**-Prakash Pant**

**There are a lot of times when many facts appear in Mathematics as a coincidence. We are often surprised by ****such discovery and always amazed by it. But, it’s just because we don’t have the big picture at that time. When we begin discovering and researching on the topic and once we get the big picture interconnecting the fact, then the coincidence at once becomes a simple truth just as 2+3 = 5.**

**Here, I have come up with some of such coincidences. First, is from a simple compound angle trigonometry. **

**Sin(A+B) sin(A-B) = (sinA + si****nB)(SinA â€“ SinB)**

**It just seems that it is following the distributive property which usually does not happen in any other case of cosine, tangent, or anything. For this, once we discover ****how it happens, it may not be of any surprise at all. **

**Sin(A+B) sin(A-B) = Â˝ [cos ((A+B) – (A-B)) – cos((A+B)+(A-B)) ]**

**= Â˝ (cos 2B â€“ cos2A)**

**= Â˝ (1-2sin ^{2} B â€“ 1 + 2sin^{2}A)**

**= Â˝ (2sin ^{2}A â€“2sin^{2}B)**

**= sin ^{2}A â€“ sin^{2}B**

**= (sin A + sinB)(sinA â€“ SinB)**

**Another such coincidences that teachers often use to amuse students in lower grades (since we discover it in higher grades) is,**

**(1 + 2 + 3 + â€¦. + n) ^{2} = (1^{3} + 2^{3} + 3^{3} + â€¦. + n^{3})**

**It is also quite simple truth if we know the summation formulas.**

**We know:**

**1 + 2 + 3 + â€¦ + n = n(n+1)/2**

**Also,**

**(1 ^{3} + 2^{3} + 3^{3} + â€¦. + n^{3}) = n^{2} ( n+1)^{2} / 4**

** = (n(n+1)/2) ^{2}**

**Thus, we can clearly see how it holds and nothing is surprising any more.**

**Next one of the famous example that almost everyone gets fascinated of is e ^{Ď€i} = -1**

**Once you discover that e ^{iÎ¸} = cos Î¸ + i sin Î¸ and e^{iÎ¸} gives the unit line on the complex plane with head of the line on origin and tilted at the angle given by theta with the x-axis, then theta = pi^c, it will just be in negative x-axis and with unit length. So, of course, the value should be â€“1.**

**Next one of the amazing things for sixth graders on divisibility test was divisibility of 3. I was just quite surprised at that time with the fast that to check if a number is divisible by 3, just check for the sum of the digits if that’s divisible. It was really a wonderful tool for us. But now with tool of congruent modulo, it isnt so electrifying now. In simpler terms, what happens is, if we have: **

** 1000a + 100b + 10c + d, then we can write it as **

** 999a + a + 99b + b + 9c + c + d **

** = 3(333a + 33b + 3c) + a+b+c+d **

**Since the first term is always divisible by 3, we just need to check if the sum, I.e a+b+c+d is divisible by 3. **

**And that’s how it works. **

**Now, just before we end, I want to show you a magic square by Ramanujan. **

**Normally we see 5×5 squares but this is 4×4 and it has even more features than those 5×5. In this, sum of numbers of any row or diagonals is 139; I know you are saying that will be in any magic square but wait, **

**Sum of corner numbers is also 139. Not only that, sum of identical colored boxes is also 139. **

**Sum of the numbers in central square is also 139. **

**If we move a 2×2 square over the main square, the sum of the numbers within the small square is also 139. **

**Can you discover some others too? And, can you think of some ways to create such magic squares as donâ€™t by Ramanujan? So, this is the question to ponder upon. And with this, I end this article. **

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