Blog #148: Something about 153 and Year 2025!

My earlier post a week ago, was around welcoming 2025 and when I started thinking about what should be the next blog about, I did not have to spend lot of time - Enter the ever versatile and amazing Armstrong number 153!

153 is the first and smallest Armstrong number and it is special in many ways and I was sure that somewhere there is a link between 153 and the perfect square year of this millenium - 2025

Let's see how special 153 is before establishing the link with 2025

a) For starters 153 is an Armstrong number and hence sum of the cubes of its digits equal the number itself

b) 153 is the sum of two squares 12 ^ 2 and 3 ^ 2 as well as difference of two squares 13 ^ 2 and 4 ^ 2!

c) 153 can be written as (9 ^ 2) + 72 as well as (15 ^ 2) - 72 - WOW!

Now let's look at what happens when 153 is flipped over - we get 351 which is not an Armstrong number but watch these similarities between 153 and 351

d) 351 can be written as  18 ^ 2 + 3 ^ 3 as well as 20 ^ 2 - 7 ^ 2

e) Anagrams of 153 are 135 and 315 as well and guess what.. 135 is difference of two squares 12 ^ 2 - 3 ^ 2 and 315 is difference of two squares as well 18 ^ 2 - 3 ^ 2!!

Now finally...

f) 153 + 351 gives 504 which is the lowest 3 digit number that has factors from 1 to 9 except 5 

When we multiply 504 with 5 we get the magic number 2520, which is the smallest 4 digit number with factors from 1 to 10. 

One of the anagram of 2520 is 2025 - Our perfect square year!

Welcome 2025 again!

Blog 147: Welcoming 2025!

Received a short note about some interesting facts around the number 2025 through social media channels earlier today and that kindled my spirits to write my next Maths blog welcoming the year 2025 a bit in advance - a number that is pretty unique and versatile and something which we could all relate to when we work as a Team in the corporate world!

 

To begin with here are some straight-forward ones

a) 2025 being a perfect square

b) Product of two squares (81 and 25)

 

I mentioned earlier that 2025 is unique and versatile and has its own standing among 4 digit squares... The square ends with 25 which lends more flavor and diversity when it comes to interesting patterns. It exhibits characteristics that we all look forward to, when we form teams to execute transformational and strategic initiatives

Let's take a look

1. 20 + 25 = 45 and square of 45 is 2025 - A team is made up of different set of individuals who bring together various skills to achieve a common goal!

2. 2025 is square of sum of all digits from 1 to 9 as well as sum of cubes of digits from 1 to 9!!!! One of the anagrams of 2025 is 2520 that shows a similar pattern - it is the lowest 4 digit number that has factors from 1 to 10.

 

3. Add 1000 or 10 ^ 3 to 2025 and we get 3025 which is also a perfect square ending with 25..

Talk about adapting to different situations with seamless ease, wearing multiple hats!

 

4. Remove 0 from 2025 and you still get a perfect square ending with 25 - 225! Deliver similar output with lesser resources...

 

5. Extending the point above - Remove 0 from 2025 and add 64 which is a perfect square to 225. The resultant is 289 which is again a perfect square. Now let's get back to 2025 and add 784 (which is a perfect square) and the resulting number is 2809 - yes it is a perfect square again!

Notice the transition - we removed 0 from 2025 to begin with and finally got 2809 (0 back in the number 289) - A classic case of making different strategic moves throughout the execution phase to arrive at the final destination!

 

6. Prefix a perfect square (4) to 2025 and guess what - We get 42025 which is a perfect square again - Continue to upskill as we move forward in the journey to deliver value!

 

7. 2025 is the only perfect square during the 21st century just like some of its predecessors like 1936, 1849 as well as its successors like 2116, 2209 but what makes it also interesting is it joins the club of these "unique years" whose last 2 digits is a perfect square starting with 1600

 

1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3600...

 

Clearly 2025 is versatile, unique and finds its place among a special set of numbers which demonstrate standout behavioral patterns, diverse characteristics and seamlessly adapts itself, as it goes through the transformation journey, while retaining its core attribute all the time of being a "perfect square" year !

 

Wishing everyone a very Happy and Prosperous 2025!

#MathsForFun #Welcome2025

Blog #146: It is all about Maatram

This blog post is dedicated to every Maatram student, Alumni and the core team led by Sujith Kumar. Here we go...

At Maatram Foundation, we had the Alumni annual leadership camp last weekend. It was a great opportunity to get together and connect with an amazing and enthusiastic group of individuals, who are always on the lookout to sharpen their skills and learn something new everyday.

