Blog #113: 333031 and what we can learn from this number

I have written a blog earlier about one of my favourite number 333031 (1729 and 2520 being the other two) and given the current lockdown phase, felt that it is apt that I write a fresh ablog around this which generates lot of positive energy for me and many of my friends.

For those of you who are still wondering why this number is special for me - This is the Zip Code of BITS, Pilani (Rajasthan) which is my Alma-Mater and the town which helped shape up my career

1. For starters 333031 is a Prime number - Another "Prime" reason for choosing this!

2. Sum of all digits of 333031 is 13 which is prime again!

3. Let's see how the "couplets" of this number can be further expressed as a combination of the numbers 1,2,5 - 33 is 2 ^ 5 + 1 ^ 1, 30 is 5 ^ 2 + 5 ^ 1, 31 is 2 ^ 5 - 1 ^ 1

4. Now stepping up gears - 130333, which is 333031 written backwards, is a product of 3 prime numbers - 7, 43, 433. Nothing very special about it till we check out this pattern
- 7 x 43 = 301 and 433 - 10 ^ 2= 333. 333 and anagram of 301 combine together to form the original number 333031!

5. Going back to #4... Look at the factors 7, 43 and 433

a) Prefix 3 to 43 and you get 343 which is a multiple of 7! (7 ^ 3)
b) Suffix 3 to 43 and you get 433 which is a prime!
c) Prefix 3 to 433 and you get 3433 which is a prime!
d) Suffix 3 to 433 and you get 4333 which is a multiple of 7!

So the end result is either prime or multiple of 7

6. Going back to 333031 - 3031 is a subset of this number which is again 433 x 7 - Voila! (These two numbers are a subset of the factors of 130333 in #2)

7. Take a look at the sequence

a) 333031 is a prime
b) 33331 is a prime (without the zero)
c) 3331 is a prime
d) 331 is a prime
e) 31 is a prime!

333031 in unique in many ways and showcases some unique lessons on teamwork  -

a) Versatility

b) Exhibits consistent behaviour in smaller chunks or groups (analogous to smaller scrum teams delivering the required outcome)

c) Each digit on its own and/or when combined together contributes to something unique for the entire ecosystem (similar to how each individual make up a great team),

d) When faced with an adverse situation (Read - reversing the number or breaking down the digits or removing one digit at a time), it still retains its core characteristics (Similar to a project going through a crisis where the team need to scale up or work with distributed teams like we are all doing now and still deliver as per target without losing the core theme)

Blog Post #112: Women's Day 2020 and Investing with long term goals in mind

Investing on financials or gaining more awareness & knowledge through specialization in specific subject, requires patience, commitment and it is an ongoing process. Investing on a continuous basis keeping the long term in mind is always important and this small post is a dedication on Women's day where we could start small like a 20-20 match during the Year 2020 and invest on an ongoing basis to reap the benefits in the long term.

Explained briefly through numbers on how multiplier, compounding effect works in simple terms - The illustrative examples are taken to show the unique number patterns that emerge as we look long term:)

In summary, the examples would show that Addition is like a SIP (Systematic Investment Plan), Multiplication is more like Compounding and Growth while Factorization is equivalent of Diversification

a) 2020 - Start small like a T20 match but don't lose focus on the long term goals of mastering a 5 day Test or Championship!

b) 2020 itself is a result of two perfect squares - 1764 (42 ^ 42) and 256 (16 ^ 16). Power of addition and multiplication

c) Now add 5 to 2020 (Equivalent of planning for the near term) - Resultant is 2025. 2025 is a perfect square (45*45) - Think of it as a multiplier effect in the medium term

d) Think longer and add 500 instead of 5 (100x) to 2020 - Resulting number is 2520 which is one of the unique numbers in Mathematics world. 2520 is the lowest four digit number which has factors from 1 to 10! Power of factorization and compounding in the long run explained. When you are chasing a long term vision, make it unique like 2520!

e) If you closely look at the above pattern 2520 is an anagram of 2025 which was the medium term outcome that we started with!

Happy Women's Day everyone!

Blog Post #111 - Series #2 - Dissect and Analyze with the help of an all too familiar number!

In my earlier post, I had written about breaking down a complex problem into multiple smaller chunks using the example of finding n-th root of a number. In this post, we would look at the following - Given a particular scenario or use case or any problem statement, how important it is to evolve different perspectives and analyze various dimensions of the subject in focus. I would use a well known number in the digital world as an example just to illustrate how one could develop this skill by being patient and observant

65536 is synonymous with computers as 64 KB translates to 65536 Bytes. 65536 has also many factors of 2's multiple - 2 ^ 16, 4 ^ 8, 256 x 256, 16 ^ 4 and so on.

