Blog Post 77: Ramanujan's number and BITS 91 batch reunion

22nd December was the birth anniversary of the Mathematical genius Ramanujan and one of his well known all time favourite number is 1729 which is a product of 19 and 91... 

Coincidentally, four days after the birthday of the brilliant mathematician, on 26th Dec, the two magical numbers 19 and 91 once again come together in the form of 1991 batch of Pilani to meet at the "Prime" location - 333031! 

BITS 1991 batch will always remain special and happy to be part of Sir Ramanujan's history as well!

Blog Post #76 Speciality of 1991

Chose this number since this is the entry year into my Alma Mater and we are celebrating our 25th anniversary this year :)

1991 is unique in few ways

a) It is a palindrome (stating the obvious :))
b) 1991 = 181 x 11 and 181 is in turn is a palindrome and 11 as well!!!
c) Observe the pattern of the digits that make up these numbers

1991 = 1^2, 3 ^ 2, 3^2, 1^2
181 = 1 ^ 2, 2 ^ 3, 1 ^ 2
11 = 1 ^ 2, 1 ^ 1

d) 19 x 91 = 1729 which is the magical Ramanujam number and that itself is a elite group to be in

e) Last but not the least, 1991 - 1729 = 262 which is again a palindrome!

As you could see 1991 is unique in so many ways and no wonder the entire batch rocks!

Blog Post #75: Anagram Square Pairs

Observe this example

196 x 196 = 38416, 196 in turn is 49 x 4 which 7 ^ 2 x 2 ^ 2

One of the anagram of 38416 is 16384 which in turn is 128 x 128

Are there other similar square anagram pairs of this nature where the original number (in this case 196) can be further broken down into smaller squares?

There is no right or wrong answer but looking for patterns and observations if any

Blog Post #73 - Identify the approach to this day to day puzzle!

This week's blog came out of an offline discussion with my school friend A Ramananda Pai on a simple Maths problem... Thought of documenting that as a simple blog here primarily to understand how we along with our kids approach the solution and also to drive home the message that whatever we study, always has an application somewhere or the other :)
All along we always wonder why we study certain topics when they may not be of any use and this is a good simple example of how we can put something that we studies during school to real use!

Two people A and B go around a circle and cover a distance of 300 metres during every revolution. Both of them start at the same point and A takes 4 mins to complete 1 round while B takes 7 mins to complete the same round. After how many rounds, will they pass each other again (assume they both clock the same average speed throughout this exercise)

More than the actual answer, remember I am looking for the approach to the solution

Blog Post 72: Breaking down a problem into a set of patterns

I came across this simple problem/riddle floating around in WhatsApp/Facebook and most of the folks got the answer after few iterations or trial and error method...

For a change instead of a puzzle from my end, I thought I will have a a Blog Post taking this example to get different perspectives of the approach taken by individuals

May sound simple for most of you but I personally feel that this exercise is very important for children and it would be good to see how they come up with the solution given how inquisitive they are. That would also help them in the long run to become more analytical/logical rather than follow a formulae/theorem based approach

Q1

       a b c
    + a b c
    + a b c
    ----------
       c c c
   -----------

Q2

        a b c
     + a b c
     + a b c
     ----------
        b b b
     ----------

Both questions are similar and probably Q1 is easier compared to Q2 but did we use the same approach or a variation?

Go ahead and share your views

Blog Post 71: Number 343 in the spotlight!

343 is well known for its following characteristics as I had published in the earlier post

343 - 7 * 7 * 7
343 can be expressed as (3 + 4) ^ 3 by inserting few symbols between the digits
343 is one the few cubic or square palindrome

Taking this further let's look at the following series

Series 1: Insert 0 to 9 as the second digit (for eg. 3043, 3143, etc) and write down their factors

Series 2 : Prefix 0 to 9 before 343 and write down their factors

Are there any observations we could make or is it exhibited by most of the numbers?

Series 1                                                               Series 2

3043 – 17, 179                                                    0343 – 7,49

3143 – 7, 449                                                      1343 – 17, 79

               

3243 – 3, 23, 47                                                  2343 – 3, 11, 71

3343 – Prime                                                      3343 - Prime

3443 – 11, 313                                                    4343 – 43 and 101

3543 – 3, 1181                                                    5343 – 3, 13, 137

3643 – Prime                                                      6343 - Prime

3743 – 19, 197                                                    7343 – 7, 1049

3843 – 9, 61, 7                                                    8343 – 9, 9, 103

3943 – Prime number                                         9343 - Prime

Blog Post #70 - Sum of successive squares - what do we observe?

