By CHANDAN GHUGHTYAL
Mathematics has always been a subject of fascination, and some may even say that it possesses an almost magical quality. We often ask, “Is math magic or is it merelya product of logical reasoning?” Today, I would like to explore this question with you anddemonstrate how math can indeed seem magical, but at its core, it is firmly rooted in logic.
Mathematics is the language of the universe. It is the tool we use to describe and understandthe world around us. From the symmetrical beauty of a snowflake to the complex algorithmsthat power modern technology, mathematics plays an integral role. It is like a magic batonthat unlocks the secrets of the universe. But where does the magic lie? This is howmathematics can describe the seemingly inexplicable and predict the unpredictable.
Consider, for instance, the Fibonacci sequence. The numbers in this sequence are simpleadditions of the two predecessors, yet they appear in countless aspects of nature, from thearrangement of leaves on a stem to the spirals in a seashell. The magic lies in how mathreveals these hidden patterns.
Another example is the “Golden Ratio”, represented by the number phi (F). It appears in art,architecture, and even in the proportions of the human body. The magic lies in how a singlenumber can hold such aesthetic significance. The ratio of the distance from the head to thetoe and navel to the toe, or the ratio of the distance from the forehead to the chin andforehead to the tip of the nose, is often referred to as the proportions or ratios found in thehuman body. These proportions have been observed and appreciated for their aestheticqualities. This ratio is called Phi (F), which is approximately equal to 1.6180339887.
Now, let us talk about logic. Mathematics is built on the foundation of pure logic. Each theorem, proof and equation are derived from a set of logical rules. If we draw any triangle,the sum of all interior angles is 180 degrees, which is purely logical. The derivative andintegration of x have the same function. The value of sin(x+2k) is the same for any integral valueof k. Math logic is a consistent, precise, and unfailingly accurate system. There is no guessworkand no ambiguity. This logical foundation is the key to the obvious magic of math.
So, is math magical or logical? It’s both! The magic of mathematics lies in its ability to take thecomplex and make it understandable, to unveil hidden patterns in nature, to model a naturalphenomenon, and to predict something based on a conjecture or model. This magic isgrounded in rigorous logic and reasoning; it is not created out of thin air.
The association of Pascal’s Triangle and Pingala’s work with mathematical concepts canindeed seem magical, as they reveal fascinating patterns and relationships withinmathematics. The numbers in Pascal’s triangle are used in different branches of mathematicsand can be elegantly applied to solve challenging issues. The triangle is a useful tool in algebrabecause it also shows the coefficients of binomial expansions. Pingala was an Indian scholarfrom antiquity who made contributions to mathematics. His work, which was completed inthe third century BCE, involved the creation of the first algorithms for determining thenumber of different rhythms in a piece of music. He represented rhythmic patterns using abinary system, which was one of the first instances of binary coding.
To illustrate this, let us engage in activities that showcase the magical and logical aspects ofmathematics. We can explore mind-boggling mathematical puzzles, discover the wonders offractals, and witness the beauty of primenumbers.
One activity is to determine whether it is magic or if there is logic. Write any number; let it be12345. Now, without writing any other number, I will tell you the sum of this number and thenumber you will tell me after I write the sum. The answer will be 112344. Just see whether itis right or wrong. Tell the other number now; let’s say it is 65432. You have said two numbers;now, it is my chance to write a number; it is my number, 34567. Now add all three numbers—two of yours and one of mine. Check! The sum is what I wrote without the other two numbers.
Is it magic? How has this happened? Some magic is there. Think and tell me. Let me tell you,there is no magic; there is clear logic. When you said 12345, I immediately wrote 112344, inwhich the unit digit is 1 less than the unit digit of the first number, i.e., 5-1 = 4. I wrote all theother digits as they were and then wrote one in the front. The third number was mine, whichwas obtained by subtracting each digit of the second number from 9. Hence, the logic is thatthe first number plus 99999 is the sum. Mathematics is full of these kinds of patterns andlogical operations, which can often appear magical when you first encounter them. However,as you have demonstrated, with careful analysis, we can unveil the logic behind the obviousmagic.
The second activity that I always thought was magic until middle school reveals thebirthday of a person after following a sequence of questions. The first question is how manytimes do you eat chocolate in a day? Multiply the number by 2 and add 5 to the product. Nowmultiply by 50. Add the current year (i.e., 2023). Subtract 250 and then subtract the year ofyour birth from it. The two right-hand numbers in your answer are your age and the left-handdigits represent the number of times you eat chocolates a day. I never analysed and thoughtthat the person who is asking is a magician. Later, when I developed an analytical mind, Irevealed the magic through algebra and understood that it is merely logical not magical.
Here is a breakdown of the steps:1. “How many times do you eat chocolate in a day?” Let us call this number X. 2. Multiply X by 2: 2X. 3. Add 5 to the product: 2X + 5. 4. Multiply the result by 50:50 (2X + 5). 5. Add the current year (2023): 50(2X + 5) + 2023. 6. Subtract 250: 50(2X + 5) + 2023 – 250 = 50(2X + 5) + 1773. 7. Finally, subtract the year of your birth from it: 50(2X + 5) + 1773 – (Year of Birth). Now, if you look closely, the left-hand digits represent the number of times you eat chocolatesa day (X), and the right-hand digits represent your age, given the current year of 2023.
Therefore, it is an algebraic manipulation, not magic. In the era of Education 4.0, which emphasises technology, innovation, and datadrivenapproaches, mathematical thinking is more important than ever. Mathematics serves as thefoundation for many advancements anddevelopments in various fields.
In conclusion, math is a perfect blend of magic and logic. It offers us the tools to explore thewonders of the universe, uncover hidden patterns, and solve real-world problems. Asstudents of science and technology, integrating math into their studies will unlock newdimensions of understanding and innovation. Embrace the magic of mathematics, but neverforget that behind every magical result is a logical and systematic process that makes it allpossible.
(Chandan Ghughtyal teaches Mathematics at The Doon School)
