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?