Magic triangle science is a concept that combines the principles of magic and science in order to explore and understand the mysteries of the universe. It takes inspiration from the idea that magic and science are not separate entities, but rather interconnected forces that can be used together to unlock new possibilities and push the boundaries of human knowledge. In the magic triangle, the three points represent magic, science, and the observer. Magic represents the mystical and supernatural aspects of the universe, while science represents the systematic study and understanding of the world through observation and experimentation. The observer is the bridge between these two realms, bringing the abilities of both magic and science together to explore the unknown. One of the key ideas of magic triangle science is that the observer is an active participant in the creation and transformation of reality.

Math Fun with a Perimeter Magic Triangle

Count your pennies! Learn a fun puzzle to test your quick computation skills—and see if you can find new strategies for getting speedy solutions.

Key Concepts
Mathematics
Addition
Counting
Puzzles

Introduction
Do you ever use math as a tool to solve interesting problems? In the 1970s math was often taught with simple worksheets. One teacher was looking for a way to help his students have more fun with math and logic. So he developed what is now known as the perimeter magic triangle puzzles. Try them out—and have some fun as you start thinking about counting in a whole new way!

Background
Counting is so common that we forget how it is connected to the broader area of mathematics that studies numbers, known as arithmetic. We can see counting as repeatedly adding one: when you add one object to another you have two objects. Add one more and you have three, and so on. Addition is the process of adding numbers. The result of the addition is called the sum. With smaller numbers you might use counting to find the sum. When you have three and want to add two, for example, you can count two numbers beyond three to get to five. With plenty of practice you can often memorize the sums of the numbers one through 10—at which point in can be fun to play with numbers to find all the ways you can make a particular sum.

Math puzzles and games can be a fun way to get practice working with numbers. Puzzles also provide entertaining ways to build strategic and logical thinking. With a little trial and error you can often start to find new strategies to complete a puzzle faster. These are the very same techniques mathematicians use: starting small and trying to find patterns in the sequence of answers. These patterns are then used to predict the answers to even bigger puzzles.

If this is all too abstract, try the puzzle presented in this activity! It might make the process of learning arithmetic clear.

Materials

• Two sheets of 9 by 12-inch paper, such as construction or craft paper (if possible, choose contrasting colors)
• Pencil or marker
• Ruler
• Scissors
• A quarter or other round object of similar size
• 21 pennies, small blocks or other small stackable objects
• More sheets of paper (optional)

Preparation

• Draw a large triangle on a sheet of paper (you can use a ruler to help make straight lines).
• Use a quarter to trace a circle on each corner of the triangle. Now trace a circle onto the middle of each side of the triangle. You should have six circles.
• On the bottom of the second sheet of paper draw six circles similar in size to the ones drawn on the triangle.
• Cut out these circles, and number them 1 through 6. These circles will be referred to as number disks.
• Keep the top part of the second sheet of paper. You will use it to write down your results.

