Interesting palindromes

Square 12. We get 144.

Reverse 12. We get 21

Square 21. We get 441

Reverse 441. We get 144.

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Ram Mohith, There is a problem that goes something like this

$12^2= 144 ; 21^2 = 441$

$122^2 = 14884; 221^2 = 48841$

$1222^2 = 1493284; 2221^2 = 4932841$

Mohammad Farhat - 2 years, 8 months ago

Yes. It is quite good.

Ram Mohith - 2 years, 10 months ago

Try to find if there are any such numbers.

Ram Mohith - 2 years, 10 months ago

Well I noticed that the 144 is 12 squared. And the prime factorization of 144 is 3^2 times 2^4 which is 4^2 and 3 and 4 are consecutive.

Mohammad Farhat - 2 years, 10 months ago

AND THAT WAY WORKS

Mohammad Farhat - 2 years, 10 months ago

Yes that is a good way finding such numbers. I too will try.

Ram Mohith - 2 years, 10 months ago

P.S. I am 10 this my Brother's account

Mohammad Farhat - 2 years, 10 months ago

@Mohammad Farhat – P.S. I am an indian living in singapore

Mohammad Farhat - 2 years, 10 months ago

@Mohammad Farhat – So your brother name is Mohammad Farhan. How come at the age of 10 you came to know all about these.

Ram Mohith - 2 years, 10 months ago

@Ram Mohith – Well, I find school outdated that is why I check up all of this and learn and ask from my father

Mohammad Farhat - 2 years, 10 months ago

@Ram Mohith – My brother's name is Mohammad FARHAN

Mohammad Farhat - 2 years, 10 months ago

@Mohammad Farhat – Oh sorry for the typo.

Ram Mohith - 2 years, 10 months ago

@Ram Mohith – OK Apology accepted

Mohammad Farhat - 2 years, 10 months ago

@Mohammad Farhat – Once check my profile. I have changed my quotation. It is quite interesting and yet it has quite depth meaning in it.

Ram Mohith - 2 years, 9 months ago

You just take two consecutive numbers. Then multiply them. and then the rest works.

Mohammad Farhat - 2 years, 10 months ago

My assumption WORKED. YAY

Mohammad Farhat - 2 years, 10 months ago

I am thinking to frame a question based on these observations.

Ram Mohith - 2 years, 10 months ago

May I assist

Mohammad Farhat - 2 years, 10 months ago

Did you get still anymore numbers like these.

Ram Mohith - 2 years, 10 months ago

@Ram Mohith – I am currently doing courses. Give me a minute and I will update you

Mohammad Farhat - 2 years, 10 months ago

@Mohammad Farhat – Ok. No problem. Take your own time.

Ram Mohith - 2 years, 10 months ago

@Ram Mohith – I already updated

Mohammad Farhat - 2 years, 10 months ago

@Ram Mohith – NO

Mohammad Farhat - 2 years, 10 months ago

@Ram Mohith – There has to be some kind of error.

Mohammad Farhat - 2 years, 10 months ago

I am trying to obtain a general form for these numbers or if there is some periodicity between the numbers .

Ram Mohith - 2 years, 10 months ago

@Ram Mohith – What does periodicity mean

Mohammad Farhat - 2 years, 10 months ago

@Mohammad Farhat – Like there is some common difference between the numbers. More clearly they should be in any progression or series.

Ram Mohith - 2 years, 10 months ago

@Ram Mohith – oh

Mohammad Farhat - 2 years, 10 months ago

You cannot frame a question

Mohammad Farhat - 2 years, 10 months ago

Why can't we ?

Ram Mohith - 2 years, 10 months ago

It only works around 20

Mohammad Farhat - 2 years, 10 months ago

All single digit integers will satisfy this condition the reason being when they are reversed the same number is obtained.

Ram Mohith - 2 years, 10 months ago

True

Mohammad Farhat - 2 years, 10 months ago

All multiples of 10 are exceptions to this condition. Reason :

$10^2 = 100$ . On reversing 10 we get $01$ . Now square it we get $01^2 = 1 = 01 = 001 = 0001 ..... and~so~on$ . Now reversing it we get $10,100,1000 .....and~so~on$ . So, the multiples of $10$ are exceptions. More clearly they will be in an undetermined form.

Ram Mohith - 2 years, 10 months ago

Make sense

Mohammad Farhat - 2 years, 10 months ago

This also works with $13$ :

$13^2 = 169$

$31^2 = 961$

Henry U - 2 years, 7 months ago

My friend told me that on Friday but I forgot

Mohammad Farhat - 2 years, 7 months ago

And if you take $14$ in base $20$ (so 24 in base 10) and square it, you get $18G_{20}$ . If you square $42_{20}$ (81 in base 10) you get $G81_{20}$ .

Henry U - 2 years, 7 months ago

Did you know that both the endings of 31 and 19 end in 61.

31 times 31 = 961

and

19 times 19 = 361

Mohammad Farhat - 2 years, 10 months ago

Square 20. We get 400.

Reverse 20. We get 02.

Square 02. We get 004

Reverse 400. We get 004

Mohammad Farhat - 2 years, 10 months ago

Good one again

Ram Mohith - 2 years, 10 months ago

It only works below 20

EDIT: The pattern is with 1 and a followed number of 2's

Mohammad Farhat - 2 years, 10 months ago

Ok. Should see about this !!!

Ram Mohith - 2 years, 10 months ago

WAIT 10 and 11 work

Mohammad Farhat - 2 years, 10 months ago

Sorry. It is around 20

Mohammad Farhat - 2 years, 10 months ago

20 means squaring and reversing

Mohammad Farhat - 2 years, 10 months ago

The idea of making a problem out of this out of question

Mohammad Farhat - 2 years, 10 months ago

Unless it is a proof stating question

Mohammad Farhat - 2 years, 10 months ago

Yes. Your point is also correct.

Ram Mohith - 2 years, 10 months ago

But my idea is not to write a proof based question.

Ram Mohith - 2 years, 10 months ago

@Ram Mohith – You can twist the question in the direction of palindromes. Say you give examples for palindromes and ask whether it works all the time ( For other numbers too). And you give three options. Yes, always , No, never and Yes , sometimes. But do not make it too obvious

Mohammad Farhat - 2 years, 10 months ago

@Mohammad Farhat – Yes I am also thinking about it.

Ram Mohith - 2 years, 10 months ago

Can you help me in my other notes

Mohammad Farhat - 2 years, 10 months ago

surely.

Ram Mohith - 2 years, 10 months ago

Lets go to interesting prime powers relationship(the name)

Mohammad Farhat - 2 years, 10 months ago

@Mohammad Farhat – I am coming one by one.

Ram Mohith - 2 years, 10 months ago

@Ram Mohith – ok

Mohammad Farhat - 2 years, 10 months ago

There is a question on this (but I forgot and cannot search). The real pattern is 1 followed by some number of 2's and when you reverse it still makes sense

Mohammad Farhat - 2 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$