Blog Post #115: The curious case of Zip codes!

This blog is dedicated to everyone at Mr Cooper and some fun using Maths to show the close collaboration across US and India offices!

US Corporate office Zip code at Coppell, TX is 75019

India Corporate office Zip code at Chennai, TN is 600089

Let's look at some striking similarities

a) Digits of 75019 adds up to 22 while that of 600089 adds upto 23 - These are like thick pals next door!

b) Both 75019 and 600089 are not prime numbers and more importantly number 7 is a common factor for both!

75019 - 1531 * 7 * 7
600089 - 1453 * 59 * 7

c) Now closely look at the first factor for both numbers in the earlier step - 1531 and 1453. Difference is 78 which is a multiple of 13. Now look at the second factor - 7 and 59 - Difference is 52 which is again a multiple of 13! That's incredible :)

d) 750 + 19 (first 3 digits + last 2 digits) is 769 which is a prime
     6000 + 89 (first 4 digits + last 2 digits) is 6089 which is also a prime!

e) 750 - 19 is 731 which is a product of two primes (17 * 43)
    6000 - 89 is 5911 which is again a product of two primes (23 * 257)

The similarities exhibited by these two numbers are amazing to see and truly reflects the collaboration between the teams at these locations!

I am sure if I extend this to some of our other sites in US and India, we will see more such patterns

Blog Post #114: All roads lead to Armstrong number!

This post is around working towards a common team goal and the path that we take to achieve that - have made an attempt to explain that through Armstrong numbers

Let's start this blog listing out the Armstrong numbers that most of us know about - 0, 1, 153, 370, 371 and 407 (Sum of the cubes of the digits equal to the number)

Now let's take a random number say 504 and add up the cubes of the digits till we get to see what happens after few steps!

Step 1:            5 ^ 3 + 0 ^ 3 + 4^ 3 = 189

Step 2:            1 ^ 3 + 8 ^ 3 + 9 ^ 3 = 1242

Step 3:            1 ^ 3 + 2 ^ 3 + 4 ^ 3 + 2 ^ 3 = 81

Step 4:            8 ^ 3 + 1 ^ 3 = 513

Step 5:            5 ^ 3 + 1 ^ 3 + 3 ^ 3 = 153!!

153 is an Armstrong number and the iteration stops here!!! That's sheer magic - Isn't it?

Let's take another number 841

Step 1:            8 ^ 3 + 4 ^ 3 + 1 ^ 3 = 577

Step 2:            5 ^ 3 + 7 ^ 3 + 7 ^ 3 = 811

Step 3:            8 ^ 3 + 1 ^ 3 + 1 ^ 3 = 514

Step 4:            5 ^ 3 + 1 ^ 3 + 4 ^ 3 = 190

Step 5:            1 ^ 3 + 9 ^ 3 + 0 ^ 3 = 730

Step 6:            7 ^ 3 + 3 ^ 3 + 0 ^ 3 = 370

370 is an Armstrong number and iteration stops here!!

Moral of the story

When we work as a team to deliver an outcome, we all strive towards a common goal (which is the Armstrong number in this case) though the paths we take may sometime differ (as we saw in the two examples above)

Not all numbers behave this way but it is an interesting exercise to see which ones follow this pattern!

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!