Unveiling The OBMEP Folding Puzzle: Calculate The Final Area!

Nov 15, 2025 by Admin 62 views

Hey math enthusiasts! Ready to dive into a fun geometry challenge? Today, we're tackling a classic problem from the OBMEP (Brazilian Mathematics Olympics): the folding of a square sheet of paper. This isn't just about finding an answer; it's about sharpening your spatial reasoning and problem-solving skills. So, grab your virtual paper and let's get started!

The Setup: Folding a Square

The Problem: We begin with a square sheet of paper, a perfect 8 cm by 8 cm. This is the stage. Now, we are asked to fold this sheet three times in a specific way, as shown in the original problem statement. The first and second folds are parallel to one of the diagonals of the square, and the third fold is perpendicular to that same diagonal. The question is: What is the area of the final figure after these magical folds?

This OBMEP problem looks simple, but the key is to visualize each fold and understand how the area changes. Don't worry if it sounds a bit confusing at first. That's perfectly normal! We will break down each step so that you don't miss anything. Let's make it a piece of cake. This is an excellent exercise in understanding geometric transformations and spatial visualization, both crucial in mathematics.

Step-by-Step Breakdown

First Fold: Imagine folding the square along a diagonal. This divides the square into two congruent right-angled triangles. The initial fold doesn't change the area, but it sets the stage. If you're physically doing this, you'll see a triangle on top of another.
Second Fold: Make another fold parallel to the diagonal, effectively folding the triangle again, maybe along an internal line. It is similar to the first fold. This creates more triangles, but the key is that these folds maintain the proportional relationships of the sides and angles of the triangles. At this point, we will have folded the shape along the diagonal and a parallel line to the same diagonal, and this means that our new shape will have a shape that looks like a triangle or a kite.
Third Fold: This one is crucial. The third fold is perpendicular to the diagonal. It effectively cuts off a part of the shape, creating the final figure. Now, we need to identify the exact area left after the third fold.

By carefully considering each fold, you can determine how the area changes with each step. The final figure may appear complicated. Still, by carefully considering the proportions, we can simplify this.

Solving the Puzzle: Unveiling the Area

Now, the fun part! Let's unravel how to calculate the area of the final figure, the OBMEP's goal. To solve this problem effectively, you have to follow a structured approach. Without getting lost in unnecessary complications, let's keep it simple. Remember, each fold transforms the original square into a series of smaller shapes, and we're just calculating the area of the final one.

The Geometric Approach

First, determine the nature of the final figure: It is important to know that the folds will create similar triangles and quadrilaterals. By visualizing these shapes, you can predict what the final one looks like. Since the folds are strategic, it will result in a shape with sides and angles related to the original square.
Analyze each Fold: Remember that the first and second folds don't directly change the area. However, they create divisions that are essential for the final fold. If you have the shape in front of you, you will observe the various forms created. Also, it is easy to see that these folds create several triangles with specific angles, so they could be used to facilitate calculations.
Calculate the Areas: The third fold removes a part of the shape. To calculate the final area, you can identify the shape removed by the third fold and subtract it from the area of the shape after the second fold. Or, you can calculate the individual area of the final shape by adding the different pieces.
Mathematical Relationships: Since the folds are connected to the diagonals of the squares and triangles, we have the use of specific mathematical relationships, such as the Pythagorean theorem and trigonometric ratios. These will give us the base and height dimensions necessary to calculate the areas of the various forms.

The Solution: The correct answer will be a fraction of the original square's area. After completing the folds, the final area will be 1/8 of the original square. Since the original square has an area of 64 cm² (8cm x 8cm), the area of the final figure is 8 cm².

Mastering the Concepts: Why This Matters

This OBMEP problem teaches more than just calculating the area; it teaches the ability to think mathematically, which is important for your future. The skills you get solving this problem can be useful for other competitions or even daily activities.

Why Practice with OBMEP Problems?

Improve Problem-Solving: OBMEP problems are designed to challenge your way of thinking and enhance your ability to devise problem-solving strategies. The exercise of this type of problem improves analytical thinking.
Enhance Spatial Reasoning: By visualizing each fold, you boost your spatial reasoning. The ability to manipulate shapes in your mind is an invaluable skill in mathematics and other areas.
Boost Your Confidence: Successfully solving an OBMEP problem can significantly boost your confidence in your math abilities. This increase in confidence can have a positive impact on your learning experience.

Conclusion: Your Next Steps

So, there you have it! The solution to the folding square OBMEP problem. Remember, the beauty of mathematics lies in the journey of understanding, and each problem is an opportunity to learn and grow. Keep practicing, keep exploring, and most importantly, keep having fun with math!

Feel free to revisit the problem, try different variations, and explore the related topics. With each step, you'll be one step closer to mastering the art of problem-solving. Keep the math spirit alive, and good luck! Also, be sure to keep an eye out for more math challenges.