MAATRAM stands for Transformation and let's see how the numbers and digits go through a Transformation below

Each letter of the word MAATRAM when mapped to numbers add upto 67 - 13+1+1+20+18+1+13.

A) 67 is a prime number to begin with and the digits add upto 13 which is again a prime

B) MAATRAM starts with M and ends with M which maps to the number 13 again!

C) MA at the beginning and AM at the end act as a mirror image and flank the three letters in the middle - ATR. A,T, R adds upto 39 which is a multiple of two prime numbers 13 and 3!

D) Prefix numbers 1 to 9 to 67 and this is the pattern we see

    1) 167 - A prime again

    2) 267 - Product of two primes (89 and 3)

    3) 367 - A prime

    4) 467 - A prime

    5) 567 - Product of a perfect square 81 and a prime (7)

    6) 667 - Product of two primes (29 and 23)

    7) 767 - Product of two primes (59 and 13)

    8) 867 - Product of a perfect square 289 and a prime (3)

    9) 967 - A prime!!

The pattern demonstrated above is relatively unique comprising of only primes and squares!

E) And finally MAATRAM means TRANSFORMATION and when Transformation is mapped to numbers and we add those digits (20+18+1+14+19+6+15+18+13+1+20+9+15+14), the resultant is 183 which is again a multiple of two primes! 61 and 3. Not only that, for Indian cricket lovers the number 183 is very special as it was the winning score in the 1983 World Cup victory :)

Clearly the numbers tell the narrative about what MAATRAM stands for. "Prime" reason why our Foundation is very different and strive to bring in a positive transformation in the lives of many students. Extremely glad to be associated with Maatram for more than a decade now, giving back to the student community in the field of education, a noble cause that is very near and dear to me. Looking forward to the next 10 as our Maatram Alumni team gets stronger year by year led by many amazing professionals!

Sujith Kumar, Punitha Anthony, ADITYA SETHURAMAN, Naren Sundram, Rajagopalan R, Vinu Viswanathan, Saravana Subramanian SAFe® SPC, Princely Samuel, Vinoth Kumar, Sarath R

#MaatramFoundation #EducationForAll #Transformation

Blog #145: Welcoming 2024

As we step into a brand New Year, here's a short Mathematical note welcoming 2024

1. 2024 is a difference of two perfect squares - 2025 and 1 and #2 and #3 below illustrates that through alternate means

2. Add digits 1 to 9 and we get 45. (45 + 1) * (45 - 1) is 2024

3. Add the cubes of the digits from 1 to 9 and the resultant is 2025  (45 ^ 2). Subtract 1 ^ 2 as in #1 and we get 2024 again!

4. 2024 is also sum of 1936 + 88. In turn 1936 is nothing but a perfect square of 44. 

Hence 2024 can be written as (44 ^ 2) + (44 * 2). Look at how 2 and 4 which are the primary digits that make up 2024 come into play again! Add 20 and 24 and the resultant is 44 again!

5. Last but not the least..2024 x 2024 = 4096,576 where 4096 is a perfect square and 576 is also a perfect square (24 in action again!)

Wishing everyone a very Happy, Healthy and Prosperous 2024!

Blog #144: Tribute to the Maths Genius on his 136th birth anniversary

Today is 136th birth anniversary of Dr. Srinivasa Ramanujan, the Man who knew Infinity!

Here's a small Mathematical tribute to someone who loved Numbers and patterns and weaved magic and also why Year 2023 was special in that context

1. 1887 was his birth year and a multiple of three prime numbers 17, 37, 3

2. Coincidentally Year 2023 happens to be a multiple of 17 as well- 17, 17, 7 (all prime again)

3. 17 is the common thread across 1887 and 2023 and it is also the first 2 digits of Ramanujan's famous magic number 1729!

And for the grand finale!

Here's what makes both 2023 and Ramanujan's 136th birth anniversary year extra special. Look at the pattern below for 2023..

a)  2+0+2+3 = 7 is a prime

b) 2 ^ 2 + 0 + 2 ^ 2 + 3 ^ 2 = 17. Prime again!

c) 2 ^ 3 + 0 + 2 ^ 3 + 3 ^ 3 = 43. One more prime!

d) 2^4 + 0 + 2 ^4 + 3 ^4 yields 113 which is again a prime!!!

e) 2 ^ 5 + 0 + 2 ^ 5 + 3 ^ 5 = 307. Prime galore!

f) 2 ^ 6 + 0 + 2 ^ 6 + 3 ^ 6 = 857. Yes you guessed it right. Prime again 

g) 2 ^ 7 + 0 + 2 ^ 7 + 3 ^ 7 = 2443. Finally..We get a number which is a product of two primes! The streak is broken but not before displaying an amazing pattern

This is why numbers are so special and on that note, this is me signing off for 2023 with this humble tribute to the Man who achieved a lot in the field of Mathematics in a short span of time and continues to make a profound impact even today

#SrinivasaRamanujan #MathematicalGenius

Blog Post #143: Case of Amicable Squares!