But a lesser known fact of 65536 is what I would focus on now given the theme of the current topic and it opens up a plethora of surprises. Let's dissect and break down 65536 digit by digit or carve out subsets, to reveal some amazing patterns

a) 65 - 13 x 5 (product of two primes)

b) 655 - 131 x 5 (product of two primes)

c) 6553 - Prime (product of two primes)

d) 553 - 7 x 79 (product of two primes)

e) 55 - 11 x 5 (product of two primes)

f) 53 - Prime

g) 653 - Prime

Go one step further and add or subtract combination of digits which are a subset

h) 65 + 536 = 601 - Prime

i) 655 + 36 = 691 - Prime

j) 655- 36 = 619 - Prime

k) 6 + 553 is 559 which in turn is product of two primes 13 x 43

l) 6553 + 6 is 6559 which in turn can be written as 7 x 937

m) 6553 - 6 = 6547 - Prime

n) 553 - 66 = 487 - Prime

Wow... A number which has 2 and its multiples as factors and is outright EVEN, exhibits a completely different behavior, when we dissect and analyze its digits or subset

This approach is pretty helpful in real life scenario, where given a use case or problem definition, it is imperative to break them down into multiple user stories or features and look at commonalities and patterns

Blog Post #110 - The art of doing mental mathematics - Series #1

I was inspired by couple of videos I saw earlier today on kids doing mental mathematics calculations and hence decided to write a blog of slightly different nature and probably a series if this interests more folks. My intention is to remove the fear of Numbers and Maths in general and also make it more fun!

Background

I used to be fascinated by numbers from my junior school days (6th grade to be precise) and used to do mental Additions and Multiplications back then on a regular basis (Thanks to my classmate Ramesh who triggered this out of the blue noticing something that i had back then). After I enrolled into BITS, Pilani for my graduation, I started exploring mental mathematics to do n-th root of a number and logarithms - BITS was the perfect platform to explore as there was constant encouragement from my batch-mates and seniors, to do more

Now you may wonder why in the world someone has to do mental mathematics and calculate n-th root of a number when you can key in the same on a calculator or mobile these days and get the response! Yes there are enough options now and even 15 to 20 years back but performing mental maths really helped me to break-down complex problems and patterns and make you more inquisitive when you see patterns or numbers. Last but not the least, it also help remove the "fear" one has towards Maths as a subject

Enough of the preamble now and let's start with one simple example of how doing mental maths can probably help tickle our brain cells and help sharpen our analytical skills gradually as you experiment further


Example : Find the 31st root of 24 (Yes 24 ^ 1/31)!!

Must be wondering why I have taken an example of this nature.. The intent is not to teach someone how to find the root without using a calculator as it would require practice and time/effort but let's see how to break this down to simpler form and just work out the approach. The path to a solution is more important than the solution itself!

31 is a prime number and so there is no way find the 31st root as is. Hence let's look at the number nearby which has maximum factors

32 is closest to 31 and 31 can be factorized as 2x2x2x2x2

This would mean that we need to find five square roots of 24 in order to arrive at the answer. Now that sounds relatively simple compared to 31st root of a number isn't it?

Let's get into action

Step 1: Square root of 24 - 25 is the perfect square nearby and square root of that is 5. Knowing that, i would go with the law of approximation that square root of 24 would be ~ 4.9 (it doesn't matter if we need to get it very accurate as we still have more steps to go)

Step 2: Square root of 4.9... Not that easy but we know 2^2 is 4 and 2.5 ^ 2 is 6.25 (25x25). Since we have to find square root of 4.9, I would go with a number closer to 2.2 and see where we stand (22x22 is 484 and so 2.2 ^ 2 is 4.84). We are almost there. Let's go with 2.22 or 2.23 as the answer

Now drill down 3 more steps and as you go further this path, we will see that finding square root gets easier

Step 3: Square root of 2.23 is approximately 1.49 (2.25 square root is 1.5 and hence I went with 1.49)

Step 4: Square root of 1.49 is approximately 1.22 (since 144 square root is 12 and 1.44 would be 1.2)

Step 5: Square root of 1.22 is approximately 1.105 (since 121 square root is 11 and 1.21 would be 1.1)

Now what did we observe in Step 3 to 5? Most of us would remember squares upto 20 or 25 since we would have used it in some form or shape while doing Maths at school.. Invariably as you come down the chain while finding square roots, this trick would help us to arrive at the number

There is still one more step to be performed - Remember we started with 32nd root instead of 31 to ease the process and so we need to "adjust" the final answer. This would come by sheer experience but one thing we need to remember is that as n increases, 1/n decreases and approximation rule comes in handy

32nd root of 24 was approximately 1.105 and since I need 31st root, I will round it off to 1.108 or 1.109 (which is the final answer!)

In summary, while this technique may not give you near perfect answer but what you have managed to achieve is to breakdown a complex ask into smaller chunks and got your creative brain cells working.

This is especially important for students appearing in aptitude tests where they are not tested always to give answers upto 3 or 4 digits but given 3 or 4 choices, they need to identify the closest match - This is one area where practicing this comes very handy!


