Guide to Math with Paper for ways to keep going and to support children’s math thinking during all of these activities. You can also get creative and think of your own ways to do math with paper.

Related posts

Stories About Young Children’s Amazing Math Thinking

Celebrate Culture and Math with Papel Picado

Exploring the Number Zero: Helping Children Understand the Empty Set

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Non Verbal Reasoning : Paper Folding

Many competitive exams such as SSC CGL commonly ask paper folding-based questions. In this type of reasoning-based question, a piece of paper with a pattern on it is folded in half or so, and candidates need to determine how the paper will look after folding or opening. 

How to Solve Paper Folding Questions: Step-by-Step Guide

Follow the below steps to solve any kind of paper folding-based questions.

• Step 1: Observation: Observe the paper carefully, and note the shape of the paper.
• Step 2: Visualization: In your mind, carefully visualize the fold. Mostly you will find one-fold questions, which are easy to visualize but sometimes you may encounter more than one-fold questions. 
• Step 3: Elimination Method: Once you visualize the final outcome, try to use the elimination method wrong answer which clearly can’t be the answer. Narrow down your choices.
• Step 4: Choose the best match that falls exactly the way you visualize.  

Paper Folding: Non-verbal Reasoning Question and Answers

Direction to solve: In each of the following problems, a square transparent sheet (X) with a pattern is given.  Determine which of the four options would result in the pattern looking the way you want it to when the transparent sheet is folded at the dotted line.

Q1. In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option D is the correct answer. By carefully examining it, we can see that the red marked area will become common when the paper is folded. Both the parts of the curves come together to form a circle in option D.

Q2. In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option B is the correct answer. As we can see the diameter of the circle and side of the square are of equal length so options A and D are directly eliminated. Option c gets eliminated as here square in rotate 90 degrees from the original image needs to be formed. 

Q3. In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option A is the correct answer. The first half of the paper will complete the circle in the second half. When the upper part of the square sheet is folded downward part of the semi-circle gets inverted and both form a circle. 

Q4 . In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option 3 is the correct answer. The triangle on the upper half will not change but the triangle on the second half will flip horizontally.

Q5. In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option 3 is the correct answer. When the circle with an arrow is folded arrows in both halves of the paper will face in opposite directions as in option 3. 

Q6.  In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: The lower portion will not change, as the lines in the upper side are vertical, again there will be no change in the direction of the lines. Option 4 is the correct answer.

Q7.  In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option 2 is the correct answer. 

Q8. In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option 2 is the correct answer. As we know X is drawn on transparent paper, so we can clearly observe the overlapping of the lines. The leaf of the second part will overlap and cover the leaf of the first part. 

Q9. In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option C is the correct answer. The first part of X will be the same square and will have a dot inside it in the first part, a circle with a dot inside it in the second part and the square will have a dot inside it in the third part. 

Q10.  In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option A is the correct answer. The figure has three curves so, after folding as well it should have three curves hence option A is correct. 

Q11.  In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option B is the correct answer. Circle is in the middle and when we fold it from the middle we get a semicircle which is clearly shown in Option B. 

Q12. In the following problems, a square transparent sheet with a pattern is given. Figure out from amongst the four alternatives how the pattern would appear when the transparent sheet is folded at the dotted line.

Solution: Option C is the correct answer. When the inclined slanted line is folded it will form a triangle with a straight line hence option C is correct. 

Keep practicing. All the best !!!. 

Related Links

Coding and Decoding Logical Reasoning Q & A Order and Ranking Chain Rule 

Frequently Asked Questions

How do you solve paper folding and cutting questions.

Use the steps mention in the above article, first carefully examine the paper, then visualize the way it is folded, use the elimination method to eliminate the wrong options.

What is the concept of paper cutting and folding?

Paper cutting and folding concepts involves a transparent or white paper which is been folded and some designs been cut into it and you have to visualize this in your mind in order to choose the correct option. 