Procedure

• On the paper with the triangle use the 21 pennies to build towers on each circle. Each circle must have at least 1 penny, but no two towers can be of the same height. Can you do it?
• Keep trying until you find a solution!
• Count the number of pennies in each tower. Write down each sum in order from the smallest to the largest number. What do you notice about this set of numbers?
• Shift the towers around or rebuild them until you can fulfill one more requirement: The total number of pennies used to build the three towers on each side of the triangle must be the same. If you build towers of 1, 5 and 3 pennies in the circles lining up on one side of the triangle, for example, you used 1 + 5 + 3 = 9 pennies on that side. Lining up towers of 1, 2 and 4 pennies on the adjacent side would not work because 1+ 2 + 4 = 7 —not 9 like the first side. (Notice the tower of 1 penny was placed on the corner of this triangle, so it contributes to two sides.) If you tried 1, 2 and 6 for the adjacent side instead, that works because 1 + 2 + 6 = 9. Now you can place the one tower that is left and check if 9 pennies are used in the three towers on the third side of this triangle. Try it out! Did you find a solution?
• If this is not a solution, think. Can you rearrange a few towers and get a solution?
• If working with abstract numbers is easier for you, replace the towers with the number disks. Each number disks then represents a tower of pennies. The number written on the number disks informs you of the number of pennies in that tower.
• Using 9 pennies per side is possible! Did you find the solution? Are there several ways you can arrange the towers so there are 9 pennies used per side?
• Can you arrange the pennies so you use 10, 11 or even 12 pennies per side?
• Extra: Show that there are no solutions that use 8 or fewer pennies per side—or show that there are no solutions with a total of 13 or more pennies per side.
• Extra: The puzzle presented in this activity is called a “perimeter magic triangle of order three.” To extend it to a higher-order perimeter magic triangle start by drawing a new triangle. Add circles on the corners like you did the first time, but this time add two more circles on each side in between the corners. For this puzzle you will need nine number disks. Number them 1 through 9. Just like in the previous puzzle you need to find ways to place the disks on the circles so the sums of the numbers on each side of this triangle are identical. Mathematicians call this triangle a triangle of order four as it has four numbers on each side. Once you have solved this puzzle continue with a triangle of order five (add three more circles between the corners and cut 12 number disks), then order six, and so on.
• Extra: Can you create a strategy to find solutions for this type of puzzle quickly?

Observations and Results
Did you find that you can only arrange the 21 pennies in towers of 1, 2, 3, 4, 5 and 6 pennies if you need to make six towers of different heights? Could you come up with ways to arrange the towers so the sum of pennies used on each side of the triangle is identical for all three sides? It is possible for a total of 9, 10, 11, and 12 pennies per side.

To use a total of 9 pennies on each side, you place the towers with 1, 2, and 3 pennies on the corners of the triangle. The tower of 6 pennies goes in between the towers of 1 and 2 pennies because 1 + 2 + 6 = 9. The tower of 5 pennies stands between the tower of 1 and the tower of 3 pennies, as 1 + 3 + 5 also equals 9. The towers with 2, 4 and 3 pennies fill up the third row. Notice how the smallest towers are placed on the corners for this solution.

To arrange the towers so that you use 12 pennies on each side start by arranging the tallest towers (those with 6, 5 and 4 pennies) on the corners of your triangle and fill in the circles in between. Place the smallest tower you have left (1 penny tall) in between the two tallest towers (5 and 6 pennies each). Do you see that the smallest one you are left with (2 pennies tall) goes in between the tallest ones that need a tower in between (the towers with 6 and 4 pennies each)?

A strategy you could use to find the solution that has 10 pennies on each side is listing all the ways you can make 10 by adding three different numbers. You will find 3 + 2 + 5 = 10, 5 + 4 + 1 = 10, and 1 + 6 + 3 = 10. Can you see that 3, 5 and 1 are part of two of these sums? This means these go on the corners of your triangle. You can use the same strategy to find out how to place the pennies so there are 11 or 12 pennies used on each side.

Are you wondering how you can know that using 8 pennies per side is not possible? With 8 pennies per side you use 3 X 8, or 24, pennies for the triangle. Because you reuse the pennies on the corner towers you at most use 1 + 2 + 3 (the sum of the three smallest towers) or 6 pennies fewer. In other words you can use at most 18 pennies. The puzzle asks you to use 21.

This activity brought to you in partnership with Science Buddies

Magic Triangle: A Mathematical Marvel

Mathematics is a branch of science that focuses on numbers, shapes, and relationships. Within the depths of this discipline lie many interesting and cleverly designed concepts and examples. One of them is the magic triangle.

Mathematics is a branch of science that focuses on numbers, shapes, and relationships. Within the depths of this discipline lie many interesting and cleverly designed concepts and examples. One of them is the magic triangle.

A magic triangle is a mathematical geometry term that refers to a triangle with certain properties. These triangles are geometric shapes where numbers or number sequences are arranged in a specific pattern. Magic triangles are popular among mathematicians and math enthusiasts, and they possess many intriguing characteristics.

