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27 replies to this topic

### #1 bonanova

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Posted 23 March 2008 - 08:48 AM

My 1000th post.
Maybe I should give my keyboard a rest.

Anyway Happy Easter, and here's a little gem for you.

Take a number N, reverse its digits and add it to N.
Repeat if necessary. Eventually you will get a palindrome.

For example, starting with 39, we have 39 + 93 = 132.
Then 132 + 231 = 363 = palindrome.

After a moment's thought, this is not surprising.
But sometimes you do have to be patient.
Consider the starting number 89:
```89	  ------>   159487405
187	 |		  664272356
968	 |		 1317544822
1837	 |		 3602001953
9218	 |		 7193004016
17347	 |		13297007933
91718	 |		47267087164
173437	 |		93445163438
907808	 |	   176881317877
1716517	 |	   955594506548
8872688	 |	  1801200002107
17735476	 |	  8813200023188  =  palindrome!
85189247 --->```

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #2 grey cells

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Posted 23 March 2008 - 09:43 AM

My 1000th post.
Maybe I should give my keyboard a rest.

Anyway Happy Easter, and here's a little gem for you.

Take a number N, reverse its digits and add it to N.
Repeat if necessary. Eventually you will get a palindrome.

For example, starting with 39, we have 39 + 93 = 132.
Then 132 + 231 = 363 = palindrome.

After a moment's thought, this is not surprising.
But sometimes you do have to be patient.
Consider the starting number 89:

```89	  ------>   159487405
187	 |		  664272356
968	 |		 1317544822
1837	 |		 3602001953
9218	 |		 7193004016
17347	 |		13297007933
91718	 |		47267087164
173437	 |		93445163438
907808	 |	   176881317877
1716517	 |	   955594506548
8872688	 |	  1801200002107
17735476	 |	  8813200023188  =  palindrome!
85189247 --->```

HAPPY EASTER and HAPPY 1000th POST DAY BONONOVA.
And what a mind-boggling post that is.
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### #3 neida

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Posted 23 March 2008 - 03:16 PM

HAPPY EASTER and HAPPY 1000th POST DAY BONONOVA.
And what a mind-boggling post that is.

Nice post, but I would be amazed if anyone can prove this!!

There are actually some numbers that are thought never to resolve to a palindrome in this way - 196 is the smallest example. These numbers are called "Lychrel numbers". However, no one has ever proven that they exist, simply no one has managed to prove that they don't.

For example, 196 has been processed through supercomputers, but even when the resultant number is over 300 million digits long, it still isn't a palindrome.
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### #4 grey cells

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Posted 23 March 2008 - 03:28 PM

Nice post, but I would be amazed if anyone can prove this!!

There are actually some numbers that are thought never to resolve to a palindrome in this way - 196 is the smallest example. These numbers are called "Lychrel numbers". However, no one has ever proven that they exist, simply no one has managed to prove that they don't.

For example, 196 has been processed through supercomputers, but even when the resultant number is over 300 million digits long, it still isn't a palindrome.

Hi neida.I just wanted to compliment bononova on his 1000th post.You bet I don't intend to prove it .Atleast not now.But your challenge is definitely an interesting one . Maybe I can prove the contrary. But putting all maybe's aside , please post your answer on The palindromic puzzle of yours . It is frustrating to know the question and not the answer for 2 whole days.So please post your answer soon.Sorry for the initial lack of courtesy.(forgot to include please).

Edited by grey cells, 23 March 2008 - 03:33 PM.

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### #5 sunrise

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Posted 23 March 2008 - 03:29 PM

Nice post, but I would be amazed if anyone can prove this!!

There are actually some numbers that are thought never to resolve to a palindrome in this way - 196 is the smallest example. These numbers are called "Lychrel numbers". However, no one has ever proven that they exist, simply no one has managed to prove that they don't.

For example, 196 has been processed through supercomputers, but even when the resultant number is over 300 million digits long, it still isn't a palindrome.

Wow!

I was looking for some proof by induction. I found out that some numbers can be expressed as sum of two numbers, n and n', such that n' is obtained by revesing the digits of n, and others which cannot be expressed as such. Both classes contain palindromes. 111 is a palindrome which cannot be expresses as sum of n and n'.

However, the above post has helped me in deciding to abandon this problem now.
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### #6 chanakyavg

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Posted 23 March 2008 - 04:07 PM

Nice post, but I would be amazed if anyone can prove this!!

There are actually some numbers that are thought never to resolve to a palindrome in this way - 196 is the smallest example. These numbers are called "Lychrel numbers". However, no one has ever proven that they exist, simply no one has managed to prove that they don't.