In my last blog post, I had written about "Amicable numbers" which has been well documented and researched

Last week, we met for a Team lunch and when I received the bill, I noticed that it was a perfect square! (1764). I have written about 1764 in one of my earlier blog almost 5 to 6 years ago but given the recent topic on amicable numbers, I was wondering if some of the squares like 1764 have something in common and started exploring that further

Guess what - Numbers don't disappoint us any time and I did get to see a peer/counterpart for 1764 which had similar attributes and patterns if not a 100% match

Here we go and look at some interesting points about 1764 and 2916 which i would like to call as "Amicable Squares" for the purpose of this blog

1. To begin with 1764 and 2916 are perfect squares of 42 and 54

2. Both squares are formed by a combination of prime number and another perfect square - 17 and 64, 29 and 16

3. One of the anagram of 1764 is 4761 and similarly 9216 is an anagram of 2916 - Both turn out to be perfect squares again (of 69 and 96) and on top of that, as we could see 69 and 96 are reverse of each other!

4. Another anagram of 1764 turns out to be the famous "Kaprekar constant" - 6714 (More about it could be found through this link (https://en.wikipedia.org/wiki/6174_(number)#:~:text=6174%20is%20known%20as%20Kaprekar's,Kaprekar.)

5. On similar lines, 9261 is an anagram of 2916 and is a perfect cube while 1296 which is another anagram, is a perfect square of 36 which in turn is a perfect square of 6!

6. Replace 64 with 29 to get 1729 which is Ramanujan number again! Cannot leave this magic number out of any number game :)

7. Add the digits of 1764 and 2916 and we would get 18 in both cases!

1764 and 2916 are a great example of perfect squares which demonstrate similar behavior and it would be great fun to see if there are similar square pairs (formed by a combination of prime number and a square)

#NumbersAreFun #PerfectSquares

Blog Post #142: Numbers which complement each other as a friend

Thanks to the recent post by Dilip and my friend Sukumar for tagging me on that, which acted as the trigger for this next blog of mine. The numbers in limelight are 220 and 284 are part of what Mathematicians call as “Amicable Numbers” and these two are the smallest of that series

https://en.m.wikipedia.org/wiki/Amicable_numbers

Given the friendly nature of these amicable numbers, I was wondering if there are more attributes or qualities that these two numbers stand for and started exploring further this weekend. Here are few of my observations and findings

1. Add 220 and 284 and the resultant is 504 which is the smallest three digit number with factors from 1 to 9 except 5 ofcourse. When multiplied by 5 the resulting number is 2520 which is a special number - the lowest whole number which has factors from 1 to 10!

2. Subtract 220 from 284 and we get a perfect square

3. Both 220 and 284 are resultant of a perfect square minus 5. 220 is 225 - 5 while 284 is 289 - 5

4. Swap the last digit of these two numbers to form 224 and 280 which are successive multiples of 56! Truly amicable and friendly even when they swap a digit!

5. Let’s reverse both numbers and we get 022 and 482. Look at these closely and we realise that both are multiples of a prime number with 2! 11 x 2 and 241 x 2 !!

6. Suffix 9 to 220 and we get 2209 which is a perfect square of 47. So what’s big in that one may wonder…Replace the last digit 4 in 284 with 09 and we get 2809 which is a perfect square of 53. Guess what 53 and 47 in turn adds up to 100!!!

7. I am always fascinated by 1729 (Ramanujan number) and strongly believe 220 and 284 should have some relation with 1729. Guess what… 220 + 284 is 504 as we saw earlier and 504 + a perfect square (1225) yields 1729. Voila! That’s the brilliance of Ramanujan number

No wonder 220 and 284 are part of the Amicable or Friendly number list given they share so many common attributes as seen above. Draw a parallel to real life and/or corporate world, these are the qualities that we would look up for amongst friends or colleagues when they work together as a team - flexibility, fungibility, work towards a common goal, adapt to any ambiguous situation and produce results.

Coincidentally this is my 142nd blog which happens to be half of 284 :)