How many times we can cut the paper?

We can cut the paper, exactly 7 times.

What is the objective of paper cutting?

The objective behind asking the paper cutting based questions in exams to encourage creativity, concentration.

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Problem Solving through Paper Folding

2014, Australian senior mathematics journal

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Paper Folding and Cognitive Development

Developing spatial skills is an important part of steam education..

Posted January 25, 2023 | Reviewed by Ekua Hagan

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• Paper folding improves cognitive development and can increase STEAM skills in children.
• Research shows that paper folding emerges when children are about 2 and becomes more accurate with age.

Paper folding is an engaging and fun activity for the whole family that can increase STEAM skills in children. STEAM stands for science, technology, engineering, art, and mathematics and refers to a range of skills needed to be successful in today's world.

Developing spatial skills is an important part of STEAM education , as it enables students to think spatially and manipulate objects with their hands. Research shows that paper folding emerges when children are about 2 years old and becomes more accurate with age. Parents can foster these developmentally important STEAM skills by engaging children in paper folding activities. Paper folding can be done at home or on holidays or birthdays; it is a great way to keep children off media devices and strengthen their creative thinking and problem solving abilities. It also provides opportunities for families to work together, helping them bond and become closer as a unit.

When engaging in paper folding activities, children can learn to recognize and identify shapes and patterns as well as develop problem-solving skills. This can help them understand concepts of geometry, physics, and engineering that are used in everyday life. In addition, it helps them to think more critically and creatively while improving their hand-eye coordination. It's also a great way for parents to encourage creative self-expression in young children by allowing them to explore the limitless possibilities of paper folding art.

Paper folding activities are fun for the whole family. Not only will they foster STEAM skills in your child, but they also provide an opportunity for you all to share quality time together. So why not give it a try? You and your child will be amazed at the possibilities. With a few simple materials, such as paper and scissors, you can enjoy this creative activity while building STEAM skills in your children. So don't hesitate—get folding today.

Maithri Sivaraman, Ph.D. , is a researcher affiliated with Ghent University, Belgium. Tricia Striano Skoler, Ph.D. , is the author of Doing Developmental Research and formerly with Max Planck Institute.

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The Journal of Problem Solving

Home > Libraries > LIBRARIESPUBLISHING > PUPOAJ > JPS > Vol. 8 > Iss. 1 (2015)

Conceptual Transformation and Cognitive Processes in Origami Paper Folding

Thora Tenbrink , Bangor University Follow Holly A. Taylor , Tufts University Follow

Research on problem solving typically does not address tasks that involve following detailed and/or illustrated step-by-step instructions. Such tasks are not seen as cognitively challenging problems to be solved. In this paper, we challenge this assumption by analyzing verbal protocols collected during an Origami folding task. Participants verbalised thoughts well beyond reading or reformulating task instructions, or commenting on actions. In particular, they compared the task status to pictures in the instruction, evaluated the progress so far, referred to previous experience, expressed problems and confusions, and—crucially—added complex thoughts and ideas about the current instructional step. The last two categories highlight the fact that participants conceptualised this spatial task as a problem to be solved, and used creativity to achieve this aim. Procedurally, the verbalisations reflect a typical order of steps: reading—reformulating—reconceptualising—evaluating. During reconceptualisation, the creative range of spatial concepts represented in language highlights the complex mental operations involved when transferring the two-dimensional representation into the real world. We discuss the implications of our findings in terms of problem solving as a multilayered process involving diverse types of cognitive effort, consider parallels to known conceptual challenges involved in interpreting spatial descriptions, and reflect on the benefit of reconceptualisation for cognitive processes.

