**What is a Palindrome?**

A palindrome is a word, which reads backwards the same as it does forwards. Well known examples are Anna or radar.

You can apply this principle to numbers. For instance 1001 or 69896 are palindromes.

**Counting the Palindromes **

All the digits are palindromes (1,2,3,…,9).

There are also **9 palindromes** with two digits** **(11,22,33, …,99).

You can find to every two-digit number one, and only one number with three digits and with four digits.

For example: For the number 34 there are 343 and 3443.

You can conclude that there are **90 palindromes **with** **three and also **90 palindromes** with four digits.

You can find to every three-digit number one, and only one number with five digits and with six digits.

For example: To the number 562 there are 56265 and 562265.

You can conclude that there are **900 palindromes **with** **five and **900 palindromes** with six digits.

You have 9+9+90+90+900+900 = 1998 palindromes up to one million. That’s 0,1998 %. About every 500th number is a palindrome.

**Position of the Palindromes**

But they are not spread over all numbers regularly. This shows the picture below, which includes the first 1000 numbers.

**Products with the Digit 1**

11×11 = 121

111×111 = 12321

1111×1111 = 1234321

…

111 111 111 x 111 111 111=12345678987654321

again…

11×111 = 1221

111×1111 = 123321

1 111×11111 = 12344321

…

111 111 111 x 1 111 111 111=123456789987654321

I suppose that all products with the digit 1 are palindromes, if one.factor has at the most 9 digits.

All palindromes have the shape 123…..321.

** **

**Prime Numbers among the Palindromes**

All palindromic primes with 3 digits:

101 131 151 181 191 |
313 353 373 383 . |
727 757 787 797 . |
919 929 . . . |

There are no primes with 4 digits. They all have the factor 11. (Example:4554=4004+550=4×1001+550=4x91x11+11×50=11x(4×91+50)

There are 93 primes with 5 digits.

There are no primes with 6 digits. They all have the factor 11.

There are 668 primes with 7 digits.

** **

**196-Problem**

Pick a number. Add the number, which you must read from the right to the left (mirror number), to the original number. Maybe the sum is a palindrome. If the sum isn’t a palindrome, add the mirror number of the sum to the sum. Maybe you have a palindrome now, otherwise repeat the process. Nearly all numbers have a palindrome in the end.

Example: 49 49+94=143 143+341= 484 !

There are some numbers, which have no palindromes. The lowest one is 196. But the proof is still missing.

196 er moton r ki ki numb ache,seigula bol…interesting post….bt pic ta bujhlam na..

Great post. One thing I found when thinking about palindromes is, if u reverse any number and find the difference between the two you always get a number that is dividable by 9.

21 – 12 = 9

321 – 123 = 198

Ofcourse it’s not true for same palindromes that are same.

Just a thought, does it have any significance with palindromes?

OOps,it’s true even for same palindromes. Wish I could edit that post.

At first I’m really sorry for this late reply 🙂 btw I am curious about what you said for palindromes. Palindromes remain same if you reverse them. Then the difference would always be 0, isn’t it?

Yes, that’s true, I just didn’t look at it right then. But what is really amazing is that Number ~ Reverse(Number) = Always divisible by 9 or more specifically by the prime 3. Does that hold for all cases?

Ya it’s a principle and surely holds for all cases. It’s just amazing that you found it yourself 🙂