Take a look at the series below - first 20 set of sum of successive squares

1 ^ 2 + 2 ^ 2 = 5
2 ^ 2 + 3 ^ 2 = 13
3 ^ 2 + 4 ^ 2 = 25
4 ^ 2 + 5 ^ 2 = 41
5 ^ 2 + 6 ^ 2 = 61
6 ^ 2 + 7 ^ 2 = 85
7 ^ 2 + 8 ^ 2 = 113
8 ^ 2 + 9 ^ 2 = 145
9 ^ 2 + 10 ^ 2 = 181
10 ^ 2 + 11 ^ 2 = 221
11 ^ 2 + 12 ^ 2 = 265
12 ^ 2 + 13 ^ 2 = 313
13 ^ 2 + 14 ^ 2 = 365
14 ^ 2 + 15 ^ 2 = 421
15 ^ 2 + 16 ^ 2 = 481
16 ^ 2 + 17 ^ 2 = 545
17 ^ 2 + 18 ^ 2 = 613
18 ^ 2 + 19 ^ 2 = 685
19 ^ 2 + 20 ^ 2 = 761
20 ^ 2 + 21 ^ 2 = 841

What all do we observe and are there specific inferences?

Blog Post #69 - Is this pattern a unique behaviour or just a coincidence?

Consider these numbers

131 - Prime number, Its anagrams 113 and 311 are prime as well
919 - Prime number, Its anagrams 199 and 991 are prime as well
373 - Prime number, Its anagrams 733 and 337 are prime as well

Is there a pattern or does it break somewhere? Does it also extend to 4 digit numbers?

Even if it breaks somewhere is there something that is common to all these numbers?

Blog Post #68 - What's unique about this series

What are the unique attributes about this series

576, 676, 5476, 5776, 15376, 15876

List all possible observations and is there any other series similar to this

Blog Post #67 - Find the 3 digit number

1. Let's call the number as XYZ
2. XYZ + a ^ 2 = Perfect square while XYZ - a^2 is a Prime number
3. XYZ + b ^ 2 = Perfect square while XYZ - b ^2 is again a Prime number
4. Suffix a digit W and XYZW is a perfect square again
5. XYZ itself is unique in some sense 

Blog Post #66 - What's the next series and why?

Looking at the series below, find the missing numbers

64, 81, 125, 144, 216, 289, ?, ?

Anything else unique about the result?

Blog Post # 65 - What is unique about these sequence of numbers?

I just happened to notice this pattern when I saw two cars with registration numbers 2809 and 1521 earlier today. Needless to say both number caught my attention since they were perfect squares and then tried to see if few more numbers exhibit that pattern

Here's the sequence I am looking at

1521, 2025, 2809, 4225, 4900...

It is obvious that they are perfect squares but there is something unique they exhibit apart from that in a sequence

What is the pattern they exhibit and also give me the next 2 or 3 four digit numbers in that sequence?

Blog #64 - Dedicated to Viswanathan Anand, the Chess Grandmaster!

Blog # 64 dedicated to the evergreen GOD of Chess - Viswanathan Anand!

New planet named after him with the number 4538 and the number does seems to be pretty special!

4538 - Square (53) + 1729 (Maths Genius Ramanujam Number!)
4538 - Square (67) + Square (7)
4538 - Square (64) + Square (21) + Square (1)

Anand has been the "King of 64 squares" in the Chess world and truly the number above does reflect that as well

And coincidentally this is also my 64th Blog!

Blog #63 - Find this number

Back with another question around a 4 digit number (ABCD) - again the digits can repeat and not necessarily distinct

1. AB + CD results in a number which is part of an unique number series
2. This number satisfies the condition X ^ Y where X and Y are positive integers
3. This number is a product of two numbers C and D and in turn if we add C + D it again results in a number which is part of a special series
4. C and D can also be expressed as X1 ^ Y1 and X2 ^ Y2
4. DCBA (reverse of this number) + a perfect square results in another well known number!

Find ABCD

Blog #62 - Is there a pattern in this number sequence?

1024, 2401, 4096, 9604 - Do these numbers exhibit any pattern and if yes, is there any other sequence which shows similar behaviour?

Feel free to share your inputs

Blog Post #61 - Five digit Number fun

Sunday musings!

Find this five digit number - ABCDE (Digits can repeat and not necessarily distinct)

- It can be expressed as (a ^ b) x (c ^ b) where a, b, c are integers
- (b ^ a) x (b ^ c) is a special number in its own sense
- Remove the 2nd digit (B) and it is a perfect square
- Add 1 to B and swap with A - (BACDE) and it is still a perfect square!