Recommended Citation

Tenbrink, Thora and Taylor, Holly A. (2015) "Conceptual Transformation and Cognitive Processes in Origami Paper Folding," The Journal of Problem Solving : Vol. 8 : Iss. 1, Article 1. DOI: 10.7771/1932-6246.1154 Available at: https://docs.lib.purdue.edu/jps/vol8/iss1/1

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An interactive game for training reasoning about paper folding

• Published: 18 October 2020
• Volume 80 , pages 6535–6566, ( 2021 )

Cite this article

• Zoe Falomir   ORCID: orcid.org/0000-0002-6398-8488 1 , 2 ,
• Ruben Tarin 1 , 2 ,
• Aurelio Puerta 1 , 2 &
• Pablo Garcia-Segarra 1 , 2  

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Paper folding-and-punching tests are used to measure spatial abilities in humans. This paper presents a qualitative model for paper folding (QPF) and a computer game ( Paper Folding Game ) developed to apply and show the reasoning capabilities of the QPF model. This interactive game presents paper-folding activities intended to help users train and understand how to fold a paper to get a specific shape. Then, it presents paper-folding-and-punching tests to the players. The Paper Folding Game can automatically generate paper-folding-and-punching questions with varying degrees of difficulty depending on the number of folds and holes made, thus producing additional levels for training. The reasoning mechanisms in the QPF model are used by the Paper Folding Game to infer the right answer to each paper-folding-and-punching question. This reasoning capability allows the game to provide feedback to the players when they are wrong and also to create other plausible answers automatically, so that random question-answers are shown to the players in the master-mode. The Paper Folding Game has been implemented using Unity engine and it is available to download from GooglePlay and AppleStore for everyone to train their spatial reasoning skills.

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Paper Folding Game in GooglePlay: https://play.google.com/store/apps/details?id=com.spatialreasoninggames.PaperFolding

Paper Folding Game in AppleStore: https://apps.apple.com/hu/app/paper-folding/id1474625529

Spatial reasoning games: https://spatialreasoninggames.weebly.com/

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An alternative method is applying cloth physics to a paper. However to represent a static paper one should negate the gravity physics involved, thus most of the performance of the physics would not be used, producing a waste in computational resources which would slow the game.

If shapes and holes are not predefined, then a more general approach is needed, like physically modifying the models by creating new vertices and connections, essentially in the same way a 3D editing software would do. However, this is a more computationally expensive solution, not useful in this use case.

This penalty is applied because from 5 answers, any user picking a random answer has a 20% chance of getting the correct answer. If pressing the button was an optimal option then halving the reward would require at least doubling the chance of picking the right answer (so 40% or higher), but that is not the case because the user has 3 remaining answers after pressing the button and then, the probability is roughly 33.3%, which means this is a button for emergencies only, and not an optimal thing to do in every situation.

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Acknowledgments

Zoe Falomir acknowledges the project Cognitive Qualitative Descriptions and Applications Footnote 15 (CogQDA) funded by the University of Bremen through the 04-Independent Projects for Postdocs actions and the Ramon y Cajal fellowship (RYC2019-027177-I / AEI / 10.13039/501100011033) awarded by the Spanish Ministry of Science, Innovation and Universities.

Ruben Tarin, Aurelio Puerta, and Pablo Garcia-Segarra acknowledge their Erasmus+ Scholarships gathered at University Jaume I and provided by the EU and also the support by the Bremen Spatial Cognition Centre.

We also acknowledge all reviewers’ comments which helped to improve this paper.

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Zoe Falomir, Ruben Tarin, Aurelio Puerta & Pablo Garcia-Segarra

Universitat Jaume I, ES Tecnologia i Ciéncies Experimentals, Av. Vicent Sos Baynat s/n, E-12071, Castelló, Spain

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Falomir, Z., Tarin, R., Puerta, A. et al. An interactive game for training reasoning about paper folding. Multimed Tools Appl 80 , 6535–6566 (2021). https://doi.org/10.1007/s11042-020-09830-5

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Received : 26 December 2019

Revised : 31 July 2020

Accepted : 09 September 2020

Published : 18 October 2020

Issue Date : February 2021

DOI : https://doi.org/10.1007/s11042-020-09830-5

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