One of the most famous magic triangles is the Pascal’s triangle, discovered by Blaise Pascal and named after him. Pascal’s triangle has the property that when numbers are placed at each corner of the triangle, the sum of the numbers in the upper row and the lower row is always the same. This triangle plays a significant role in combinatorics and probability theory. Each number in Pascal’s triangle is equal to the sum of the two numbers above it. For example, the numbers at the edges of the triangle are 1, and the numbers inside are the sum of the two numbers above.

Pascal’s triangle presents many interesting examples with mathematical operations and patterns. Each row of the triangle represents combinations and can be used in solving combinatorial problems. The numbers in Pascal’s triangle can also be used to observe various mathematical and statistical relationships.

Another famous magic triangle is the Sierpinski triangle. The Sierpinski triangle was introduced by Polish mathematician Wacław Sierpinski. This triangle is a fractal shape where at each level, the lower triangles are completely empty, and the upper triangle is completely filled. The Sierpinski triangle is an intriguing shape studied in fractal geometry. Fractals are complex objects with intricate structures that repeat themselves in similar ways. The Sierpinski triangle is created by infinitely repeating smaller copies of the triangle.

Magic triangles can also be used for mathematical games, puzzles, and problems. For instance, numbers are placed inside or along the edges of the triangle, and specific mathematical rules are followed. These types of games help enhance mathematical thinking skills and require mental effort to discover relationships between numbers.

The properties and applications of magic triangles hold great importance in mathematical research and problem-solving processes. Mathematicians work with magic triangles to solve problems or make new discoveries in number theory, probability theory, combinatorics, and other mathematical disciplines.

In conclusion, magic triangles are a mathematical marvel, offering intriguing geometric structures that capture the interest of mathematicians and math enthusiasts. Magic triangles like Pascal’s triangle and the Sierpinski triangle encompass numerous applications involving mathematical patterns, relationships, and rules. These triangles can serve as tools to develop mathematical thinking skills, solve problems, and make mathematical discoveries. Magic triangles, as captivating elements of the mathematical world, will continue to contribute to mathematical explorations and understanding.

Magic Triangle: Math Puzzle for Kids

Are you ready to give your kids a fun math puzzle and brain teaser to streeeaaaatch their brains? I'm ready to give you one! Have you ever heard of the magic triangle puzzle? The concept is similar to the magic square, which was part of our diy Camp Mathematics last year. In that case, the numbers were arranged in a grid, but here the kids will be working on the triangle's perimeter.

Both my 7 year old and my 11 year old really enjoyed trying to solve the puzzler. My 11 year old found the smaller triangle to be a breeze but the larger one turned out to be the perfect challenge. My younger son persevered through several errors, but was so proud of himself when he figured it out!

Enough delay. Let's get started!

There are several different kinds of magic triangles and I am sharing 2 of them with you. The great part, is that each puzzle has several solutions so the fun can last and last, and last. and you can choose a puzzle based on your child's age and abilities.

What you need:

• A kid! (one or more will do)
• Our printable! Download the printable here. (Note: by downloading you agree to our terms of service*)

• Make your own with pencil and paper. The number counters I created are not strictly necessary, but it is easier to correct mistakes by moving the counters than by constantly erasing.

Instructions:

Arrange the numbers for each triangle (1-6 for the 3 x 3 x 3 triangle; 1-9 for the 4 x 4 x 4 triangle) so that the sum of numbers on each side is equal to the sum of numbers on every other side.

For the small triangle, arrange the numbers so that the sum of each side equals 9. There are also solutions for 10, 11 and 12.

For the large triangle arrange the numbers so that the sum of each side equals 17. You can also find solutions for 19, 20, 21, and 23.

Watch the video to see the magic triangle in action:

Extensions:

For advanced young mathematicians, do not tell kids what the sum for each magic triangle side will be. Simply give the instruction to arrange the numbers so that the sum on each side is equal to the others. Let them try and figure out that each side should add up to 9, for example.

This puzzle can be done with even larger triangles -- with 5 numbers, or 6 numbers on each side. (You'll have to make your own game board, however!) Can you figure out what the numbers on each side should add up to?