For example, 196 has been processed through supercomputers, but even when the resultant number is over 300 million digits long, it still isn't a palindrome.

Neida, I am not a Math major, the last time I read Math was almost 17/18 years ago. My knowledge rotates mostly around common sense.

I looked up what "Lychrel numbers" were and it showed up on Wikipedia. With respect to 196, they seem to stick with the defenition of how to rotate numbers and get palindromes, but with the rest they dont seem to. Can you explain why? (As i said, i am not a math major)

`A Lychrel number is a natural number which cannot form a palindrome through the iterative process of repeatedly reversing its base 10 digits and adding the resulting numbers.`

Example:
`# 56 becomes palindromic after one iteration: 56+65 = 121.# 57 becomes palindromic after two iterations: 57+75 = 132, 132+231 = 363.# 59 is not a Lychrel number since it becomes a palindrome after 3 iterations: 59+95 = 154, 154+451 = 605, 605+506 = 1111`

Now their Example for 196:
`196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997`

Should it not go like this instead?
`196 887 1675 7436....`
I know it amounts to the same, but not having studied Abstract Math, I do not understand this. please explain or point to a good source to read up on this.

Thanks. (PM me if u need my email address so we can take this offline. Thanks)
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### #7 unreality

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Posted 23 March 2008 - 04:23 PM

Why would you take this offline? The whole point is to discuss it here

my idea:

if the number is made up of digits that in total add up to less than 10 (ie, add up to a single digit), than you get it in the first round:

2+1 = 3, which is a single digit, so:
21 + 12 = 33

4+1+1+2+1 = 9, single digit, so:

41121 + 12114 = 53235

1117 adds up to 10, but it doesnt necessarily mean it WONT be a palindrome on the first round:

1117 + 7111 = 8228

1218 + 8121 = 9339

So that leads to a more general (and obvious) rule, that if the corresponding flip-pairs each add up to a single digit, there is a palindrome in the first round, because there are no carry-overs

for example, take 72133641
its flip pairs are 7&1, 2&4, 1&6, and 3&3. All of those add up to single digits, so there are no carry-overs, hence a perfect palindrome after round 1

note you can take this number in two ways:

716

716 + 617 = 1333

or 0716 + 6170 = 6886

This can obviously be done to any number to manipulate it so that its flip-pairs all add up to single digits... so bonanova, what's your rule on zeroes?

btw congrats on post #1000 ;D
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### #8 neida

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Posted 23 March 2008 - 06:05 PM

Now their Example for 196:
196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997

Should it not go like this instead?
196 887 1675 7436....

Hi chanakyavg - I've just looked at the wikipedia page and I think you've just misread the page a little. That first list of numbers above is provided as a list of suspected Lychrel numbers (or "candidates") rather than an example of how 196 progresses.
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### #9 chanakyavg

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Posted 23 March 2008 - 06:15 PM

Hi chanakyavg - I've just looked at the wikipedia page and I think you've just misread the page a little. That first list of numbers above is provided as a list of suspected Lychrel numbers (or "candidates") rather than an example of how 196 progresses.

Thanks, i read that page again after ur post, seems like I did mis-read it.
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### #10 ALFRED

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Posted 23 March 2008 - 10:45 PM

bona - congrats on the 1,000th post.

I'm afraid I don't have much to contribute to this puzzle. I do however, have a theory I'd like to contribute to the thread in the hopes that someone else might be able to run with it. I do this while running the risk of giving away the answer to one of my own posts but seeing as how this one is so much more intriguing I think it's worth it.

I worked as an accountant for a little while after college and before I left that career path running and screaming I did pick up a little trick every good accountant uses. Whenever you're reconciling an account and you end up with a difference, the first thing you do is divide the difference by 9. If the answer is divisible by 9, there's a 99% chance the reason you're off is a transposition error. A lot of people (especially those who are good at number puzzles) outside the accounting world may already know of this math fact but I have also learned that if you ever have a coincidence between two numbers there's a 99% chance it's not a coincidence. And seeing as how this puzzle has a lot to do with number transpositions I can't help but think this little math fact might not just be a coincidence.

For example:

86 - 68 = 18/9 = 2
651 - 165 = 486/9 = 54
7824 - 8274 = -450/9 = -50

Notice it doesn't even matter what sort of transposition takes place, as long as all the digits of the original number are still present, the difference will be a multiple of 9.

Now I know the OP has a strict forward to backward transposition and the numbers are added to together instead of subtracted but like I said, I can't help but feel that this coincidence is more than a coincidence. I'd love to know if anyone has any thoughts on this.
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