Have fun and try out few examples and I will back with another example in my next Blog post

Blog Post #109: Number in focus - 1221, the palindrome

Starting 2020 with a post on a 4 simple digit number, which happens to be a palindrome - 1221

On the face of it it just like any other palindrome number - Divisible by 11 like any palindrome for e.g

Let's dig a little deeper and unearth some of the unique characteristics that's otherwise not so obvious

1. 1221 is product of 407 and 3 and 407 in turn is the largest 3 digit Armstrong number. Not only that 407 is sum of 343 and 64 - 343 is a palindrome while 64 is a perfect square or a cube. 407 is a multiple of 37 and 11 and we would see more of it later

2. 12 multiplied by 21 gives 252 which is a palindrome again

3. Now let's insert number 3 right in the middle of 12 and 21 which results in the five digit palindrome 12321 - No surprises there as only the middle digit has been introduced. Let's look further though

4. 12321 is a perfect square itself - 111 * 111. 111 like 407 earlier is a multiple of 37!

5. Let's add 123 + 21 (first 3 digits and last 2 digits) and the resulting number is 144 which is a perfect square and 144 written in reverse (441) is a perfect square. More importantly 144 is 12 * 12 (which are the first two digits of 1221) while 441 is 21 * 21 (which are the last two digits of 1221)!

6. Now add 12 + 321 (first 2 digits and last 3 digits) and the resulting number is 333 which is a multiple of 37 again!

7. Now let's look at a subset of 1221 which is 121 - again a perfect square and a palindrome as well!

1221 or 12321 or 121 form a unique pattern and as I mentioned earlier, we need to dig deeper and analyze to unearth some of these invisible patterns

Would be back with more in my subsequent posts!

Blog Post #108 - Year end tribute to Srinivasa Ramanujan!

As we near the end of the 2nd decade of the 21st century, here's a short blog post a week after the birthday of an eminent personality whom I admire the most - Mathematician Srinivasa Ramanujan

Dec 22nd 2019 marked the 132nd birth anniversary of the Mathematical genius and needless to say it is all about numbers when we talk about Ramanujan!

Let's look at few patterns

a) 1887 + 2019 is 3906 which is the product of two consecutive numbers 62 * 63 and in turn their sum (62+63) is a perfect cube (125)

b) 2019 - 1887 is 132 which is the product of two consecutive numbers again! 12 * 11

c) Now add the digits of 1887 and 2019 in pairs - 18 + 87 + 20 + 19 which gives us 144 and a perfect square

d) Now add the individual digits - 1 + 8 + 8 + 7 + 2 + 0 + 1 + 9 which gives us 36 and a perfect square again (36 is one fourth of 144)

Now for the Final Home run which is completely out of the Blue!

Extract the 1st and last digits of both 1887 and 2019 - 17 and 29. 17 and 29 together form 1729 which is the famous Ramanujan Number! Isn't it awesome that 2019, which turns out to be the last year of this decade along with the birth year of Ramanujan produces the magic number that he came up with

That's the sheer beauty of Ramanujan and Numbers - Cannot separate both!

It does not stop there.. Hold on! The remaining two digits are 88 and 01 after extracting the other two and the resulting number is 8801. So what? It is yet another 4 digit number - Yes but like 1729 it is also a multiple of 13! 1729 is 13 and 133 while 8801 is 13 * 677 (Notice two 3s and two 7s within the factors which is unique again!)

Blog Post # 107: The world of 5 digit squares

Let's take a peek into the world of 5 digit squares and you will be surprised on some of the characteristics and patterns that is in play

I have chosen two numbers 59049 and 65536 since they are multiples of the two smallest prime numbers - 2 and 3!

59049 is 3 ^ 10 and 65536 is 2 ^ 16

a) 59049

- 243 ^ 2 or 9 ^ 5
- One of the anagram of 59049 is the number itself
- Last 2 digits is a perfect square
- Digits 4 and 5 are perfect squares as well!
- 9049 (last 4 digits) is a prime number

Now for the fascinating part - Replace the first digit (5) with numbers from 1 to 9 and we would notice that none of them are prime and at the same time product of 2 or 3 prime numbers

- 19049 is 43 * 443
- 29049 is 3 * 23 * 421
- 39049 is 17 * 2297
- 49049 is 7^ 3 * 11 * 13
- 69049 is 29 * 2381
- 79049 is 137 * 577
- 89049 is 3 * 29683
- 99049 is 37 * 2677

b) 65536

- Well know number in computer parlance
- It has so many factors of multiples of 2 - 256 ^ 2 or 16 ^ 4 or 4 ^ 8 or 2 ^ 16
- Its immediate neighbour 65537 is a prime number though!
- Again one of the anagram of 65536 is the number itself
- Last 2 digits again is a perfect square


Contrast to 59049, this is an even square but look at the following combinations

- 655 + 36 is 691 which is a prime
- 65 + 536 is 601 which is again a prime
- 6553 + 6 is 6559 which is a product of two prime numbers 7 and 937


Further let's add 59049 and 65536 - the end result is 124585

- 124585 is product of two prime numbers - 5 and 24917
- 124 + 585 is a prime number
- 12 + 4585 is a prime number

Last but not the least the last two digits of both numbers are both perfect squares (49 ad 36) and their difference is 13 which is a prime number. The sum of the square roots is also 13!

Square root of 59049 is 243 and that of 65536 is 256 - Guess what their difference is also 13!

And we can go on and on!