Want more triangle fun? Math Geek Mama has a printable so kids can explore patterns in Pascal's Triangle.

About magic triangles:

I first learned about the magic triangle in my son's book, See Inside Math (affiliate link). Upon further investigation I learned that perimeter magic was first written about by mathematician Terrel Trotter, Jr. It quickly became part of many a recreational math repertoire. You can learn more about the math behind the triangles on this website.

Why do this puzzle?

Solving magic triangles exercises kids' basic addition skills, but also critical thinking, mental math and logical thinking.

More fun math puzzlers:

Solution:

I always strongly encourage you not to look at the solution? Can't solve the magic triangle today? Try again tomorrow. Your kids will learn that it is MUCH more satisfying to solve a problem on their own, than to "cheat." Nevertheless, I always get the question, "where is the solution??" So I'll let you in on a secret: watch the video above and I reveal the solution at the end for all sums.

See more of our favorite math games in action:

*This printable is for personal or educational use only. Commercial use is prohibited. You may print out as many copies as you like. If you wish to share it with others, you must link to this blog post, not the pdf file directly.

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One of the key ideas of magic triangle science is that the observer is an active participant in the creation and transformation of reality. By using both the scientific method and magical practices, the observer has the potential to influence and shape the universe in ways previously thought impossible. Through a combination of scientific inquiry and magical rituals, the observer is able to tap into hidden dimensions of reality and harness their power for the betterment of humanity.

Magic triangle science draws on a variety of disciplines including physics, biology, psychology, and metaphysics. It seeks to integrate these various fields of study in order to create a more holistic understanding of the universe and our place within it. By using the tools and techniques of both magic and science, practitioners of magic triangle science aim to expand our understanding of the world and discover new realms of possibility. Critics of magic triangle science argue that it is a pseudoscience, lacking in empirical evidence and rigorous scientific methodology. They argue that the inclusion of magic in scientific inquiry undermines the credibility and validity of the research. However, proponents of magic triangle science argue that it is a necessary and valuable approach to exploring the mysteries of the universe, and that it has the potential to unlock new scientific discoveries and advancements. In conclusion, magic triangle science is a concept that seeks to combine the principles of magic and science in order to explore and understand the mysteries of the universe. It recognizes the interconnected nature of these two forces and the potential for their combination to unlock new realms of possibility. While it remains a controversial topic, magic triangle science offers a unique and innovative approach to scientific inquiry and the pursuit of knowledge..

Reviews for "A Scientific Approach to the Magic Triangle: Unlocking its Secrets"

- Jane Smith - 2/5 - I was really disappointed with "Magic Triangle Science". I found it to be overly simplistic and lacking in depth. The explanations given were very basic and didn't provide enough information for me to fully understand the concepts. Additionally, I felt like the experiments were too easy and didn't challenge me at all. Overall, I wouldn't recommend "Magic Triangle Science" for anyone looking for a more advanced or in-depth exploration of science.

- John Doe - 1/5 - I have to say that "Magic Triangle Science" is one of the worst science books I have ever read. The content was too vague and didn't provide any real scientific explanations. It felt more like a children's book than a serious science resource. The experiments suggested were also not well-designed and didn't yield any meaningful results. I was really disappointed with this book and would not recommend it to anyone interested in serious science education.

- Sarah Thompson - 2/5 - As someone with a background in science, I found "Magic Triangle Science" to be too basic and lacking in scientific rigor. The explanations were oversimplified to the point where they were borderline inaccurate. The experiments were also too simplistic and didn't challenge me as I had hoped. I understand that this book is targeted towards a younger audience, but I believe that there are better resources out there that can provide a more accurate and engaging science education.

- Michael Johnson - 3/5 - While "Magic Triangle Science" had some interesting concepts and experiments, I felt that it was too rudimentary for my liking. The explanations were clear, but lacked depth and scientific backing. I was hoping for a more comprehensive and well-researched book on science, but this fell short. It may be suitable for beginners or younger audiences, but anyone with a basic understanding of science would find this book to be too simplistic.

Deepening our Understanding of the Magic Triangle through Scientific Research