Blog #60 - X ^ Y - Y ^ X - An observation

I was randomly thinking through some of the scenarios for the formulae (X ^ Y) - (Y ^ X) where one is odd and other is even. For eg. ( 4 ^ 7) - (7 ^ 4)

Surprisingly, most of the resultant answer is either prime or a number whose factors are predominantly prime numbers and odd as well

3 ^ 4 - 4 ^ 3= 17
2 ^ 7 - 7 ^ 2 = 79
2 ^ 9 - 9 ^ 2= 431
4 ^ 5 - 5 ^ 4 = 399 (19 x 7 x 3)

Is there any pattern others see? And any justification using a mathematical equation (I did try logarithms but don't think it helps)

Blog #59 - Find the 4 digit number

- It is a 4 digit number (call it as ABCD)
- (AB + CD) ^ 2 is one of it anagrams
- One of its three digit anagram is also a perfect square
- ABCD - 1 results in a number which has 11 as one of its factor
- ABCD + 1 results in a number which is prime
- ABCD + 9 ^ 2 is again a perfect square
- Last not but not the least this number itself is pretty unique and famous for a reason (Clue: Uniqueness in terms of its factors)

Blog Post #58 - Armstrong number and their patterns

The well known Armstrong numbers are 153, 370, 371, 407 (apart from 0 and 1). Here are few more patterns exhibited by these numbers around squares, cubes and prime numbers

While most of the natural numbers will exhibit pattern of these sorts, it is interesting to see how fe number pairs play a larger role for these 4 numbers

a)  153

12 ^ 2 + 3 ^ 2
13 ^ 2 - 4 ^ 2

One of the anagram of 153 is 513 is 8 ^ 3 + 1 ^ 3

b) 370

7 ^ 3 + 3 ^ 3

19 ^ 2+ 3 ^ 2

One of the anagrams of 370 is 703 (which is not an Armstrong number but check the pattern)

703 = 19 x 37 (19 is there in earlier pattern exhibited by 370 and 37 is a subset of 370!)

c) 371

7 ^ 3 + 3 ^ 3 + 1 ^ 3 (sub of 3 different cubes)

One of the anagram of 371 is 731 which is formed by the digits 7, 3, 1 again

d) 407

7 ^ 3 + 4 ^ 3 or 7 ^ 3 + 8 ^ 2

Its anagram 704 is 7 ^ 3 + 19 ^ 2 (again notice the role number 19 plays in some of these numbers)

Last but not the least the Armstrong numbers show intricate relationship pattern among themselves

407 in turn is 370 + 37 (effectively 37 * 10, 37 * 11)
407 is also 371 + 6 ^ 2

Blog Post #57 - Prime numbers around 1729

This is a recap of one of my earlier post but with few more details around on prime factorization for one of the most famous number - "RAMANUJAM NUMBER - 1729"

Here are more amazing facts about this number that's not very obvious...

1729 is 10 ^ 3 + 9 ^ 3 = 12 ^ 3 + 1 ^ 3 (This is the Ramanujam number as we all know)

1729 in turn is 13 x 7 x 19 (13, 7, 19 are all prime numbers!)

Now look at the odd number anagrams on 1729...

1279 - Prime number
1297 - Prime number as well as
1927 - 41 x 47 - Both are prime in turn
7129 - Prime number
7219 - Prime number
7291 - 23 x 317 - Both are prime in turn
7921 - 89 x 89 and 89 in turn is a prime
9127 - Prime number
9217 - 13 x 709 - Both are prime in turn
9271 - 73 x 127 - Both are prime in turn
9721 - Prime number

Note that odd anagrams are either prime or have factors which are prime and cannot be broken down further

Dig deeper... (Some of these in turn are sum of squares +/- prime number)

1279 is (36 ^ 2) - 17
1296 is (36 ^ 2) + 1
1927 is (44 ^ 2) + 17
9127 is (96 ^ 2) - 89
9217 is (96 ^ 2) + 1

Hats off to the Number 1729 again!

Blog Post #56 - The curious world of Prime numbers

After a small hiatus again here's a post on Prime numbers... No better place to start than my Alma Mater - Pilani Pin Code - 333031!!!

333031 is a prime number and I decided to look at its reverse 130333 which has 7, 43, 433 as factors
So I thought of looking at prime numbers around 333031

Listed below the prime numbers starting 333019 till 333049 as a sample

333019 – Prime, 910333 – 17, 53549

333023 – Prime, 320333 – 13, 41, 601

333029 – Prime, 920333 - Prime

333031 – Prime, 130333 – 7, 43, 433

333041 – Prime, 140333 - Prime

333049 – Prime, 940333 – 373, 2521

When I look at the factors of the numbers which are reverse of the original number, I see that either they are prime or have factors which are prime themselves!!!

Now that Pilani gave me some good for thought to start off on a Saturday morning, I started looking at the 4 digit numbers in random... Take a look below

1913 - Prime, 3191 - Prime

1747 - Prime, 7471 - 31, 241

3041 - Prime, 1403 - 23, 61

3371 - Prime, 1733 - Prime

3863 - Prime, 3683 - 29, 127

5981 - Prime, 1895 - 5, 379

7879 - Prime, 9787 - Prime

Isn't that wonderful to see? 

Needless to say if the original number starts with 2, 4, 6, 8 (or sometimes even 5), the number in reverse may not exhibit this pattern always